Infinite-Dimensional Dynamical Systems: Volume 2 Attractors and Methods 9783110587265, 9783110586992

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Boling Guo, Liming Ling, Yansheng Ma, and Hui Yang Infinite-Dimensional Dynamical Systems

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Boling Guo, Liming Ling, Yansheng Ma, and Hui Yang

Infinite-Dimensional Dynamical Systems |

Volume 2: Attractors and Methods

Mathematics Subject Classification 2010 35B20, 35B40, 35B41, 35B42, 35C08, 35Q35, 35Q55, 35Q56, 37L25, 37L30, 37L65 Authors Prof. Boling Guo Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics 6 Huayuan Road Haidian District 100088 Beijing People’s Republic of China [email protected] Prof. Liming Ling South China University of Technology School of Mathematics Wushan RD., Tianhe District 381 510640 Guangzhou People’s Republic of China [email protected]

Dr Yansheng Ma Northeast Normal University School of Mathematics and Statistics 5268 Renmin Street Jilin Province 130024 Changchun People’s Republic of China [email protected] Prof. Hui Yang Yunnan Normal University School of Mathematics 768 Junxian Road Yunnan Province 650500 Kuming People’s Republic of China [email protected]

ISBN 978-3-11-058699-2 e-ISBN (PDF) 978-3-11-058726-5 e-ISBN (EPUB) 978-3-11-058708-1 Library of Congress Control Number: 2018934551 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck Cover image: Chong Guo www.degruyter.com

Preface This book introduces the mathematical theory, research methods and results for infinite-dimensional dynamical systems. In 1993, the author gave a brief introduction to the conceptual framework, methods and research progress on infinite-dimensional dynamical systems in his monograph “Nonlinear evolution equations”. However he didn’t have an extensive and in-depth discussion on the topic due to limited space. As numerous new and important research achievements began to accumulate during the past decades, the authors have made a decision to write a monograph about infinite-dimensional dynamical systems. The aim of this book is to introduce the rudimentary knowledge, some interesting problems and important and new results including the authors’, cooperators’ and other scholars’ resent research on infinite-dimensional dynamical systems through some concise but heuristic methods. The main emphasis in the first volume is on the mathematical analysis of attractors and inertial manifolds. In Chapter 1 “Attractor and its dimension estimation” we mainly introduce the global attractor and estimation of Hausdorff and fractal dimensions for some dissipative nonlinear evolution equations in modern physics. Chapter 2 “Inertial manifold” deals with inertial manifolds for a range of generalized differential equations and the spectral gap conditions. Moreover, we study the existence, smoothness and normal hyperbolic properties of inertial manifolds. Chapter 3 “Approximate inertial manifold” constructs the inertial manifolds and investigates the convergence of approximate inertial manifolds, which provides a constructive method to establish existence of inertial manifolds. The second volume devotes to the modern analytical tools and methods in infinite dimensional dynamic system. In Chapter 1 “Discrete attractor and approximate calculation” we introduce results on the existence of discrete attractor and approximate calculation which are closely related to infinite-dimensional dynamical systems. Employing numerical calculations, we provide images of global attractors and approximate inertial manifolds. Chapter 2 “Some properties of a global attractor” introduces some properties of a global attractor, including oscillatory properties and asymptotic behavior. The asymptotic behavior of inertial manifolds can be determined only by the properties of a few points and it is closely related to the unstable manifold of the hyperbolic fixed point. We estimate an upper bound of Hausdorff length of level sets through the geometric measure method and provide a new method to give a lower bound on the dimensional estimate for the attractor. In Chapter 3 “Structures of small dissipative dynamical systems” we mainly introduce the structure of stable and unstable manifolds with small perturbations and the chaotic behaviors by employing the geometric singular perturbation theory, the center manifold theory in infinite dimensional setting and Melnikov method. The structure of stable and unstable manifolds is related to the first Chern number on the fiber bundle. Chapter 4 “Existence and stahttps://doi.org/10.1515/9783110587265-201

VI | Preface bility of solitary waves” uses the concentration-compactness principle to study the existence of solitary wave solutions and discusses the nonlinear stability, instability, orbital stability and asymptotic stability for the solitary waves by the energy functional method and spectral analysis. As the content of infinite-dimensional dynamical systems is quite rich and extensive, it is closely related to many subjects, such as fluid mechanics, functional analysis, topology, geometric measure theory, numerical mathematics, and so on. There are numerous new methods and results due to the quick developments of infinitedimensional dynamical systems. Owing to the limited time and knowledge of the authors, there must be some inadvertent errors and omissions in the book. Any suggestions and comments are welcomed. Beijing, China July 2017

Boling Guo

Contents Preface | V 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Discrete attractor and approximate calculation | 1 Generalized Ginzburg–Landau equations | 1 Zakharov system | 17 Inertial manifolds under time-discretization | 36 Landau–Lifschitz equations | 70 Nonlinear Galerkin method | 82 Stability analysis and numerical results | 108 Two-dimensional Newton–Boussinesq equation | 116 Numerical computation and analysis of cubic Ginzburg–Landau equation | 130 One-dimensional Kuramoto–Sivashinsky equation | 133

2 2.1 2.2 2.3 2.4 2.5 2.6

Some properties of global attractor | 147 Kuramoto–Sivashinsky equation | 147 Generalized Ginzburg–Landau equation | 152 Upper bound estimate for the winding number | 160 Oscillations of solutions of the Kuramoto–Sivashinsky equation | 170 Hausdorff length of level sets | 174 The structure of the global attractor and lower bound estimate of its dimension | 188

3 3.1 3.2 3.3 3.4

Structures of small dissipative dynamical systems | 193 Quintic Ginzburg–Landau equation | 193 The derivative Ginzburg–Landau equations | 210 The perturbed nonlinear Schrödinger equation | 219 Center manifold theory in infinite dimensions | 268

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Existence and stability of solitary waves | 281 Orbital stability | 282 The nonlinear derivative Schrödinger equation | 308 Long wave–short wave resonance equations | 324 The generalized Kadomtsev–Petviashvili equations | 334 The generalized Davey–Stewartson system | 348 Nonlinear Schrödinger–Kadomtsev–Petviashvili equations | 366 Asymptotic stability of solitary waves for BBM equation | 375

VIII | Contents Bibliography | 399 Index | 405

1 Discrete attractor and approximate calculation We have done research on the global attractors, inertial manifolds and approximate inertial manifolds, and derived a series of significant and interesting results which reveal many properties of infinite-dimensional dynamical systems. However, there are still some interesting and essential problems of infinite-dimensional dynamical systems, which we know little about. For example, we have only obtained the existence of a global attractor and its rough dimensional estimate, but what geometric properties and topological invariants it possesses? How to distinguish the fixed points, periodic solutions, quasi-periodic solutions and singular attractors? And what is the relations between the above phenomena and initial value conditions, boundary value conditions, parameters? To solve these problems, we study discrete PDEs which are ODEs (namely going from infinite dimension to finite dimension) and employ the numerical methods. In this chapter, by the discretization method, we will study discrete attractors, initial manifolds, approximate inertial manifolds for some classical equations (such as Ginzburg–Landau equation, Zakharov system, and so on), and will give numerical results which provide some new ideas and inspirations. For the validity as t → ∞, we need a numerical method usable for long time, which brings far-reaching consequences to computational mathematics. Thus we introduce the nonlinear Galerkin method which shows great superiority in long time numerical computations.

1.1 Generalized Ginzburg–Landau equations In 1996, Guo and Chang [39] studied discrete attractor and dimension estimation of a kind of generalized Ginzburg–Landau equations. In 1989, Brand and Deissler [8] put forward the following generalized Ginzburg– Landau equation: 𝜕t u + νux = χu + (γr + iγi )uxx − (βr + iβi )|u|2 u

− (δr + iδi )|u|4 u − (λr + iλi )|u|2 ux − (μr + iμi )|u|2 ux ,

(1.1.1)

where γi , δi , χ are positive constants; i = √−1; ν, γi , βr , βi , δr , δi , λr , λi , μr , μi are real constants. In 1995, Guo and Gao [41] studied the periodic initial value problem for equation (1.1.1) with condition u(x + 1, t) = u(x, t), u(x, 0) = u0 (x),

x ∈ ℝ, t > 0,

x ∈ ℝ.

(1.1.2) (1.1.3)

For the case ν = δi = λr = λi = μr = μi = 0, there are several results. When βr < 0, γr = 0, in 1988, Yang [100] considered that the solution of problem (1.1.1)–(1.1.3) may blow up. In 1995, Guo and Gao proved that there exist a global smooth solution and https://doi.org/10.1515/9783110587265-001

2 | 1 Discrete attractor and approximate calculation global attractor of problem (1.1.1)–(1.1.3) and they obtained upper bounds of Hausdorff and fractal dimensions of the global attractor in [41]. Now we will study the uniform a priori estimates of discrete solutions for the problem (1.1.1)–(1.1.3). Suppose that J ∈ ℕ, h = 1J , and the approximate function u(x) ∈ L2 (0, 1) is

T

1 2 u = (u1 , u1 , . . . , uJ )T = (u( ), u( ), . . . , u(1)) . J J

(1.1.4)

Set 2 −1 ( .. ( A = J 2 A1 = J 2 ( . (. . . ... (−1

−1 2 .. . ... ... ...

0 −1 .. . ... ... ...

... ... .. . ... ... ...

... ... .. . 2 −1 0

... ... .. . −1 2 −1

−1 0 .. ) .) ) 0) −1 2 )J×J

(1.1.5)

and 1 1 1 1 (u − uj ) = Δ + uj , ujx = (uj − uj−1 ) = Δ − uj , h j+1 h h h 1 1 ujx̂ = (u − uj−1 ) = (Δ u + Δ − uj ). 2h j+1 2h + j ujx =

Applying the finite difference method to Ginzburg–Landau equation (1.1.1), we derive 𝜕t uj + νujx̂ = χuj + (γr + iγi )ujxx̄ − (βr + iβi )|uj |2 uj

− (δr + iδi )|uj |4 uj − (λr + iλi )P(uj−1 , uj , uj+1 ) − (μr + iμi )Q(uj−1 , uj , uj+1 ), (1.1.6)

where 1 1 P(uj−1 , uj , uj+1 ) = uj (u2j )x̂ = u (u2 − u2j−1 ), 2 4h j j+1 1 1 󵄨 󵄨 Q(uj−1 , uj , uj+1 ) = uj (󵄨󵄨󵄨u2j 󵄨󵄨󵄨)x̂ − uj (u2j )x̂ = [2u (|u |2 − |uj−1 |2 ) − uj (u2j+1 − u2j−1 )], 2 4h j j+1

uj (t) = uj+1 (t),

uj (0) = u0 (xj ),

(1.1.7)

j = 1, 2, . . . , J.

(1.1.8)

The inner product of discrete complex periodic functions uh = {uj | j = 1, 2, . . . , J} and vh = {vj | j = 1, 2, . . . , J} is J

(u, v)h = ∑ uj vj h, j=1

1.1 Generalized Ginzburg–Landau equations | 3

where vj denotes the complex conjugate of vj . We denote the norms of discrete function uh and its kth order difference quotients δk uh as 1/p

J−k 󵄩 󵄩󵄩 Δk+ uj 󵄩󵄩󵄩p 󵄩󵄩 k 󵄩󵄩 󵄩󵄩δ uh 󵄩󵄩p = ( ∑ 󵄩󵄩󵄩 k 󵄩󵄩󵄩 ) 󵄩 󵄩󵄩h j=1 󵄩 h

,

1 ≤ p ≤ ∞,

󵄨󵄨 Δk uj 󵄨󵄨 󵄨󵄨 + 󵄨󵄨 󵄩󵄩 k 󵄩󵄩 = max δ u 󵄨󵄨 󵄩󵄩 h 󵄩󵄩∞ 󵄨󵄨, j=1,2,...,J−k 󵄨󵄨 hk 󵄨󵄨 where k ≥ 0 is a nonnegative integer and p is a real number. In the following, we state some interpolation inequalities for the discrete function uk (see [101] for details). Lemma 1.1.1. For any discrete function uh = {uj | j = 1, 2, . . . , J}, 1/2

‖uh ‖∞ ≤ K1 ‖uh ‖1/2 2 (‖δuh ‖2 + ‖uh ‖2 ) , 󵄩󵄩δ2 uh 󵄩󵄩󵄩 + ‖uh ‖2 )1/2 , ‖δuh ‖2 ≤ K2 ‖uh ‖1/2 󵄩 󵄩2 2 (󵄩 1− 󵄩󵄩 k 󵄩󵄩 󵄩󵄩δ uh 󵄩󵄩2 ≤ K3 ‖uh ‖2

k+ 21 − p1

1− 󵄩 󵄩 (󵄩󵄩󵄩δl uh 󵄩󵄩󵄩2 + ‖uh ‖2 )

l

(1.1.9) (1.1.10) k+ 21 − p1 l

,

(1.1.11)

where the constants K1 , K2 , K3 are independent of the step length of uh , 2 ≤ p ≤ ∞, 0 ≤ k < n. Denote ‖uh ‖2 = ‖uh ‖, ‖uh ‖H 1 = ‖uh ‖ + ‖δuh ‖, ‖uh ‖H 2 = ‖uh ‖ + ‖δ2 uh ‖, u = uh for simplicity. Lemma 1.1.2. Suppose that the complex discrete functions fh = {fj | j = 1, 2, . . . , J} and gh = {gj | j = 1, 2, . . . , J} satisfy the periodicity condition fj = fj+J , gj = gj+J , then J

Re ∑ fj (fj+1 − fj−1 ) = 0, j=1

(fj , gj )x = fj+1 gjx + fjx gj ,

(fj , gj )x̂ = fj+1 gjx̂ + fjx̂ gj−1 ,

(f , gx̄ ) = −(fx , g). Proof. It follows from J

J

J−1

j=1

j=1

j=0

Re ∑ fj (fj+1 − fj−1 ) = Re[∑ fj fj+1 − ∑ f j+1 fj ] and the periodicity condition J−1

J

j=0

j=1

∑ f j+1 fj = ∑ f j+1 fj

that J

J

j=1

j=1

Re ∑ fj (fj+1 − fj−1 ) = ∑ Re(f j fj+1 − f j+1 fj ) = 0, 1 (fj , gj )x = (fj+1 gj+1 − fj gj ) = fj+1 gjx + fjx gj . h

(1.1.12)

4 | 1 Discrete attractor and approximate calculation Similarly, (fj , gj )x̂ =

1 (f g − f g ) = fj+1 gjx̂ + fjx̂ gj−1 , 2h j+1 j+1 j−1 j−1 J

J

J

J−1

j=1

j=1

j=0

(f , gx̄ ) = ∑ fj g jx h = ∑ fj (g j − g j−1 ) = ∑ fj g j − ∑ fj+1 g j j=1 J

J

j=1

j=1

= ∑ fj g j − ∑ fj+1 g j = −(fx , g). Lemma 1.1.3. Suppose that γr > 0, δr > 0, χr > 0 and 4γr δr > (γi − μi )2 , then for any solution of equations (1.1.6)–(1.1.8) we have 2 󵄩󵄩 󵄩2 󵄩2 P −2χt −2χt 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ e 󵄩󵄩󵄩u(0)󵄩󵄩󵄩 + (1 − e ), 2χ

∀t ≥ 0,

(1.1.13)

∞

󵄩 󵄩2 ∫ 󵄩󵄩󵄩ux (t)󵄩󵄩󵄩 dt < ∞,

(1.1.14)

0

where P = 2χ +

(βr +1/2)2 , 2β

β is an absolute constant.

Proof. Calculating the inner product of (1.1.6) with u leads to J J J 1 d ‖u‖2 = −ν Re ∑ ujx̂ uj h + χ‖u‖2 − γr ‖ux ‖2 − βr ∑ |uj |4 − δr ∑ |uj |6 h 2 dt j=1 j=1 j=1 J

− Re{(λr + iλi ) ∑ P(uj−1 , uj , uj+1 )uj h} j=1 J

− Re{(μr + iμi ) ∑ Q(uj−1 , uj , uj+1 )uj h}. j=1

(1.1.15)

It follows from Lemma 1.1.2 that J

Re ∑ ujx̂ uj h = j=1

J 1 Re ∑ uj (uj−1 − uj+1 )h = 0, 2 j=1

J

J 1 Re ∑[2|uj |2 (|uj+1 |2 − |uj−1 |2 ) − u2j (u2j+1 − u2j−1 )] = 0. 4 j=1

Re ∑ P(uj−1 , uj , uj+1 )uj h = j=1

Thus J

− Re{(λr + iλi ) ∑ P(uj−1 , uj , uj+1 )uj h} j=1

J

= − Re{(μr + iμi ) ∑ Q(uj−1 , uj , uj+1 )uj h} j=1

1.1 Generalized Ginzburg–Landau equations | 5

=

J J 1 1 λi Im ∑(uj )2 (u2j+1 − u2j−1 ) − μi Im ∑(uj )2 (u2j+1 − u2j−1 ) 4 4 j=1 j=1

=

J 1 (λi − μi ) Im ∑(uj )2 (uj+1 − uj−1 )(uj+1 − uj + uj − uj−1 ) 4 j=1 J

≤ (λi − μi ) ∑ |uj |2

|uj+1 + uj−1 | |ujx + ujx̄ | 2

j=1

󵄨 ≤ a1 b1 (∑ |uj | 󵄨󵄨 󵄨󵄨 j=1 J

≤

4 󵄨󵄨󵄨 uj+1

2

h

1

+ uj−1 󵄨󵄨󵄨2 2 󵄨󵄨 h) ‖ux ‖ 󵄨󵄨 2 󵄨

a21 J b2 ∑ |uj |6 h + 1 ‖ux ‖2 , 2 j=1 2

(1.1.16)

where a1 b1 = |λi − μi |. Young inequality fg ≤ ε

pf

p

p

( ε1 g)p

󸀠

+

p󸀠

,

1 1 = 1, + p p󸀠

f , g, ε > 0

(1.1.17)

and the periodicity condition have been used in inequality (1.1.16). Substituting the above inequality into (1.1.15) and selecting a1 , b1 such that α = 2γr − b21 > 0,

β = 2γr − a21 > 0,

we derive J J d ‖u‖2 ≤ 2χ‖u‖2 − α‖ux ‖2 + 2βr ∑ |uj |4 h − β ∑ |uj |6 h. dt j=1 j=1

(1.1.18)

From − β|uj |6 + 2β|uj |4 + 2χ‖uj ‖2 = −β(|uj |3 −

βr + 2

β

1 2

|uj |2 ) − |uj |4 + (

(βr + 21 )2 β

+ 4χ)|uj |2 − 2χ|uj |2

≤ −(|uj |2 − P) + P 2 − 2χ|uj |2

≤ P 2 − 2χ|uj |2 , where

P = 2χ +

(βr + 21 )2 2β

,

and by (1.1.18), we obtain d ‖u‖2 + 2χ‖u‖2 + α‖ux ‖2 ≤ P 2 . dt

(1.1.19)

Applying Gronwall inequality leads to (1.1.13). Therefore, (1.1.14) can be obtained by integrating (1.1.19) with respect to t and inequality (1.1.13).

6 | 1 Discrete attractor and approximate calculation Corollary 1.1.1. Under the conditions of Lemma 1.1.3, suppose that ‖u0 ‖ ≤ R, R > 0, then there exist global solutions for the discrete system (1.1.6)–(1.1.8). Lemma 1.1.4. Under the conditions of Lemma 1.1.3, we derive d 2 ‖ux ‖2 + γr ‖uxx̄ ‖2 ≤ E1 (1 + ‖ux ‖2 ) , dt

(1.1.20)

where the constant E1 is independent of the discrete function uh and its step length h. Proof. Calculating the inner product of (1.1.6) with uxx and taking the real part, we obtain J d ‖ux ‖2 + γr ‖uxx̄ ‖2 = χ‖ux ‖2 + Re{(βr + iβi ) ∑ |uj |2 uj ujxx̄ h} dt j=1 J

+ Re{(δr + iδi ) ∑ |uj |4 uj ujxx̄ h} j=1 J

+ Re{(λr + iλi ) ∑ P(uj−1 , uj , uj+1 )ujxx̄ h} j=1 J

+ Re{(μr + iμi ) ∑ Q(uj−1 , uj , uj+1 )ujxx̄ h}. j=1

(1.1.21)

Employing Lemma 1.1.2, we derive J

J

j=1

j=1

∑ |uj |2 uj ujxx̄ = − ∑(|uj |2 uj )x uj ujx J

J

J

j=1

j=1

− ∑ |uj |2 |ujx |2 h − ∑ |uj−1 |2 |ujx |2 h − ∑ uj uj+1 ū 2jx j=1

J

J

j=1

j=1

= − ∑(|uj |2 + |uj−1 |2 )|ujx |2 − ∑ uj uj+1 ū 2jx , J

J

j=1

j=1

∑ |uj |4 uj ujxx̄ h = − ∑(|uj |4 uj )x uj ujx h J

J

j=1

j=1

J

J

j=1

j=1

= − ∑ |uj |4 |ujx |2 h − ∑ uj+1 (|uj |4 − |uj−1 |4 )ū jx h = − ∑ |uj |4 |ujx |2 h − ∑(|uj+1 |2 + |uj |2 )uj+1 ū jx (|uj+1 |2 − |uj |2 )h J

J

j=1

j=1

= − ∑(|uj+1 |4 + |uj+1 |2 |uj |2 + |uj |4 )|ujx |2 − ∑(|uj+1 |2 + |uj |2 )uj+1 uj ū 2jx h.

1.1 Generalized Ginzburg–Landau equations | 7

Thus J J d ‖ux ‖2 + γr ‖uxx̄ ‖2 ≤ χ‖ux ‖2 − βr ∑(|uj |2 + |uj−1 |2 )|ujx |2 h + √βr2 + βi2 ∑ |uj+1 ||uj ||ujx |2 h dt j=1 j=1 J

− δr ∑(|uj+1 |4 + |uj+1 |2 |uj |2 + |uj |4 |ujx |2 )h j=1

J

+ √δr2 + δi2 ∑(|uj+1 |2 + |uj |2 )|uj+1 ||uj ||ujx |2 h j=1

J 1 + √(λr − μr )2 + (λi − μi )2 × ∑ |uj |(|uj+1 | + |uj−1 |)|ujx ||ujxx̄ |h 2 j=1 J 1 + √μ2r + μ2i ∑ |uj |(|uj+1 | + |uj−1 |)|ujx ||ujxx̄ |h. 2 j=1

(1.1.22)

Employing Lemma 1.1.1, we have 1/2

‖u‖∞ ≤ K1 ‖u‖1/2 (‖δu‖ + ‖u‖) , hence we get J

∑(|uj+1 |2 + |uj |2 )|ujx |2 h ≤ 2‖u‖∞ 2‖ux ‖ j=1

≤ 2K1 ‖u‖(‖u‖ + ‖ux ‖)‖ux ‖2 ≤ C(‖ux ‖2 + ‖ux ‖3 )

≤ C(‖ux ‖2 + ‖ux ‖4 ).

(1.1.23)

Analogously, we derive J

∑ |uj+1 ||uj ||ujx |2 h ≤ C(‖ux ‖2 + ‖ux ‖4 ), j=1 J

∑(|uj+1 |4 + |uj+1 |2 |uj |2 + |uj |4 )|ujx |2 h ≤ ‖u‖4∞ ‖ux ‖4 ≤ C(‖ux ‖2 + ‖ux ‖4 ), j=1 J

∑(|uj+1 |2 + |uj |2 )|uj+1 ||uj ||ujx |2 h ≤ C(‖ux ‖2 + ‖ux ‖4 ), j=1

(1.1.24) (1.1.25) (1.1.26)

J

∑ |uj |(|uj+1 | + |uj−1 |)|ujx ||ujxx̄ |h j=1

≤ ≤

γr 1 J 2 ‖uxx̄ ‖2 + ∑ |u |2 (|uj+1 | + |uj−1 |) |ujx |2 h 2 2γr j=1 j γr ‖u ̄ ‖2 + C(‖ux ‖2 + ‖ux ‖4 ). 2 xx

(1.1.27)

8 | 1 Discrete attractor and approximate calculation Therefore, substituting (1.1.23)–(1.1.27) into (1.1.22), we obtain (1.1.20), i. e., d 2 ‖u ‖2 + γr ‖uxx̄ ‖2 ≤ E1 (1 + ‖ux ‖2 ) . dt x

(1.1.28)

Lemma 1.1.5 (Uniform Gronwall lemma, [91]). Let g, h, y be three locally integrable functions on [t, +∞) that satisfy dy ≤ gy + h, dt

t ≥ t0 ,

t+r

t+r

∫ g(s)ds ≤ a1 ,

∫ h(s)ds ≤ a2 ,

t

t

t+r

∫ y(s)ds ≤ a3 ,

t ≥ t0 ,

t

where r, a1 , a2 , a3 are positive constants, then y(t + r) ≤ (

a3 + a2 ) exp(a1 ), r

t ≥ t0 .

(1.1.29)

Lemma 1.1.6. Under the conditions of Lemma 1.1.3, suppose that ‖u0x ‖ ≤ R, R > 0, then we obtain the following estimates of any solution for the discrete system (1.1.6)–(1.1.8): 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩u(t)󵄩󵄩󵄩 + 󵄩󵄩󵄩ux (t)󵄩󵄩󵄩 ≤ E2 ,

(1.1.30)

t+r

󵄩 󵄩2 ∫ 󵄩󵄩󵄩uxx̄ (s)󵄩󵄩󵄩 ds ≤ E2󸀠 ,

∀r > 0,

(1.1.31)

t

where the constants E2 , E2󸀠 are independent of the discrete function uh and its step length h. Proof. It follows from Lemma 1.1.5 and (1.1.20) that d 2 ‖u ‖2 ≤ E1 (1 + ‖ux ‖2 ) ≤ 2E1 (1 + ‖ux ‖4 ). dt x

(1.1.32)

Inequality (1.1.14) implies t+1

∫ ‖ux ‖2 ds ≤ a1 ,

∀t ≥ 1.

t

Thus using Lemma 1.1.5 and setting g = y = 2E1 ‖ux ‖2 , h = c, we derive 󵄩󵄩 󵄩2 󵄩󵄩ux (t + 1)󵄩󵄩󵄩 ≤ (a1 + c) exp a1 ,

∀t ≥ 1.

(1.1.33)

1.1 Generalized Ginzburg–Landau equations | 9

In the case 0 ≤ t ≤ 1, we obtain the following inequality from Gronwall inequality and (1.1.32) 󵄩2 t󵄩 󵄩2 󵄩󵄩 ∫ 󵄩u (s)󵄩 ds 󵄩󵄩ux (t)󵄩󵄩󵄩 ≤ Ce 0 󵄩󵄩 x 󵄩󵄩 ≤ C1 ,

0 ≤ t ≤ 2.

(1.1.34)

Therefore, using (1.1.33), (1.1.34) and Lemma 1.1.3, we derive (1.1.30), which implies (1.1.31) by inequality (1.1.32). Lemma 1.1.7. Under the conditions of Lemma 1.1.6, suppose that ‖u0xx̄ ‖2 ≤ R, R > 0, then 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩u(t)󵄩󵄩󵄩 + 󵄩󵄩󵄩ux (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩uxx̄ (t)󵄩󵄩󵄩 ≤ E3 ,

(1.1.35)

where constant E3 is independent of the discrete function uh and its step length h. Proof. We conclude that the inequality d ‖u ̄ ‖2 + γr ‖uxxx̄ ‖2 ≤ C(1 + ‖uxx̄ ‖4 ) dt xx

(1.1.36)

holds. In fact, calculating the inner product of (1.1.6) with uxxx̄ x̄ , we derive J d ‖uxx̄ ‖2 + γr ‖uxxx̄ ‖2 = χ‖uxx̄ ‖2 − Re[(βr + iβi ) ∑ |uj |2 uj ujxxx̄ x̄ h] dt j=1 J

− Re[(δr + iδi ) ∑ |uj |4 uj ujxxx̄ x̄ h] j=1 J

− Re[(λr + iλi ) ∑ P(uj−1 , uj , uj+1 )ujxxx̄ x̄ h] j=1 J

− Re[(μr + iμi ) ∑ Q(uj−1 , uj , uj+1 )ujxxx̄ x̄ h]. j=1

It follows from Lemma 1.1.2 that J

J

j=1

j=1

∑ |uj |2 uj ujxxx̄ x̄ h = ∑(|uj |2 uj )xx̄ ujxx̄ h J

= ∑[|uj |2x uj+1 + (uj )2 ujx ]x ujxx̄ h j=1 J

= ∑[(|uj |2 )xx̄ uj + ujx̄ (|uj |2 )x + (|uj |2 )x̄ ujx + |uj |2 ujxx̄ ]ujxx̄ h. j=1

Similarly, J

J

j=1

j=1

∑ |uj |4 uj ujxxx̄ x̄ h = ∑[(|uj |4 )xx̄ uj + ujx̄ (|uj |4 )x + (|uj |4 )x̄ ujx + |uj |4 ujxx̄ ]ujxx̄ h,

(1.1.37)

10 | 1 Discrete attractor and approximate calculation 󵄨󵄨 󵄨󵄨 J 󵄨 󵄨󵄨 󵄨󵄨∑ P(uj−1 , uj , uj+1 )ujxxx̄ x̄ h󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨j=1 󵄨󵄨 󵄨󵄨 J 󵄨󵄨 1 󵄨󵄨󵄨 2 ≤ 󵄨󵄨∑(ū j (uj )x̂ )x ujxx̄ h󵄨󵄨󵄨 ≤ 󵄨󵄨 2 󵄨󵄨󵄨j=1 󵄨 γr ≤ ‖ujxx̄ ‖2 + C‖ujxx̄ ‖2 , 3 󵄨󵄨 J 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∑ Q(uj−1 , uj , uj+1 )ujxxx̄ x̄ h󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨j=1 󵄨 󵄨󵄨 J 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨󵄨∑(ū j (u2j )x̂ )x ujxx̄ h󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨j=1 󵄨󵄨󵄨 γ ≤ r ‖ujxx̄ ‖2 + C‖ujxx̄ ‖2 . 3

1 J 󵄨󵄨 󵄨 ∑ 󵄨ū (u2 ) + ū jx (u2j )x̂ 󵄨󵄨󵄨|ujxx̄ |h 2 j=1 󵄨 j+1 j xx̂ (1.1.38)

1 J 󵄨󵄨 󵄨 ∑ 󵄨ū (u2 ) + ū jx (u2j )x̂ 󵄨󵄨󵄨|ujxx̄ |h 2 j=1 󵄨 j+1 j xx̂ (1.1.39)

Thus we get the following inequality from (1.1.37)–(1.1.39): 1 d ‖u ̄ ‖2 + γr ‖uxxx̄ ‖2 2 dt xx J

󵄨 󵄨 ≤ χ‖uxx̄ ‖2 + C ∑ 󵄨󵄨󵄨[(|uj |2 )xx̄ uj + ujx̄ (|uj |2 )x + (|uj |2 )x̄ ujx + |uj |2 ujxx̄ ]ujxx̄ 󵄨󵄨󵄨h j=1

J

󵄨 󵄨 + C ∑ 󵄨󵄨󵄨[(|uj |4 )xx̄ uj + ujx̄ (|uj |4 )x + (|uj |4 )x̄ ujx + |uj |4 ujxx̄ ]ujxx̄ 󵄨󵄨󵄨h j=1

2γ + r ‖uxxx̄ ‖2 + C‖uxx̄ ‖2 . 3

(1.1.40)

Applying Lemma 1.1.1, we get 1

1

1

‖ux ‖∞ ≤ C(‖ux ‖2 + ‖uxx ‖2 ) 2 ‖ux ‖ 2 ≤ C(1 + ‖uxx ‖) 2 and ‖u‖∞ ≤ C, thus J

󵄨 󵄨 ∑ 󵄨󵄨󵄨[(|uj |2 )xx̄ uj + ujx̄ (|uj |2 )x + (|uj |2 )x ujx + |uj |2 ujxx̄ ]ujxx̄ h󵄨󵄨󵄨 j=1

J

≤ C ∑[|ujxx̄ |(|uj |2 + |uj ||uj−1 | + |uj ||uj+1 |) j=1

+ (|ujx |2 + |ujx̄ |2 + |ujx ||ujx̄ ||uj |)|ujxx̄ |]h

≤ C(1 + ‖uxx̄ ‖2 + ‖uxx̄ ‖4 ),

(1.1.41)

J

󵄨 󵄨 ∑ 󵄨󵄨󵄨[(|uj |4 )xx̄ uj + ujx̄ (|uj |4 )x + (|uj |4 )x ujx + |uj |4 ujxx̄ ]ujxx̄ 󵄨󵄨󵄨h j=1

J

4

≤ C ∑[|ujxx̄ |(|uj | + |uj ||uj−1 | + |uj ||uj+1 |) j=1

1.1 Generalized Ginzburg–Landau equations | 11

2

+ |uj |3 (|ujx | + |ujx̄ |) ]|ujxx̄ |h

≤ C(1 + ‖uxx̄ ‖2 + ‖uxx̄ ‖4 ).

(1.1.42)

Substituting the above inequalities (1.1.41), (1.1.42) into (1.1.40) leads to (1.1.36), i. e., d ‖u ̄ ‖2 + γr ‖uxxx̄ ‖2 ≤ E2 (1 + ‖uxx̄ ‖4 ), dt xx

(1.1.43)

where E2 is a constant. From Lemma 1.1.6, inequality (1.1.31) and the uniform Gronwall lemma, we derive inequality (1.1.35). With the above a priori estimates, we obtain the existence of a discrete global attractor. Theorem 1.1.1. Suppose that γr > 0, δr > 0, χ > 0 and 4γr δr > (γi − μi )2 , then for every J

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ initial value u0j ∈ C , where C = C × C × ⋅ ⋅ ⋅ × C, C being a complex domain, there exists a global solution of problem (1.1.16)–(1.1.18). Moreover, if u0 satisfies J

J

‖u0 ‖ ≤ R0 , then for Γ0󸀠 > Γ0 =

R0 > 0,

(1.1.44)

P2 , the solution of equation (1.1.6) with initial value u(0) 2χ

󵄩󵄩 󵄩 󸀠 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ Γ0 ,

∀t ≥ T0 =

R2 1 lg 󸀠 2 0 2 . 2χ (Γ0 ) − Γ0

= u0 satisfies (1.1.45)

Proof. Inequality (1.1.45) can be derived from inequality (1.1.43). Theorem 1.1.2. Under the conditions of Theorem 1.1.1, there exists a semiflow of (1.1.6) which possesses a global attractor with respect to the L2 -norm. The attractor AJ lies in B(0; Γ0 ) ⊂ C J , where B(0; Γ0 ) denotes the ball of radius Γ0 centered at 0 ∈ C J . Theorem 1.1.3. Assume the conditions of Theorem 1.1.1 hold and suppose that the initial value u0 satisfies ‖u0 ‖ ≤ R0 ,

R0 > 0.

(1.1.46)

Then for 1

1

Γ1󸀠 > Γ1 = (a1 + 2E1 ) 2 (exp a1 ) 2 , 1 a1 = [P 2 + (2χ + 1)Γ02 ], α we have the following estimate of the solution u = u(t) of the discrete equations (1.1.6)–(1.1.8): 󵄩󵄩 󵄩 󸀠 󵄩󵄩u(t)󵄩󵄩󵄩H 1 ≤ Γ1 ,

∀t ≥ T1 = max{T0 , 2}.

(1.1.47)

12 | 1 Discrete attractor and approximate calculation The semiflow of (1.1.6) possesses a global attractor with respect to the H1 -norm. The attractor AJ lies in B(0; Γ1 ) ⊂ C J , where B(0; Γ1 ) denotes the ball of radius Γ1 centered at 0 ∈ CJ . Proof. It follows from inequality (1.1.19) that t+1

t+1

t

t

󵄩2 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩󵄩 2 󵄩󵄩u(t + 1)󵄩󵄩󵄩 − 󵄩󵄩󵄩u(t)󵄩󵄩󵄩 + 2χ ∫ 󵄩󵄩󵄩u(s)󵄩󵄩󵄩 ds + α ∫ 󵄩󵄩󵄩ux (s)󵄩󵄩󵄩 ds ≤ P .

(1.1.48)

So by Theorem 1.1.1, we have 󵄩󵄩 󵄩2 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ Γ0 ,

∀t ≥ T0 .

(1.1.49)

Thus t+1

1 󵄩 󵄩2 ∫ 󵄩󵄩󵄩ux (s)󵄩󵄩󵄩 ds ≤ a1 = [P 2 + (2χ + 1)Γ02 ]. α t

And from (1.1.33), we get 1 1 󵄩󵄩 󵄩2 󵄩󵄩ux (t + 1)󵄩󵄩󵄩 ≤ Γ1 = (a1 + 2E1 ) 2 (exp a1 ) 2 ,

∀t ≥ 1.

(1.1.50)

Therefore we obtain inequality (1.1.17), which completes the proof of Theorem 1.1.3. Similarly, we derive the following theorem from (1.1.28), (1.1.43) and the uniform Gronwall inequality. Theorem 1.1.4. Assume the conditions of Theorem 1.1.1 hold and suppose that the initial value u0 satisfies ‖u0 ‖H 2 ≤ R0 ,

R0 > 0.

(1.1.51)

Then for Γ0󸀠 > Γ2 = (a2 + 2E2 ) ln a2 , a2 =

Γ12 + 2E1 (1 + Γ14 ) , γr

the solution u(t) of (1.1.6)–(1.1.8) satisfies 󵄩󵄩 󵄩 󸀠 󵄩󵄩u(t)󵄩󵄩󵄩H 2 ≤ Γ2 ,

∀t ≥ T2 > T1 .

(1.1.52)

The semiflow of (1.1.6) possesses a global attractor with respect to the H2 -norm. The attractor AJ lies in B(0; Γ2 ) ⊂ C J , where B(0; Γ2 ) denotes the ball of radius Γ2 centered at 0 ∈ CJ . In the following, we study the dimension estimation of the attractor of system (1.1.6)–(1.1.8). Consider the linearized equation

1.1 Generalized Ginzburg–Landau equations | 13

ivt + νvx = χv + (γr + iγi )vxx̄ − (βr + iβi )(3|u|2 v + u2 v)̄ { { { { { 1 { { ̄ 2 )x̂ + (uv)x̂ u) ̄ { − (δr + iδi )(3|u|4 v + 2|u|2 u2 v)̄ − (λr + iλi )( v(u { { 2 { { 1 { { ̄ 2 )x̂ + (vū + uv)̄ x̂ − (uv)x̂ u], ̄ − (μr + iμi )[|u|2x̂ v − v(u { { { 2 { { { J {v(0) = v0 ∈ C ,

(1.1.53)

where u = S(t)u0 is the solution of problem (1.1.6)–(1.1.8). Let Q(t)v0 = Q(t; u0 , v0 ) = v(t)

(1.1.54)

be the semiflow of (1.1.53) in C J . Proposition 1.1.1. For any t ∈ ℝ, lim

ε→0

sup

u0 ,u1 ∈AJ 0 0.

From Lemma 1.1.1, we have 󵄩 (k) 󵄩2 󵄩 (k) 󵄩 2 󵄩󵄩 (k) 󵄩󵄩2 󵄩 (k) 󵄩 󵄩 (k) 󵄩 1 󵄩 (k) 󵄩 3 󵄩󵄩ϕ 󵄩󵄩4 ≤ C1 (󵄩󵄩󵄩ϕ 󵄩󵄩󵄩 + 󵄩󵄩󵄩ϕx 󵄩󵄩󵄩) 2 󵄩󵄩󵄩ϕ 󵄩󵄩󵄩 2 ≤ C1 (󵄩󵄩󵄩ϕ 󵄩󵄩󵄩 + ε3 󵄩󵄩󵄩ϕx 󵄩󵄩󵄩) , 1 2

1 2

1 2

‖u‖∞ ≤ C(‖u‖ + ‖ux ‖) ‖u‖ ≤ C‖u‖H 1 .

ε3 > 0,

1.1 Generalized Ginzburg–Landau equations | 15

Thus Re(F 󸀠 (u(τ))ϕ(k) , ϕ(k) ) ≤ −[γr − √λr2 + λi2 (2ε3 ‖u‖∞ ‖ux̂ ‖ + ε1 ‖u‖2∞ ) 󵄩 󵄩󵄩2 − √μ2r + μ2i (2ε3 ‖u‖∞ ‖ux̂ ‖ + 3ε2 ‖u‖2∞ + 3ε3 ‖u‖∞ ‖ux̂ ‖)]󵄩󵄩󵄩ϕ(k) x 󵄩 󵄩 󵄩 󵄩2 + k2 (‖u‖H 1 + ‖u‖2H 1 + 1)󵄩󵄩󵄩ϕ(k) 󵄩󵄩󵄩 .

(1.1.61)

Therefore, since ε1 , ε2 , ε3 are small enough, we have Re(F 󸀠 (u(τ))ϕ(k) , ϕ(k) ) ≤ −

γr 󵄩󵄩 (k) 󵄩󵄩2 2 󵄩 (k) 󵄩2 󵄩󵄩ϕ 󵄩󵄩 + k3 (‖u‖H 1 + ‖u‖H 1 )󵄩󵄩󵄩ϕ 󵄩󵄩󵄩 . 2

(1.1.62)

Similarly, we derive 󵄩 󵄩󵄩2 󵄩󵄩 (k) 󵄩󵄩2 Re((F 󸀠 (u(τ))ϕ(k) )x , ϕ(k) ) = χ 󵄩󵄩󵄩ϕ(k) x 󵄩 󵄩 − γr 󵄩󵄩ϕxx 󵄩󵄩

− Re(βr + iβi )((3|u|2 ϕ(k) + u2 ϕ(k) )x , ϕ(k) x )

− Re(δr + iδi )((3|u|4 ϕ(k) + 2|u|u2 ϕ(k) )x , ϕ(k) x )

1 − Re(λr + λi )(( ϕ(k) (u2 )x̂ ) + ((uϕ(k) )x̂ u)x , ϕ(k) x ) 2 x 1 − Re(μr + iμi )((|u|2x̂ ϕ(k) )x − ((u2 )x̂ ϕ(k) )x 2

− ((ϕ(k) u + uϕ(k) )x̂ u)x − ((uϕ(k) )x̂ u)̄ x , ϕ(k) x ),

(1.1.63)

where 2 (k) 2 2 (k) (k) (3|uj |2 ϕ(k) + u2j ϕ(k) )x = 3|uj+1 |2 ϕ(k) jx + 3|uj |x ϕj + uj+1 + ϕjx + (uj )x ϕj , j

󵄨󵄨 2 󵄨󵄨 2 (k) 󵄨󵄨 2 󵄨󵄨 2 2 (k) 4 (k) 4 (k) (k) (3|uj |4 ϕ(k) j + 2|uj |uj ϕj )x = 3|uj+1 | ϕjx + 3|uj |x ϕj + 2󵄨󵄨uj+1 󵄨󵄨uj+1 ϕjx + 2(󵄨󵄨uj+1 󵄨󵄨uj+1 )x ϕj , (k) (k) [(uj ϕ(k) ̂ uj , j )x̂ uj ]x = (uj+1 ϕj+1 )x̂ ujx + (uj ϕj )xx

󵄨󵄨 2 (k) 2 (k) (k) 󵄨 󵄨󵄨Re(βr + iβi )((3|u| ϕ + u ϕx )x , ϕx )󵄨󵄨󵄨 ≤ √λr2 + λi2 [(‖u‖∞ ‖uxx̂ ‖ + 3‖u‖∞ ‖ux̂ ‖∞ +

1 󵄩 󵄩󵄩2 2 󵄩 󵄩 (k) 󵄩󵄩2 ‖u‖2∞ )󵄩󵄩󵄩ϕ(k) x 󵄩 󵄩 + ε4 ‖u‖∞ 󵄩󵄩ϕxx 󵄩󵄩 ], 4ε4

ε4 > 0,

󵄨 󵄨󵄨 1 2 󵄨󵄨 (k) (k) (k) 󵄨󵄨 2 (k) 󵄨󵄨Re(μr + iμi )((|u|x̂ ϕ )x − ((u )x̂ ϕ(k) )x − ((ϕ u + uϕ(k) )x̂ u)x − ((uϕ )x̂ u)̄ x , ϕx )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄩 󵄩󵄩2 5 󵄩󵄩 k 󵄩󵄩2 2 ≤ √μ2r + μ2i {[9‖u‖∞ ‖ux̂ ‖∞ + 2ε5 ‖u‖2∞ 󵄩󵄩󵄩ϕ(k) xx 󵄩 󵄩 + 2 (‖u‖∞ ‖uxx̂ ‖ + ‖ux ‖∞ )]󵄩󵄩ϕx 󵄩󵄩 5 1 󵄩 󵄩 󵄩 󵄩2 + [ (‖u‖2∞ 󵄩󵄩󵄩uxx̂ + ‖ux ‖2∞ 󵄩󵄩󵄩) + ‖u‖2∞ ]󵄩󵄩󵄩ϕ(k) 󵄩󵄩󵄩 }, ε5 > 0. 2 2ε5

16 | 1 Discrete attractor and approximate calculation It follows from (1.1.63) that 󵄩󵄩 (k) 󵄩󵄩2 󵄩󵄩 (k) 󵄩󵄩2 Re((F 󸀠 (u(τ))ϕ(k) )x , ϕ(k) x ) ≤ χ󵄩 󵄩ϕx 󵄩󵄩 − γr 󵄩󵄩ϕxx 󵄩󵄩

󵄩 󵄩󵄩2 + (ε4 √λr2 + λi2 ‖u‖2∞ + 2ε5 √μ2r + μ2i ‖u‖2∞ 󵄩󵄩󵄩ϕ(k) xx 󵄩 󵄩)

󵄩󵄩2 󵄩 󵄩2 󵄩 + k4 (‖u‖∞ + ‖ux ‖ + ‖ux ‖∞ + ‖uxx ‖) × (󵄩󵄩󵄩ϕ(k) 󵄩󵄩󵄩 + 󵄩󵄩󵄩ϕ(k) x 󵄩 󵄩 ). Thus, as ε4 , ε5 are sufficiently small , we have Re((F 󸀠 (u(τ))ϕ(k) )x , ϕ(k) x )≤−

γr 󵄩󵄩 (k) 󵄩󵄩2 󵄩 (k) 󵄩2 󵄩 (k) 󵄩2 󵄩ϕ 󵄩 + k5 ‖u‖H 2 (󵄩󵄩󵄩ϕ 󵄩󵄩󵄩 + 󵄩󵄩󵄩ϕx 󵄩󵄩󵄩 ). 2 󵄩 xx 󵄩

(1.1.64)

It follows from (1.1.59), (1.1.62) and (1.1.64) that l

∑ Re(F 󸀠 (u(τ))ϕ(k) , ϕ(k) )H 1 ≤ −

k=1

≤−

l γr l 󵄩󵄩 (k) 󵄩󵄩2 󵄩 󵄩2 󵄩 󵄩󵄩2 ∑ 󵄩󵄩ϕxx 󵄩󵄩 + k6 ‖u‖H 2 ∑ (󵄩󵄩󵄩ϕ(k) 󵄩󵄩󵄩 + 󵄩󵄩󵄩ϕ(k) x 󵄩 󵄩) 2 k=1 k=1 l γr l 2 ∑ λk + k6 ‖u‖H 2 (l + ∑ λk ), 2 k=1 k=1

(1.1.65)

where λk = 4J 2 sin2

J k = 1, 2, . . . , [ ] 2

kπ , J

are the eigenvalues of matrix A = J 2 A1 . Since sin x/x ≥ following inequality from (1.1.64):

2 , π

x ∈ (0, π2 ], we obtain the

l

∑ Re(F 󸀠 (u(τ))ϕ(k) , ϕ(k) )H 1

k=1

≤−

5

4

3

2

γr l l l l l (( ) − 10( ) + 40( ) − 80( ) + 80( ) − 32) + 2lk6 √E3 , 3 θ2 θ2 θ2 θ2 θ2

where θ1 J < l < θ2 J,

θ1 , θ2 > 0.

Thus, for l such that l ≥ l0 = [

2k6 √E3 θ2 + 12]θ2 , γr

there exists J0 ≥ l such that t

1 qj = lim ( inf ∫ Re tr(F 󸀠 (τ) ⋅ QJ )dτ) < 0. t→∞ u0 ∈AJ t 0

(1.1.66)

1.2 Zakharov system | 17

Therefore we obtain dH (AJ ) ≤ J0 ,

dF (AJ ) ≤ 2J0

(1.1.67)

from [91], where dH (AJ ), dF (AJ ) denote Hausdorff and fractal dimensions of the global attractor AJ , respectively. Theorem 1.1.5. Under the conditions of Theorem 1.1.4, the Hausdorff and fractal dimensions of the global attractor for the discrete equations (1.1.6)–(1.1.8) are finite, namely inequalities (1.1.67) hold.

1.2 Zakharov system Consider the dissipative Zakharov system 1 n + αnt − Δ(n + |E|2 ) = f (x), λ2 tt iEt + ΔE − nE + iγE = g(x),

(1.2.1) (1.2.2)

where the complex-valued variable E(x, t) represents the envelope of the electric field, and the real-valued variable n(x, t) represents the deviation of the ion density from its equilibrium value. The parameter λ is proportional to the ion acoustic speed, α ≥ 0, γ ≥ 0. In 1972, Zakharov proposed the system (1.2.1)–(1.2.2) which is now known as Zakharov system. When α = γ = 0, f (x) = g(x) = 0, Zakharov computed the soliton solution for this system, and explained clearly the long unresolved phenomenon in laser target shooting, namely, the density hollow phenomenon near the critical surface. Hence Zakharov system caused great concern among the international physics community. In 1974, this system has been discussed in detail in an international physics conference in Kiev (the Soviet Union). Sulem and Sulem [90] were the first who originally showed in the existence and uniqueness of global solutions to the one-dimensional Zakharov system. Global existence of solution for the initial-boundary value problem for 1D Zakharov system was first obtained by Guo and Shen [48] in 1979. In 1994, Guo [37] established the existence and uniqueness of global solution of the initial-boundary value problem for 2D and 3D generalized Zakharov system. Existence of global attractors and estimates of finite Hausdorff dimension of system (1.2.1)–(1.2.2) were explored by Flahaut [25] in 1991. In 1995, Guo and Chang [11] verified the existence of a discrete attractor and obtained an estimate of its dimension. Letting m = nt +εn, ε > 0 in (1.2.1)–(1.2.2), we have three unknown variables which satisfy the following system: m = nt + εn, 2

(1.2.3) 2

2

2

2

mt + (αλ − ε)m − ε(αλ − ε)n − λ Δ(n + |E| ) = λ f ,

(1.2.4)

18 | 1 Discrete attractor and approximate calculation iEt + ΔE − nE + iγE = g.

(1.2.5)

The initial–boundary value problem for equations (1.2.3)–(1.2.5) with the initial conditions n(x, 0) = n0 (x),

(1.2.6)

E(x, 0) = E0 (x)

(1.2.8)

m(x, 0) = n1 (x) + ε1 n0 (x),

(1.2.7)

and the boundary conditions n(x, t) = 0,

x ∈ 𝜕Ω,

(1.2.9)

m(x, t) = 0,

x ∈ 𝜕Ω,

(1.2.10)

E(x, t) = 0,

x ∈ 𝜕Ω,

(1.2.11)

is considered, where Ω denotes the open set (0, L) ⊂ ℝ, 0 < L < ∞. We consider a finite difference scheme for the problem (1.2.3)–(1.2.11). As usual, continuous Sobolev spaces Hp (Ω) and H01 (Ω) are used. In order to consider the discrete case, the following notations are used: xj = jh,

L 0 ≤ j ≤ J = [ ], h

Ej (t) ∼ E(xj , t), Nj (t) ∼ N(xj , t), mj (t) ∼ m(xj , t), 1 1 wjx = (wj+1 (t) − wj (t)), wjx = (wj (t) − wj−1 (t)), h h J

J

j=1

j=1

󵄩 󵄩2 󵄨 󵄨2 (u, v) = h ∑ uj vj 󵄩󵄩󵄩w(t)󵄩󵄩󵄩 = h ∑ 󵄨󵄨󵄨wj (t)󵄨󵄨󵄨 , 󵄩󵄩 󵄩 󵄨 󵄨󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩w(t)󵄩󵄩󵄩∞ = sup 󵄨󵄨󵄨wj (t)󵄨󵄨󵄨󵄩󵄩󵄩w(t)󵄩󵄩󵄩1 = 󵄩󵄩󵄩w(t)󵄩󵄩󵄩 + 󵄩󵄩󵄩wx (t)󵄩󵄩󵄩 . 1≤j≤J

We discretize the initial–boundary value problem (1.2.3)–(1.2.11) as mj = njt + εnj , 2

(1.2.12) 2

2

2

2

2

mjt + (αλ − ε)mj − ε(αλ − ε)nj − λ njxx − λ (|Ej | )xx = λ f ,

(1.2.13)

iEjt + Ejxx − nj Ej + iγEj = gj ,

(1.2.14)

mj (0) = n1 (xj ) + ε1 n0 (xj ),

(1.2.16)

n0 (t) = nJ (t) = 0,

(1.2.18)

nj (0) = n0 (xj ),

Ej (0) = E0 (xj ),

m0 (t) = mJ (t) = 0,

(1.2.15) (1.2.17) (1.2.19)

1.2 Zakharov system | 19

E0 (t) = EJ (t) = 0,

(1.2.20)

where 0 < j < J, t > 0. The potential function u is defined by (uj )xx = mj ,

u0 = 0, uJ = 0.

(1.2.21)

Lemma 1.2.1. Assume that ξj and ηj are discrete functions defined on the set 0 ≤ j ≤ J and ξ0 = ξJ = 0, η0 = ηJ = 0. Then we have (ξ , ηx ) = −(ξx , η),

(ξj ηj )x = ξj+1 ηjx + ξjx ηj = ξj ηjx + ξjx ηj+1 ,

(1.2.22)

(ξj ηj )x = ξj ηjx + ξjx ηj−1 = ξj−1 ηjx + ξjx ηj ,

(ξj ηj )xx = ξj ηjxx + ξjx ηjx + ξjx ηjx + ξjxx ηj . Proof. Observe that J−1

J

(ξ , ηx ) = ∑ ξj (ηj+1 − ηj ) = −h ∑ ηj ξjx = −(ξx , η), j=1

(ξj ηj )x =

ξj+1 ηj+1 − ξj ηj h

j=1

=

ξj+1 ηj+1 − ξj+1 ηj + ξj+1 ηj − ξj ηj h

= ξj+1 ηjx + ξjx ηj .

The other inequalities can be proved similarly. Lemma 1.2.2. For any discrete functions uh = {uj | 0 ≤ j ≤ J} on the finite interval [0, L], 1− 󵄩󵄩 k 󵄩󵄩 󵄩󵄩δ uh 󵄩󵄩p ≤ C‖uh ‖2

k+ 21 − p1 n

‖u ‖ 󵄩 󵄩 (󵄩󵄩󵄩δk uh 󵄩󵄩󵄩2 + hn 2 ) L

k+ 21 − p1 n

,

where 2 ≤ p ≤ ∞, 0 ≤ k < n, δk uh denotes the difference quotient of order k ≤ 0. Lemma 1.2.3. Suppose that E0 ∈ L2 (Ω) and g ∈ L2 (Ω). Then for any solution of the problem (1.2.12)–(1.2.20), we have 2 󵄩󵄩 󵄩2 󵄩 󵄩2 −γt ‖g‖ −γt 󵄩󵄩E(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩E(0)󵄩󵄩󵄩 e + 2 (1 − e ). γ

Proof. Computing the inner product of (1.2.14) with E yields i(Et , E) + (Exx̄ , E) − (nE, E) + iγ(E, E) = (g, E).

(1.2.23)

Then using Lemma 1.2.1 and taking the imaginary part of this equality imply d ‖g‖2 ‖E‖2 + γ‖E‖2 ≤ 2 . dt γ Using Gronwall lemma, we have 2 󵄩󵄩 󵄩2 󵄩 󵄩2 −γt ‖g‖ −γt 󵄩󵄩E(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩E(0)󵄩󵄩󵄩 e + 2 (1 − e ). γ

(1.2.24)

20 | 1 Discrete attractor and approximate calculation Lemma 1.2.4. Suppose that n0 (x) ∈ L2 (Ω), n1 (x) ∈ H −1 (Ω), E0 (x) ∈ H01 (Ω), g(x) ∈ H01 (Ω), f (x, t) ∈ L∞ (ℝ+ ; H −1 (Ω)). Then for the solution of the problem (1.2.12)–(1.2.20), we have ‖ux ‖2 +

λ2 ‖u‖2 + λ2 ‖Ex ‖2 ≤ H0 (0)e−β0 t + C, 2

(1.2.25)

where H0 (t) ≡ ‖ux ‖2 + λ2 ‖n‖2 + 2λ2 ‖Ex ‖2 + 2λ2 (n, |E|2 ). Proof. Computing the inner product of (1.2.13) with u yields 1 d ‖u ‖2 + (αλ2 − ε)‖ux ‖2 + ε(αλ2 − ε)(n, u) + λ2 (n, m) + λ2 (|E|2 , m) = −λ2 (f , u), 2 dt x (1.2.26) where (1.2.21) is used. It should be noticed that, due to equation (1.2.12), 1 d ‖n‖2 + ε‖n‖2 , 2 dt wjx̄ = nj , w0 = 0, j = 1, 2, . . . , J. (n, m) =

(1.2.27)

Let μ1 be the first eigenvalue of the algebraic system of equations (1.2.27). Then we have the estimate 1 󵄨󵄨 󵄨 󵄨 󵄨 . 󵄨󵄨(n, m)󵄨󵄨󵄨 = 󵄨󵄨󵄨(w, ux )󵄨󵄨󵄨 ≤ ‖w‖‖ux ‖ ≤ ‖ux ‖‖n‖ |μ | 1

Consequently, it follows that A ≡ (αλ2 − ε)‖ux ‖2 + ε(αλ2 − ε)(n, u) + ελ2 ‖n‖ ≥ (αλ2 − ε)‖ux ‖2 −

2(αλ2 − ε) ‖ux ‖‖n‖ + ελ2 ‖n‖. |μ1 | 2

μ2

Now, the parameter ε is chosen as ε ≤ min( αλ4 , 2αλ1 2 ). We have A ≥ ελ2 ‖n‖ + (αλ2 − ε)‖ux ‖2 − √

ε (αλ2 − ε)‖ux ‖‖n‖ 2αλ2

ε αλ2 ‖ux ‖2 . ≥ (λ2 ‖n‖2 + ‖nx ‖2 ) − 2 2 Hence we derive from (1.2.26)

1 d 2 2 ε αλ2 (λ ‖n‖ + ‖ux ‖2 ) + (λ2 ‖n‖2 + ‖ux ‖2 ) + ‖ux ‖2 + λ2 (|E|2 , m) 2 dt 2 2 αλ2 λ2 ‖ux ‖2 + ‖ϕ‖2 , ≤ 2 2α

1.2 Zakharov system | 21

where ϕj is defined as ϕjx = fj ,

ϕ0 = 0,

j = 1, 2, . . . , J.

(1.2.28)

Finally, we have λ2 ‖ϕ‖2 1 d 2 2 (λ ‖n‖ + ‖ux ‖2 ) + ε(λ2 ‖n‖2 + ‖ux ‖2 ) + 2λ2 (nt , |E|2 ) + 2ελ2 (nt , |E|2 ) ≤ . 2 dt α (1.2.29) Now, computing the inner product of (1.2.14) with Et and taking the real part yield 1 1 d ‖E ‖2 + (n, |E|2t ) − γ Im(E, Et ) = − Re(g, Et ). 2 dt x 2

(1.2.30)

On the other hand, taking the real part of (1.2.23) implies Im(E, Et ) = − Im(Et , E) = ‖Ex ‖2 + (n, |E|2t ) + Re(g, Et ). Moreover, taking the imaginary part of the inner product of (1.2.14) with g, we have Re(g, Et ) = Im(Ex , gx ) + Im(nE, g) − γ Re(E, g). Hence (1.2.30) can be rewritten as d ‖E ‖2 + (n, |E|2t ) + 2γ‖Ex ‖2 + 2γ(n, |E|2t ) = −2 Im(Ex , gx ) − 2 Im(nE, g). dt x

(1.2.31)

Using the definition of H0 (t), it follows from (1.2.29), to which twice λ2 multiplied by (1.2.31) is added, that 1 d H (t) + ε(λ2 ‖n‖2 + ‖ux ‖2 ) + 2λ2 (ε + 2γ)(n, |E|2 ) + 4γλ2 ‖Ex ‖2 2 dt 0 λ2 ≤ ‖ϕ‖2 − 4λ2 Im(Ex , gx ) − 4λ2 Im(nE, g), α

(1.2.32)

while λ2 ‖ϕ‖2 − 4λ2 Im(Ex , gx ) − 4λ2 Im(nE, g) α 2 2 2 γ λ2 1 ε‖n‖2 2k0 ‖g‖1 ‖E‖ + ), ≤ ‖ϕ‖2 + 4λ2 ( ‖Ex ‖2 + ‖g‖21 ) + 4λ2 ( α 2 2γ 8 ε where k0 is a constant such that ‖g‖∞ ≤ k0 ‖g‖1 . Consequently, one gets from (1.2.32) that 8k 2 λ2 ‖g‖21 ‖E‖2 d λ2 2λ2 H0 (t) + A0 (t) ≤ ‖ϕ‖2 + ‖g‖21 + 0 , dt α γ ε

(1.2.33)

22 | 1 Discrete attractor and approximate calculation where A0 (t) = ε‖ux ‖2 +

ελ2 ‖n‖2 + 2λ2 (ε + 2γ)(n, |E|2 ) + 2γλ2 ‖Ex ‖2 . 2

Now, we take β0 = min( 4ε , γ2 ) and estimate ελ2 󵄨󵄨 2 2 2 󵄨 ‖n‖ + γλ2 ‖Ex ‖2 + C, 󵄨󵄨[2β0 λ − 2λ (ε + 2γ)](n, |E| )󵄨󵄨󵄨 ≤ 4 where Lemmas 1.2.2 and 1.2.3 were used. Thus, β0 H0 (t) − A0 (t) ≤ C. Therefore, we have 8k 2 λ2 ‖g‖21 ‖E‖2 2λ2 2λ2 d H0 (t) + β0 H0 (t) ≤ ‖ϕ‖2 + ‖g‖21 + 0 + C ≡ k1 . dt α γ ε Using Gronwall lemma, we obtain H0 (t) ≤ H0 (t)e−β0 t +

k1 , β0

(1.2.34)

where k1 is a constant independent of t. Finally, we estimate J 1 2λ2 (n, |E|2 ) ≤ 2λ2 ( ‖n‖2 + h ∑ |Ej |4 ). 4 j=1

By Lemmas 1.4.2 and 1.2.3, we have 2λ2 (n, |E|2 ) ≤

λ2 ‖n‖2 + λ2 ‖Ex ‖2 + k22 . 2

The definition of H0 (t) implies λ2 ‖n‖2 + λ2 ‖Ex ‖2 − k22 , 2 H0 (t) ≤ C(‖ux ‖2 + ‖n‖2 + ‖Ex ‖2 + 1).

H0 (t) ≥ ‖ux ‖2 +

(1.2.35)

Using Lemma 1.2.3 and inequality (1.2.34), we have ‖ux ‖2 +

k λ2 ‖n‖2 + λ2 ‖Ex ‖2 ≤ H0 (0)e−β0 t + 1 + k22 . 2 β0

(1.2.36)

1.2 Zakharov system |

23

Now we consider more a priori estimates for a solution of problem (1.2.12)–(1.2.20) by using discrete Sobolev interpolation theorems. The procedure is the same as in Lemma 1.2.3. In order to estimate the difference quotient of higher order, we assume that Ej = 0, { nj = 0,

j < 0 and j > J, j < 0 and j > J.

(1.2.37)

Lemma 1.2.5. Suppose that n1 (x) ∈ L2 (Ω), n0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), E0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), g(x) ∈ H01 (Ω), f (x, t) ∈ L∞ (ℝ+ ; L2 (Ω)). Then for any solution of problem (1.2.12)–(1.2.20), we have ‖m‖2 + λ2 ‖nx ‖2 + λ‖Exx ‖2 ≤ H0 (0)e−β1 t + C, where H1 (t) ≡ ‖m‖2 + λ2 ‖nx ‖2 + 2λ2 ‖Exx ‖2 − 2λ2 (|E|2xx , n) + 2λ2 (|E|2x + |Ex |2x , n) − 4λ2 Re(g, Exx ). Proof. The lemma can be proved by computing the inner product of (1.2.14) with (Etxx + γExx ) and taking the real part, and then computing the inner product of (1.2.13) with mj . We have 1 d ‖E ̄ ‖2 + γ‖Exx ‖2 − Re(nE, Etxx̄ + γExx̄ ). 2 dt xx

(1.2.38)

And d λ2 λ2 λ2 ‖f ‖2 (‖m‖2 + ‖nx ‖2 ) + ε(‖m‖2 + ‖nx ‖2 ) − 2λ2 (|E|2xx̄ , m) ≤ , dt 2 2 λ 2

(1.2.39)

μ2

where the parameter ε is chosen as ε ≤ min( αλ4 , 2αλ1 2 ). Computing the product of 4λ2 and (1.2.38) and adding to (1.2.39) yields d (‖m‖2 + λ2 ‖nx ‖2 + 2λ2 ‖Exx ‖2 ) + 4γλ2 ‖Exx ‖2 + ε(‖m‖2 + λ2 ‖nx ‖2 ) dt − 4λ2 Re(nE, Etxx̄ + γExx̄ ) − 2λ2 (|E|2xx̄ , n) ≤ 4λ2 Re(g, Etxx̄ + γExx̄ ) +

λ2 2 ‖f ‖ . α

(1.2.40)

Thus we get d H (t) + 4γλ2 ‖Exx̄ ‖2 + ε(‖m‖2 + λ2 ‖nx ‖2 ) dt 1 − 4γλ2 Re(g, Exx̄ ) − 2ελ2 (|E|2xx̄ , n) + 2ελ2 (n, |Ex |2 + |Ex̄ |2 ) ≤

λ2 2 ‖f ‖ − 4λ2 Re(nExx̄ , Et ) + 4γλ2 Re(nE, Exx̄ ) + 2λ2 (m, |Ex |2 + |Ex̄ |2 ). α

(1.2.41)

24 | 1 Discrete attractor and approximate calculation From Lemmas 1.2.2, 1.2.3 and 1.2.4, we have

γ ε 󵄨󵄨 2 2 󵄨 2 󵄨󵄨(m, |Ex | )󵄨󵄨󵄨 ≤ 2 ‖m‖ + ‖Exx ‖ + C, 12 8λ γ ε 󵄨󵄨 2󵄨 2 2 󵄨󵄨(m, |Ex̄ |) 󵄨󵄨󵄨 ≤ 2 ‖m‖ + ‖Exx ‖ + C, 12 8λ 󵄨 γ 󵄨󵄨 2 2 󵄨󵄨(n Exx , E)󵄨󵄨󵄨 ≤ ‖Exx ‖ + C. 12

Thus d H (t) + β1 H1 (t) ≤ C. dt 1

(1.2.42)

C . β1

(1.2.43)

Employing Gronwall lemma, H1 (t) ≤ H1 (0)e−β1 t + Therefore from (1.2.43), we obtain ‖m‖2 + λ2 ‖nx ‖2 + λ2 ‖Exx ‖2 ≤ H1 (0)e−β1 t +

C + C, β1

(1.2.44)

which completes the proof of Lemma 1.2.5. Lemma 1.2.6. Suppose that f (x, t) ∈ L∞ (ℝ+ ; H01 (Ω)), g(x) ∈ H01 (Ω), n1 (x) ∈ H01 (Ω), n0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), E0 (x) ∈ H 3 (Ω) ∩ H01 (Ω). Then for any solution of problem (1.2.12)–(1.2.20), we have λ2 λ2 1 ‖mx ‖2 + ‖nxx ‖2 + ‖Exxx ‖2 ≤ H2 (0)e−β2 t + C, 2 4 2 where λ2 λ2 1 H2 (t) ≡ ‖mx ‖2 + ‖nxx ‖2 + ‖Exxx ‖2 + λ2 (nxx̄ , |E|2xx ) + λ2 (n, |Exx̄ |2 ) 2 4 2 + 4λ2 Re(nx Ex + nx̄ Ex̄ Exx̄ ) − 2λ2 Re(gx , Exxx ). Proof. Performing a similar procedure in the proof of Lemma 1.2.5, we can prove Lemma 1.2.6 by computing the inner product of (1.2.13) with −mxx , and computing the inner product of (1.2.13) with (−Etxxx̄ x̄ − γExxx̄ x̄ ) and then taking the real part: 1 d ε (‖mx ‖2 + λ2 ‖nxx ‖) + (‖mx ‖2 + λ2 ‖nxx ‖) 2 dt 2 αλ2 λ2 αλ2 + ‖mx ‖2 + λ2 (|E|xx , mxx ) ≤ ‖mx ‖2 + ‖fx ‖2 . 2 2 2α

(1.2.45)

Computing the inner product of (1.2.14) with (−Et xx̄ xx̄ − γExx̄ xx̄ ) and taking the real part, we have 1 d ‖E ‖2 + γ‖Exxx̄ ‖2 + Re(nE, Et xx̄ xx̄ + γExx̄ xx̄ ) 2 dt xxx = − Re(g, Et xx̄ xx̄ + γExx̄ xx̄ ).

(1.2.46)

1.2 Zakharov system | 25

Thus we have from (1.2.46) that d 1 1 [ ‖E ‖2 + (n, |Exx̄ |2 ) + 2 Re(nx̄ Ex̄ + nx Ex , Exx̄ ) − Re(gx , Exxx )] dt 2 xxx 2 1 + γ‖Exxx̄ ‖2 + (|E|2txx̄ , nxx̄ ) − Re(nxx̄ Exx̄ , Et ) 2 − Re(2nx̄ Et x̄ + 2nx Etx − nx̄ Etx − nx Etx , Exx̄ ) − 2 Re(Ex̄ mx̄ + Ex mx , Exx̄ ) + (2ε + γ) Re(Ex̄ nx̄ + Ex nx , Exx̄ ) 1 − (|Exxx |2 , m) + γ(n, |Exxx |2 ) = γ Re(gx , |E|xxx ). 2

(1.2.47)

Equation (1.2.45) added to (1.2.47), which was multiplied by 2λ2 , give ε d H (t) + (‖mx ‖2 + λ2 ‖nxx̄ ‖2 ) + 2γλ2 ‖Exxx̄ ‖2 dt 2 2 + ελ2 (|E|xx̄ , nxx̄ ) + λ2 (ε + 2γ)(n, |Exx̄ |2 )

+ 2λ2 (2ε + γ) Re(Ex̄ nx̄ + Ex nx , Exx̄ ) − 2γλ2 Re(gx , Exxx̄ )

≤

λ2 ‖f ‖2 + 2λ2 Re(nxx̄ Exx̄ , Et ) 2α x + 2λ2 Re(2nx̄ Et x̄ + 2nx Etx − nx̄ Etx − nx Et x̄ , Exx̄ )

+ 4λ2 Re(Ex̄ mx̄ + Ex mx , Exx̄ ) − 2γλ2 Re(nxx̄ E, Exx̄ ) + λ2 (|Exx̄ |2 , m).

(1.2.48)

We estimate every term on the right-hand side of (1.2.48). Equation (1.2.14) implies 2λ2 Re(nxx̄ Exx̄ , Et ) = 2λ2 [Im(nxx̄ Exx̄ , Exx̄ ) − Im(nxx̄ Exx̄ , nE) − γ Re(nxx̄ Exx̄ , E) − Im(nxx̄ Exx̄ , g)].

(1.2.49)

Similarly, we have J

Re(nx̄ Et x̄ , Exx̄ ) = − Im(nx̄ Exxx̄ , Exx̄ ) + h Im ∑ njx̄ (njx̄ Ej + nj−1 Ejx̄ )E jxx j=1

− γ Re(nx̄ Ex̄ , Exx̄ ) + Im(nx̄ gx̄ , Exx̄ ). We give estimates of every term in (1.2.50) and take η = implies

(1.2.50) ε , 12

ξ =

γ . 16

Equation (1.2.48)

d H (t) + A2 (t) ≤ C, dt 2 where A2 (t) =

ε (‖mx ‖2 + λ2 ‖nxx̄ ‖2 ) + γλ2 ‖Exxx̄ ‖2 4 + ελ2 (|E|2xx̄ , nxx̄ ) + λ2 (ε + 2γ)(n, |Exx̄ |2 )

2λ2 (2ε + γ) Re(Ex̄ nx̄ + Ex nx , Exx̄ ) − 2γλ2 Re(gx , Exxx̄ ).

(1.2.51)

26 | 1 Discrete attractor and approximate calculation We take β = min( 4ε , γ2 ) and estimate H2 (t) as follows:

Therefore

1 λ2 λ2 H2 (t) ≥ ‖mx ‖2 + ‖nxx̄ ‖2 + ‖Exxx̄ ‖2 − C. 2 4 2 1 λ2 λ2 ‖mx ‖2 + ‖nxx̄ ‖2 + ‖Exxx̄ ‖2 ≤ H2 (0)e−β2 t + C. 2 4 2

(1.2.52)

This completes the proof of Lemma 1.2.6.

In view of the uniform a priori estimates above, one can obtain the existence of the attractor for the discrete system (1.2.12)–(1.2.20) with respect to ‖⋅‖Σ0 , ‖⋅‖Σ1 , ‖⋅‖Σ2 -norm. Here Σ0 = H −1 (Ω) × L2 (Ω) × H01 (Ω), Σ1 = L2 (Ω) × H01 (Ω) × (H02 (Ω) ∩ H01 (Ω)), Σ2 = H01 (Ω) × (H 2 (Ω) ∩ H01 (Ω)) × (H 3 (Ω) ∩ H01 (Ω)). Theorem 1.2.1. Suppose that n0 (x) ∈ L2 (Ω), n1 (x) ∈ H −1 (Ω), E0 (x) ∈ H01 (Ω), f (x) ∈ H −1 (Ω), g(x) ∈ H01 (Ω). Then there exists a bounded absorbing set B0 for (m, n, E) in Σ0 , and for every bounded set B of Σ0 , ∀(m0 , n0 , E0 ) ∈ B , there exists t0 > 0 such that ∀t ≥ t0 , (m(t), n(t), E(t)) ∈ B0 . Proof. Assume that ‖m0 ‖2−1 + ‖n0 ‖2 + ‖E0 ‖2H 1 ≤ R2 . In particular, we have ‖E0 ‖2 ≤ But due to (1.2.24), E satisfies the inequality

R2 . λ1

2 󵄩󵄩 󵄩2 󵄩 󵄩2 −γt ‖g‖ −γt 󵄩󵄩E(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩E(0)󵄩󵄩󵄩 e + 2 (1 − e ). γ

Thus ‖g‖2 󵄩󵄩 󵄩2 2 −γt 󵄩󵄩E(t)󵄩󵄩󵄩 ≤ ‖E0 ‖ e + 2 . γ Let t0󸀠 be given such that ∀t ≥ t0󸀠 , ‖E(t)‖2 ≤ ‖m‖2−1 +

‖g‖2 . γ2

Due to inequality (1.2.36), we have

󸀠 k λ2 ‖n‖2 + λ2 ‖E‖2H 1 ≤ H0 (t0󸀠 )e−β(t0 −t0 ) + 1 + k22 , 2 β0

where k1 depends on f , g and ‖E‖. Thus ‖m‖2−1 +

󸀠 λ2 ‖n‖2 + λ2 ‖E‖2H 1 ≤ δR2 e−β(t0 −t0 ) + k0󸀠 , 2

where k0󸀠 is independent of t, initial data and step length h. Consequently, this clearly implies that there exists t0 (R) such that ‖m‖2−1 +

λ2 ‖n‖2 + λ2 ‖E‖2H 1 ≤ 2k0󸀠 , 2

∀t ≥ t0 (R).

(1.2.53)

1.2 Zakharov system | 27

Theorem 1.2.2. Under the conditions of Theorem 1.2.1, there exists a global attractor for the semiflow (1.2.12)–(1.2.20) with respect to the ‖ ⋅ ‖Σ0 norm. The attractor lies in B(0, Γ0 ) ⊂ ℝJ × ℝJ × ℂJ , where B(0, Γ0 ) is the ball centered at 0 ∈ ℝJ × ℝJ × ℂJ , with radius Γ0 = 2k0󸀠 . Theorem 1.2.3. Suppose that n1 (x) ∈ L2 (Ω), n0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), E0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), f (x) ∈ L2 (Ω), g(x) ∈ H01 (Ω). Then there exists a bounded absorbing set B1 for (m, n, E) in Σ1 . Proof. Indeed, if we assume that ‖m0 ‖2 + ‖n0 ‖2H 1 + ‖Exx̄ ‖2 ≤ R2 , then ‖m0 ‖2−1 + ‖n0 ‖2 + 3

R ‖E0 ‖2H 1 ≤ √λ . But, according to Theorem 1.2.1, there exists t0 > 0 such that ∀t ≥ t0 , 1 (m(t), n(t), B(t)) ∈ B0 , B0 is a bounded set of Σ0 , independent of the initial data. Due to (1.2.44),

‖m‖2 + λ2 ‖nx ‖2 + λ2 ‖Exx̄ ‖2 ≤ H1 (t0 )e−β2 (t−t0 ) + k1󸀠 , where k1󸀠 is a constant independent of t, initial data and step length h. Then we derive easily that there exists t1 (R) ≥ t0 > 0 such that ‖m‖2 + λ2 ‖nx ‖2 + λ2 ‖Exx̄ ‖2 ≤ 2k1󸀠 ,

∀t ≥ t1 .

(1.2.54)

We deduce the existence of a bounded absorbing set B1 for (m, n, E) in Σ1 . Theorem 1.2.4. Under the conditions of Theorem 1.2.1, there exists a global attractor for the semiflow (1.2.12)–(1.2.20) with respect to the ‖ ⋅ ‖Σ1 norm. The attractor lies in B(0, Γ1 ) ⊂ ℝJ × ℝJ × ℂJ , where B(0, Γ1 ) is the ball centered at 0 ∈ ℝJ × ℝJ × ℂJ , with radius Γ0 = 2k1󸀠 . From the inequality (1.2.52) we have Theorem 1.2.5. Suppose that n0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), n1 (x) ∈ H01 (Ω), E0 (x) ∈ H 3 (Ω) ∩ H01 (Ω), f (x) ∈ H01 (Ω), g(x) ∈ H01 (Ω). Then there exists a bounded absorbing set B2 for (m, n, E) in Σ2 . Theorem 1.2.6. Under the conditions of Theorem 1.2.1, there exists a global attractor for the semiflow (1.2.12)–(1.2.20) with respect to the ‖ ⋅ ‖Σ2 norm. The attractor lies in B(0, Γ2 ) ⊂ ℝJ × ℝJ × ℂJ , where B(0, Γ2 ) is the ball centered at 0 ∈ ℝJ × ℝJ × ℂJ , with radius Γ0 = 2k2󸀠 . In the following, we study the dimension of the attractors. Now consider the linearized system for the discrete system (1.2.12)–(1.2.20) given by vt + (αλ2 − ε)v − ε(αλ2 − ε)u − λ2 uxx − λ2 Re(Exx̄ F) − 2λ2 Re(Fxx̄ E) − 2λ2 Re(Fx Ex ) − 2λ2 Re(Fx̄ Ex ),

iFt + Fxx̄ − uE − nF + iγF = 0,

(1.2.55) (1.2.56)

28 | 1 Discrete attractor and approximate calculation v = vt + εu, v|t=0 = v0 ,

(1.2.57) u|t=0 = u0 ,

F|t=0 = F0 ,

(1.2.58)

where (m, n, E) = S(t)(m0 , n0 , E0 ), (m0 , n0 , E0 ) ∈ Σ2 and (v0 , u0 , F0 ) ∈ Σ1 . Since (m, n, E) belongs to L∞ (ℝ+ ; Σ1 ), it is easy to show that this system admits a unique solution (v, u, F) belonging to L∞ (ℝ+ ; Σ1 ). We set (v(t), u(t), F(t)) = DS(t)(m0 , n0 , E0 )(v0 , u0 , F0 ), where DS(t)(m0, n0, F0) is the differential of S(t) at (m0 , n0 , E0 ). Proposition 1.2.1. If for ∀R > 0 and ∀0 < T < +∞, there exists a constant C(R, T) such that 󵄩󵄩 (i) (i) (i) 󵄩󵄩 󵄩󵄩(m0 , n0 , E0 )󵄩󵄩Σ ≤ R, 1

i = 1, 2, ∀t, |t| < T,

then we have 󵄩󵄩 󵄩 (2) (2) (2) (1) (1) (1) (1) (1) (1) 󵄩󵄩S(t)(m0 , n0 , E0 ) − S(t)(m0 , n0 , E0 ) − DS(t)(m0 , n0 , E0 )(m0 , n0 , E0 )󵄩󵄩󵄩Σ

1

󵄩 󵄩2 ≤ C 󵄩󵄩󵄩(m0 , n0 , E0 )󵄩󵄩󵄩Σ

(1.2.59)

1

where (1) m0 = m(2) 0 − m0 ,

(1) n0 = n(2) 0 − n0 ,

E0 = E0(2) − E0(1) .

Proof. Let (2) (2) (1) (1) (1) (1) (1) (1) w(t) = S(t)(m(2) 0 , n0 , E0 ) − S(t)(m0 , n0 , E0 ) − DS(t)(m0 , n0 , E0 )(m0 , n0 , E0 )

= G1 (t) − G(t) − v(t).

Thus we have 𝜕t w(t) = L1 (G1 (t)) − L1 (G(t)) + L(G(t))v(t)

= L1 (G1 (t) + v(t) + w(t)) − L1 (G1 (t)) + L(G(t))v(t),

w(0) = 0,

(1.2.60) (1.2.61)

where Gt = L1 (G) is the operator from the discrete system (1.2.12)–(1.2.20), and vt = −L(G(t))v is the operator from system (1.2.55)–(1.2.58). Therefore, (1.2.60) can be rewritten in the form 𝜕t w(t) + L(G(t))v(t) = Λ 0 (G, v, w), where Λ 0 (G, v, w) = L1 (G(t) + v(t) + w(t)) − L1 (G(t)) + L(G(t))(v(t) + w(t)).

(1.2.62)

1.2 Zakharov system |

29

By applying the integral estimate for the linear differential equations, we get the estimate 󵄩2 󵄩 󵄩 󵄩󵄩 󵄩󵄩w(t)󵄩󵄩󵄩Σ1 ≤ C 󵄩󵄩󵄩(m0 , n0 , E0 )󵄩󵄩󵄩Σ1 . This implies that the semigroup operator S(t) is differentiable in Σ1 . Since we expect the solutions to be exponentially damped, we set p = eσt u,

q = eσt v,

G = eσt F.

Then (1.2.55)–(1.2.57) are rewritten as dp = (σ − ε)p + q, dt dq + (aλ2 − ε − σ)q − ε(aλ2 − ε) − λ2 pxx dt = 2λ2 Re(Exx G) + 2λ2 Re(Gxx E) + 2λ2 Re(Gx Ex ) + 2λ2 Re(Gx̄ Ex̄ ),

iGt + Gxx − pE − nG + i(γ − ε)G = 0.

(1.2.63)

(1.2.64) (1.2.65)

We are going to try to obtain an energy estimate of (q, p, G) with a view of studying the dimension of the attractors. To begin with, we take the scalar product in L2 of (1.2.63) and p and obtain J 1 d ‖p‖2 + (σ − ε)‖p‖2 + ∑ qj pj h. 2 dt j=1

(1.2.66)

In the same way, the scalar product of (1.2.64) and q gives J J 1 d ‖q‖2 + (αλ2 − ε − δ)‖q‖2 − ε(αλ2 − ε) ∑ pj qj h − λ2 ∑ pjxx̄ qj h 2 dt j=1 j=1 J

J

J

j=1

j=1

= 2λ2 Re ∑ Ejxx̄ Gj qj h + 2λ2 Re ∑ Gjxx̄ Ej qj h + 2λ2 Re ∑ Gjx Ejx qj h j=1

J

+ 2λ2 Re ∑ Gjx̄ Ejx̄ qj h.

(1.2.67)

j=1

Thus 1 d ε (‖q‖2 + λ2 ‖px ‖2 ) + (‖q‖2 + λ2 ‖px ‖2 ) − σ 2 (‖q‖2 + λ2 ‖px ‖2 ) 2 dt 2 J

J

J

j=1

j=1

= 2λ2 Re ∑ Ejxx̄ Gj qj h + 2λ2 Re ∑ Gjxx̄ Ej qj h + 2λ2 Re ∑ Gjx Ejx qj h j=1

J

+ 2λ2 Re ∑ Gjx̄ Ejx̄ qj h. j=1

(1.2.68)

30 | 1 Discrete attractor and approximate calculation Now consider the equation involving G and take the L2 scalar product of (1.2.65) and G. Then taking its imaginary part, we have J 1 d ‖G‖2 + (γ − σ)‖G‖2 = Im ∑ pj Ej Gj h. 2 dt j=1

(1.2.69)

In the same way, if we scalar-multiply (1.2.65) by Gxt x̄ + (γ − σ)Gxx̄ in L2 , we get 1 d ‖G ̄ ‖2 + (γ − σ)‖Gxx̄ ‖2 2 dt xx J

J

j=1

j=1

J

J

j=1

j=1

− Re ∑ pj Ej Gjxxt h − (γ − σ) Re ∑ pj Ej Gjxx h − Re ∑ nj Gj Gjxxt h − (γ − σ) Re ∑ nj Gj Gjxxt h = 0.

(1.2.70)

Thus, (1.2.67) can be rewritten in the following way: J J d 1 ( ‖Gxx̄ ‖2 − Re ∑ pj Ej Gjxx̄ h − Re ∑ nj Gj Gjxx̄ h) dt 2 j=1 j=1 J

+ (γ − σ)‖Gxx̄ ‖2 + (γ − ε) Re ∑ pj Ej Gjxx̄ h j=1

J

J

j=1

j=1

J

J

+ Re ∑ qj Ej Gjxx̄ h + Re ∑ njt Gj Gjxx̄ h + Im ∑ nj pj Gjxx̄ h + Im ∑ n2j Gj Gjxx̄ h j=1

j=1

J

− (γ − δ) Re ∑ nj Gj Gjxx̄ h = 0.

(1.2.71)

j=1

By analogy with the method used to obtain the a priori estimates, we are going to combine these energy inequalities and proceed in the following way: μ(1.2.66) + (1.2.67) + v(1.2.68) + 2λ2 (1.2.71). This leads to defining a new functional J

J

Q(t) = λ2 ‖Gxx̄ ‖2 − 2λ2 Re ∑ pj Ej Gjxx̄ h − 2λ2 Re ∑ nj Gj Gjxx̄ h j=1

j=1

2

μ v 1 λ + ‖G‖2 + ‖q‖2 + ‖px ‖2 + ‖p‖2 , 2 2 2 2

1.2 Zakharov system | 31

which satisfies d Q(t) + 2λ2 (γ − δ)‖Gxx̄ ‖2 + ν(γ − σ)‖Gxx̄ ‖2 dt ε − ( − σ)(‖q‖2 + λ2 ‖px ‖2 ) + μ(γ − σ)‖p‖2 ≤ A, 2

(1.2.72)

where J

A = 2λ2 (ε + γ − 2σ) Re ∑ pj Ej Gjxx̄ h j=1

J

J

− 2λ2 ∑ pj Ejt Gjxx̄ h − 2λ2 Re ∑ nj Gj Gjxx̄ h j=1

j=1

J

J

j=1

j=1

− 2λ2 Im ∑ nj Ej Gjxx̄ h − 2λ2 ∑ n2j Gj Gjxx̄ h J

J

+ ν Im ∑ pj Ejt Gj h + 2λ2 Re ∑ Ejxx̄ Gj qj h j=1

j=1

J

J

+ 2λ2 Re ∑ Gjx E jx qj h + 2λ2 Re ∑ Gjx E jx̄ qj h j=1

j=1

J

J

j=1

j=1

+ μ ∑ pj qj hj + 2(γ − σ) ∑ nj Gj Gjxx̄ h. Moreover, we have A ≤ 2λ2 (ε + γ − 2σ)‖p‖‖E‖∞ ‖Gxx̄ ‖

+ 2λ2 ‖p‖‖Et ‖∞ ‖Gxx̄ ‖ + 2λ2 ‖nt ‖2 ‖Gx ‖∞ ‖Gxx ‖∞

+ 2λ2 ‖n‖∞ ‖E‖∞ ‖p‖‖Gxx̄ ‖ + 2λ2 ‖n‖2∞ ‖G‖‖Gxx̄ ‖ + ν‖p‖‖E‖∞ ‖G‖ + 2λ2 ‖Exx̄ ‖‖G‖∞ ‖q‖

+ 4λ2 ‖Gx ‖|Ex |∞ ‖q‖ + μ‖p‖‖q‖ + 2|γ − σ|‖n‖∞ ‖G‖‖Gxx ‖. We have made the assumption that (m0 , n0 , E0 ) ∈ Σ2 , therefore we know that the solution (m, n, E) belongs to L∞ (ℝ+ , Σ2 ). Hence ‖Ex ‖, ‖Exx̄ ‖, ‖nx ‖, ‖nt ‖ are uniformly bounded in time. It is then obvious that there exist some constants C0 = C0 (‖Ex ‖, γ, σ, ε, ‖Etx ‖, ‖nx ‖, μ), C1 = C1 (‖nt ‖, γ, ‖mx ‖, ν, ‖Exx̄ ‖)

such that A ≤ C0 ‖p‖2 + C1 ‖Gx ‖2 + λ2 γ‖Gxx̄ ‖2 +

ε ‖q‖2 . 4

(1.2.73)

32 | 1 Discrete attractor and approximate calculation From inequalities (1.2.72) and (1.2.73), we then derive γ d ε Q(t) + 2λ2 (γ − σ − )‖Gxx̄ ‖2 + ν(γ − σ)‖G‖2 + λ2 ( − σ)‖px ‖2 dt 2 2 ε ε + ( − σ − )‖q‖2 − μ(σ − ε)‖p‖2 2 4 ≤ C0 ‖p‖2 + C1 ‖Gx ‖2 .

(1.2.74)

Until now there has been no condition imposed on σ. Hence we choose σ < min( γ2 , 4ε ), thus we have 2λ2 (γ − σ − γ2 ) = 2λ2 ( γ2 − σ) > 0 and ε ε ε − σ − = −σ + > 0. 2 4 4 Obviously, σ < γ, σ
C2C2 s , for sufficiently large K, we have wJ (L(U0 )) ≤ 1 uniformly with respect to U0 . Then it can be proved that dH (AJ ) ≤ J + 1, and therefore dimension is finite. We hence claim the following theorem: Theorem 1.2.8. The global attractor AJ of the discrete dissipative Zakharov system (1.2.12)–(1.2.20) has finite Hausdorff and fractal dimensions.

1.3 Inertial manifolds under time-discretization In 1991, Demengel and Ghidaglia [20] studied the existence and convergence of inertial manifolds for partial differential evolution equations under time-discretization. We are given on an infinite-dimensional real separable Hilbert space H, with norm | ⋅ | and scalar product (⋅, ⋅), a linear closed unbounded positive self-adjoint operator A

1.3 Inertial manifolds under time-discretization | 37

in H with domain D(A) ⊂ H. We assume that v → |Av| is a norm on D(A) equivalent to the graph-norm and that A is an isomorphism from D(A) onto H, A−1 being a compact operator on H. Hence there exists a complete orthonormal family {wj }∞ j=1 in H made of eigenfunctions of A, Awj = λj wj ,

j = 1, 2, . . . ,

0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λj → ∞,

j → ∞,

(1.3.1)

where the λj ’s are the associated eigenvalues repeated according to their multiplicity. We denote by σ(A) = {Λ k }∞ k=1 , Λ 1 < Λ 2 < ⋅ ⋅ ⋅ the set of distinct eigenvalues, Λ k being of finite multiplicity mk . The spectral projectors RA and PA are defined as PA v = ∑ (v, wj )wj , j:λj =Λ

PA = ∑ Rλ . λ≤Λ

(1.3.2)

Given f : ℂ → ℂ, the operator f (A) is defined as f (A) = ∑ f (Λ)RA v, Λ

󵄨 󵄨2 D(f (A)) = {v, ∑ 󵄨󵄨󵄨f (Λ)󵄨󵄨󵄨 |RA v|2 }. Λ

Also we consider C, a linear bounded and skew-symmetric operator, D(As0 ) → H, s0 ∈ ℝ∗+ . Assume that C ∈ L (D(Aα+s0 ), D(Aα )), α ∈ ℝ, and that C and A commute: AC = CA,

(1.3.3)

CA = RA C : RA H → RA H. Consider the following differential equation: du + Au + Cu + F(u) = 0, dt u(0) = u0 ,

(1.3.4) (1.3.5)

where F is a Lipschitz function, D(Aα ) → D(Aα−γ ), α ∈ ℝ, γ ∈ [0, t], 󵄨󵄨 −γ 󵄨 󵄨󵄨A (F(v) − F(w))󵄨󵄨󵄨α ≤ LF |v − w|α ,

v, w ∈ P(Aα ).

(1.3.6)

Denote (v, w)α = (Aα v, Aα w), |v|α = |Aα v|. Assume LF satisfies 󵄨󵄨 −γ 󵄨 󵄨󵄨A F(v)󵄨󵄨󵄨α ≤ LF (1 + |v|α ),

∀v ∈ D(Aα ).

(1.3.7)

An inertial manifold for (1.3.4) is a finite-dimensional Lipschitz manifold M ⊂ D(Aα ) which enjoys the following properties. This set is positively invariant by (1.3.4) and attracts exponentially all the solutions, i. e., u0 ∈ M

⇒

u(t) ∈ M,

∀t ≥ 0,

(1.3.8)

38 | 1 Discrete attractor and approximate calculation ∀R > 0, ∃σ > 0, C ≥ 0, such that ∀t ≥ 0, u0 ∈ D(Aα ) 󵄨 󵄨 and |u0 |α ≤ R, dα (u(t), M) = inf 󵄨󵄨󵄨u(t) − m󵄨󵄨󵄨α ≤ Ce−αt . m∈M

(1.3.9)

Due to (1.3.6), we known that for u0 ∈ D(Aα ), the Cauchy problem (1.3.4)–(1.3.5) admits a unique solution with 1

u ∈ L ((0, ∞); D(Aα )) ∩ L2 ([0, T]; D(Aα+ 2 )),

∀T > 0.

(1.3.10)

1

If u0 ∈ D(Aα+ 2 ), the solution is more regular 1

u ∈ L ((0, ∞); D(Aα+ 2 )) ∩ L2 ([0, T]; D(Aα+1 )),

∀T > 0.

(1.3.11)

1

We denote by S(t) the time t map on D(Aα ) or D(Aα+ 2 ), S(t)u0 = u(t),

∀t ≥ 0.

(1.3.12) 1

These mappings are Lipschitz continuous and bounded on D(Aα ) and D(Aα+ 2 ), since (1.3.4) is autonomous, which implies that the family S(t), t ≥ 0 forms a semigroup S(0) = I,

S(t1 + t2 ) = S(t1 )S(t1 ).

(1.3.13)

Now we consider the construction of an inertial manifold. Assume C = 0. Given u0 ∈ D(Aα ), τ > 0, we introduce the sequence {un , n ∈ ℕ} ⊂ D(Aα ): (un+1 − un )/τ + Aun+1 + F(un ) = 0,

n ≥ 0.

(1.3.14)

Since the operator (I + τA) is an isomorphism, D(Aα+1 ) → D(Aα ), we set R(τ) = (I + τA)−1 .

(1.3.15)

Thus (1.3.14) can be rewritten as un+1 = R(τ)(un − τF(un )),

n ≥ 0.

(1.3.16)

Our aim in this section is to find a function ϕτ : PH → QD(Aα ) such that Mτ = Mϕτ is an inertial manifold for (1.3.14). The mapping Sτ defined on D(Aα ) by Sτ v = R(τ)(v − τF(v)),

∀v ∈ D(Aα )

(1.3.17)

is Lipschitz continuous. Definition 1.3.1. With the previous notations, an inertial manifold for (1.3.14) is a finite-dimensional Lipschitz manifold M ⊂ D(Aα ) which satisfies Sτ M ⊂ M.

(1.3.18)

Moreover, ∀R > 0, ∃σ > 0, C ≥ 0 such that n

dα ((Sτ ) u0 , M) = dα (un , M) ≤ Ce−σnτ ,

∀u ≥ 0, |uα | ≤ R.

(1.3.19)

1.3 Inertial manifolds under time-discretization | 39

In the discrete case, too, M will be sought as the graph of a Lipschitz function ϕ and therefore the infinite-dimensional recursion formula (1.3.14) will be replaced on M = Mϕ by the finite-dimensional iteration (pn+1 − pn )/τ + Aun+1 + PF(pn + ϕ(pn )) = 0,

(1.3.20)

which can also be written as pn+1 = Sϕτ pn ,

n ≥ 0,

(1.3.21)

where Sϕτ is the Lipschitz continuous map on PH defined by Sϕτ p = R(τ)(p − τPF(p + ϕ(p))).

(1.3.22)

Take ϕ ∈ Fl , namely assume ϕ is l-Lipschitz. From (1.3.22), Sϕτ is invertible for small τ. Indeed, we have the following result. Lemma 1.3.1. Assume that ϕ ∈ Fl and −γ

−1 0 < τ < L−1 F (1 + l) Λ N .

(1.3.23)

Then the mapping Sϕτ is a homeomorphism of PH. Moreover, Sϕτ and (Sϕτ )−1 are Lipschitz on PH. When (1.3.23) holds, we deduce from this lemma that for M = Mϕ , (1.3.18) is equivalent to n

(Sτ ) M = M,

(1.3.24)

∀n ∈ ℤ,

and n

n

n

(Sϕτ ) (m + ϕ(m)) = S(Sϕτ ) + ϕ((Sϕτ ) m),

∀n ∈ ℤ, ∀m ∈ Mϕ .

(1.3.25)

For this we take u0 = p0 + ϕ(p0 ) ∈ M and project (1.3.14) on PH and QH. Setting pn = (Sϕτ )n p0 , qn = ϕ(pn ), we have pn+1 = R(τ)(pn − τPF(pn + ϕ(pn ))), q

n+1

n

n

n

= R(τ)(q − τPF(p + ϕ(p ))),

n ∈ ℤ,

(1.3.26)

n ∈ ℤ.

(1.3.27)

From (1.3.27), we know when m ≤ n, n

qn = R(τ)n−m qm − τ ∑ R(τ)n+1−k QF(pk−1 + ϕ(pk−1 )). k=m+1

(1.3.28)

40 | 1 Discrete attractor and approximate calculation We assume that Λ − λ is sufficiently large and r is sufficiently small. The first term on the right-hand side of (1.3.28) converges towards 0 as m → −∞, leading to n

qn = −τ ∑ R(τ)n+1−k QF(pk−1 + ϕ(pk−1 )), k=−∞

n ∈ ℤ.

(1.3.29)

From q0 = ϕ(p0 ), we get ∞

ϕ(p0 ) = −τ ∑ R(τ)k QF((Sϕτ ) p0 + ϕ(Sϕτ ) p0 ). −k

−k

k=1

(1.3.30)

As τ satisfies (1.3.23), we define the mapping Fτ on Fl as ∞

(Fτ ϕ)(p0 ) = −τ ∑ R(τ)k QF((Sϕτ ) p0 + ϕ(Sϕτ ) p0 ). −k

−k

k=1

(1.3.31)

Thus (1.3.30) implies that ϕ = Fτ ϕ and shows that ϕ can be searched for as a fixed point for Fτ . Theorem 1.3.1. We assume that N ≥ 1 is such that γ

2γ−1

Λ N+1 ≥ 3LF Λ 1

/2,

γ

γ

Λ N+1 − Λ N ≥ 30LF (Λ N + Λ N+1 ). Then for every τ > 0, τΛ N+1 ≤ 1, the discrete infinite-dimensional dynamical system on D(Aα ), (un+1 − un )/τ + Aun+1 + F(un ) = 0,

n≥0

possesses an inertial manifold Mτ , which is the graph of a Lipschitz mapping, PN H → QN D(Aα ). Furthermore, there exist two constants C0 and σ > 0 such that dα (un , Mτ ) ≤ Ce−σn dα (u0 , Mτ ),

∀u0 ∈ D(Aα ), ∀u ≥ 0, |uα | ≤ R.

To prove Theorem 1.3.1, we need to establish the following results: Proposition 1.3.1. Suppose that N satisfies γ

γ

Λ N+1 − Λ N ≥ 6LF (Λ N (1 + l) + Λ N+1 (1 + l−1 )).

(1.3.32)

Then for every τ > 0 such that τΛ N+1 ≤ 1,

(1.3.33)

the mapping Fτ : Fl → Fl and ‖Fτ ϕ1 − Fτ ϕ2 ‖ ≤ κ‖ϕ1 − ϕ2 ‖α , where κ = l(2 + (1 + l)−1 ).

∀ϕi ∈ Fl ,

(1.3.34)

1.3 Inertial manifolds under time-discretization

| 41

Taking l = 41 , we derive the following corollary: Corollary 1.3.1. Suppose that N satisfies γ

γ

Λ N+1 − Λ N ≥ 10LF (Λ N + Λ N+1 ). Then for every τ > 0 satisfying (1.3.33), the mapping Fτ is a strict contraction, F1/4 → F1/4 . Before giving the proof of Proposition 1.3.1, we need an estimate on pn in (1.3.26). Lemma 1.3.2. Assume that (1.3.33) holds true and set γ

−1

η = (1 + τΛ N )(1 − τΛ N LF (1 + l)) , β = τγ LF (1 + l)(1 − τLF (1 + l)) . −1

Then the solutions of (1.3.26) satisfy 󵄩󵄩 m 󵄩󵄩 −m 󵄨 2 󵄨 −m 󵄩󵄩p 󵄩󵄩 ≤ η 󵄨󵄨󵄨p 󵄨󵄨󵄨α + β(η − 1)/(η − 1),

∀m ≤ 0.

(1.3.35)

Proof. We set here and in the sequel λ = Λ N , Λ = Λ N+1 , P = PN , Q = I − PN = QN . We rewrite (1.3.26) as R(τ)−1 pn+1 = pn − τPF(pn + ϕ(pn )). The scalar product of this identity in D(Aα ) with pn leads to 󵄨󵄨 n 󵄨󵄨2 󵄨 n 󵄨 󵄨 n+1 󵄨 n n n 󵄨󵄨p 󵄨󵄨α ≤ (τPF(p + ϕ(p )), p )α + (1 + τλ)󵄨󵄨󵄨p 󵄨󵄨󵄨α 󵄨󵄨󵄨p 󵄨󵄨󵄨α .

(1.3.36)

Then using (1.3.6) and (1.3.7), we derive 󵄨󵄨 −γ 󵄨 n󵄨 n n 󵄨 󵄨󵄨A F(p + ϕ(p ))󵄨󵄨󵄨α ≤ LF (1 + l)(1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α ), since ϕ ∈ Fl . Thus 󵄨󵄨 󵄨 n󵄨 󵄨 n󵄨 n n n 󵄨 γ 󵄨󵄨PF(p + ϕ(p ), p )α 󵄨󵄨󵄨 ≤ λ LF (1 + l)(1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α )󵄨󵄨󵄨p 󵄨󵄨󵄨α . With (1.3.36), we obtain 󵄨󵄨 n 󵄨󵄨 󵄨 n+1 󵄨 󵄨󵄨p 󵄨󵄨α ≤ η󵄨󵄨󵄨p 󵄨󵄨󵄨α + β, where β, η are given in Lemma 1.3.2. Then formula (1.3.35) follows readily by induction on m ≤ 0. In deriving the necessary form of an inertial manifold, we have assumed in (1.3.35) that R(τ)n−m qm → 0 as m → −∞. It is a consequence of (1.3.32) and (1.3.35). Indeed,

42 | 1 Discrete attractor and approximate calculation by the fact that ϕ ∈ Fl , there exists a constant C such that |qm |α ≡ |ϕ(pm )| ≤ Cη−m . On the other hand, |R(τ)n−m q|α ≤ (1 + τA)m−n |q|α , q ∈ QD(Aα ). Therefore m 󵄨󵄨 n−m 󵄨 m 󵄨󵄨R(τ) q󵄨󵄨󵄨α ≤ C(1 + τΛ) ((1 + 2Λ)/η) .

(1.3.37)

Notice that τΛ ≤ 1,

Λ − λ ≥ 6LF (λγ (1 + l) + Λγ (1 + l−1 ))

⇒

1 + τΛ > η,

(1.3.38)

which proves our claim. We are now in a position to prove Proposition 1.3.1. Proof of Proposition 1.3.1. Take ϕ ∈ Fl and assume that (1.3.32) holds true, i. e., Λ − λ ≥ 6LF ((1 + l)λγ + Λγ (1 + l−1 )).

(1.3.39)

Thus we derive Λ > τ(1 + l)LF λγ . Due to (1.3.32), (1.3.35) holds true. This allows us to define Fτ by formula (1.3.31), since (1.3.38) is satisfied now. In fact, it is convenient to rewrite this formula as ∞

(Fτ ϕ)(p0 ) = −τ ∑ R(τ)k QGϕ (p−k ),

(1.3.40)

k=1

where p−k = (Sϕτ ) p0 , −k

Gϕ (p) ≡ F(p + ϕ(p)).

(1.3.41)

In order to prove Proposition 1.3.1, we have to show that (i) Fτ maps Fτ into itself; (ii) Fτ satisfies (1.3.34). This leads us to consider the expressions Fτ ϕ1 − Fτ ϕ2 and (Fτ ϕ)p01 − (Fτ ϕ)p02 . We have to estimate (δp)k ≡ (Sϕτ 1 )p01 − (Sϕτ 2 )p02 ,

k ≤ 0.

(1.3.42)

Lemma 1.3.3. With the same hypotheses and notations as in Lemma 1.3.2, (δp)n in (1.3.42) is estimated as 󵄨󵄨 󵄨 󵄨 0󵄨 −n 󵄨 0 0󵄨 −n −1 󵄨󵄨(δp)n 󵄨󵄨󵄨α ≤ η 󵄨󵄨󵄨p1 − p1 󵄨󵄨󵄨α − nη (1 + l) ‖ϕ1 − ϕ1 ‖α (1 + 󵄨󵄨󵄨p1 󵄨󵄨󵄨α ),

∀n ≤ 0.

Take ϕ ∈ Fl , p01 , p02 ∈ H. According to (1.3.40), 0

0

∞

k

−k

−k

Fτ ϕ(p1 ) − Fτ ϕ(p2 ) = −τ ∑ R(τ) Q(Gϕ (p1 ) − Gϕ (p2 )). k=1

(1.3.43)

1.3 Inertial manifolds under time-discretization

| 43

Using (1.3.6) and (1.3.43) we have ∞

󵄨󵄨 󵄨 0 󵄨 0 0 󵄨 0󵄨 k󵄨 γ 󵄨󵄨Fτ ϕ(p1 ) − Fτ ϕ(p2 )󵄨󵄨󵄨 ≤ τLF (1 + l)󵄨󵄨󵄨p1 − p2 󵄨󵄨󵄨 ∑ η 󵄨󵄨󵄨A R(τ)Q󵄨󵄨󵄨L D(Aα ) . k=1

(1.3.44)

The norm of the operator Aγ R(τ)Q in L D(Aα ) is bounded as follows: 󵄨 󵄨󵄨 γ γ −γ −γ −k 󵄨󵄨A R(τ)Q󵄨󵄨󵄨L D(Aα ) ≤ (Λ + τ k )(1 + τΛ) ,

(1.3.45)

which follows from the inequalities ξ γ (1 + τξ )−k ≤ (Λγ + τ−γ k −γ )(1 + τΛ)−k , 1 ξ ≥ Λ, k ≥ 1, γ ∈ [0, ], τ > 0. 2 Now (1.3.44) and (1.3.45) imply 󵄨󵄨 󵄨 0 0 0 󵄨 0󵄨 󵄨󵄨Fτ ϕ(p1 ) − Fτ ϕ(p2 )󵄨󵄨󵄨α ≤ ω󵄨󵄨󵄨p1 − p2 󵄨󵄨󵄨α ,

(1.3.46)

where ∞

ω = τLF (1 + l) ∑ (Λγ + τ−γ k −γ )(1 + τΛ)−k . k=1

(1.3.47)

Now our aim is to prove that inequality (1.3.39) and τΛ ≤ 1

⇒

ω ≤ l.

(1.3.48)

Set r = η(1 + τΛ)−1 and assume that r < 1. It is equivalent to the inequality which follows from Lemma 1.3.2: (1 + τλ) < (1 + τΛ)(1 − τLF (1 + l)λr ), the latter holds true due to (1.3.33) and (1.3.39). Returning to (1.3.47), we have w ≤ τLF (1 + l){

∞ 2Λγ + τ−γ ∑ k −γ r k }. 1−r k=1

The infinite sum above is bounded since ∞

∞

k=1

0

∑ k −γ r k ≤ ∫ x−γ r x dx = |ln r|γ−1 , ∞

∫ x−γ e−x dx = 2|ln r|γ−1 . 0

(1.3.49)

44 | 1 Discrete attractor and approximate calculation This leads to w ≤ τLF (1 + l){

Λγ + 2τ−γ |ln r|γ−1 }. 1−r

Due to |ln r| ≤ 1 − r, we have w ≤ τLF (1 + l){

Λγ + 2τ−γ (1 − r)γ−1 }. 1−r

From 1 − r ≤ 2Λ, we deduce w ≤ 3τLF (1 + l)Λγ (1 − r). Employing the value of r, we finally obtain w≤

3τLF (1 + l)Λγ (1 + τΛ) Λ − λ − λγ LF (1 + l)(1 + τΛ)

and (1.3.48). From (1.3.46), Lipα (Fτ (ϕ)) ≤ l. In order to prove that Fτ ϕ ∈ Fτ , we need to verify that |(Fτ ϕ)(p0 )|α ≤ l(1 + |p0 |α ), ∀p0 ∈ PH. By (1.3.7) and (1.3.40), we have ∞

󵄨 −k 󵄨 󵄨󵄨 󵄨 γ 0 󵄨 γ 󵄨 󵄨󵄨(Fτ ϕ)(p )󵄨󵄨󵄨α ≤ τ(1 + l)LF ∑ 󵄨󵄨󵄨A R(τ) Q󵄨󵄨󵄨L (D(Aα )) (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α ). k=1

(1.3.50)

From (1.3.50), we have 󵄨 󵄨 󵄨 󵄨 1 + 󵄨󵄨󵄨p−k 󵄨󵄨󵄨α ≤ ηk (1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α ),

∀k ≥ 0.

(1.3.51)

By the estimates in (1.3.50) and (1.3.47), we derive |(Fτ ϕ)(p0 )|α ≤ w(1 + |p0 |α ). Thus we obtain Fτ ϕ ∈ Fl from (1.3.51). Now we are going to prove inequality (1.3.51) in Proposition 1.3.1. Take P10 = P20 = P 0 ∈ PH, then we have 0

0

∞

k

−k

−k

Fτ (ϕ1 )(p1 ) − Fτ (ϕ2 )(p ) = −τ ∑ R(τ) Q(Gϕ1 (p1 ) − Gϕ2 (p2 )) k=1

(1.3.52)

from (1.3.40). Denoting Gϕ1 (p1 ) − Gϕ2 (p2 ) = Gϕ1 (p1 ) − Gϕ1 (p2 ) + Gϕ1 (p1 ) − Gϕ2 (p2 ), we deduce the following estimate: 󵄨󵄨 −γ 󵄨 󵄨󵄨A (Gϕ1 (p1 ) − Gϕ2 (p2 ))󵄨󵄨󵄨α ≤ LF {(1 + l)|p1 − p1 |α + ‖ϕ1 − ϕ2 ‖α (1 + |p2 |α )}. Hence by (1.3.43) and (1.3.45) with p01 = p02 = p0 , we have 󵄨󵄨 −γ 󵄨 󵄨 0󵄨 k 󵄨󵄨A (Gϕ1 (p1 ) − Gϕ2 (p2 ))󵄨󵄨󵄨α ≤ LF (1 + kβ)η ‖ϕ1 − ϕ2 ‖α (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α ).

(1.3.53)

1.3 Inertial manifolds under time-discretization | 45

It follows from (1.3.52), (1.3.45) and the above inequality that |Fτ ϕ1 − Fτ ϕ2 |α ≤ w|ϕ1 − ϕ2 |α , ∞

w = τLF ∑ (Λγ + τ−γ k −γ )(1 + τΛ)−k ηk (1 + kβ).

(1.3.54)

k=1

Comparing with (1.3.47), we find ∞

w = w(1 + l)−1 + τLF β ∑ k(Λγ + τ−1 k −γ )r k ,

(1.3.55)

k=1

k 2 −1 where r = η(1 + τΛ)−1 . Using (1.3.48) and ∑∞ k=1 kr = r(1 − r ) , we obtain

w ≤ l(1 + l)−1 + τLF β(rΛγ (1 − r 2 )

−1

+ τ−α I),

where ∞

k

I = ∑ k −(1−γ)r ≤ |ln r|γ−2 ≤ (1 − r)γ−2 . k=1

Due to the fact that rΛ ≤ 1, w ≤ l(1 + l)−1 + τLF βη−1 Λγ (r(1 − r 2 )

−1

+ (1 − r 2 ) ). −2

And it follows from βη−1 ≤ 1, τΛ ≤ 1 and 1 r ≤ , 1+r 2

τLF (1 + l)Λγ ≤

l 1−r

that w ≤ l(1 + l)−1 + 2l, which completes the proof of Proposition 1.3.1. Concerning Lemma 1.3.3, we deduce from (1.3.26) and (1.3.31) that R(τ)−1 (δp)n+1 = (δp)n + P(F(pn1 + ϕ1 (pn1 )) − F(pn2 + ϕ1 (pn2 )))

(1.3.56)

and 󵄨󵄨 󵄨 󵄨 γ 󵄨 󵄨 n󵄨 n n n n 󵄨󵄨P(F(p1 + ϕ1 (p1 )) − F(p2 + ϕ1 (p2 )))󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨λ LF (1 + l)(δp)n 󵄨󵄨󵄨α + ‖ϕ1 − ϕ2 ‖(1 + 󵄨󵄨󵄨p1 󵄨󵄨󵄨α ). Hence calculating the inner product of (1.3.56) with (δp)n leads to 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 n󵄨 γ γ 󵄨󵄨(δp)n 󵄨󵄨󵄨α (1 − τλ LF (1 + l)) ≤ (1 + τλ)󵄨󵄨󵄨(δp)n+1 󵄨󵄨󵄨α + τλ LF ‖ϕ1 − ϕ2 ‖α (1 + 󵄨󵄨󵄨p1 󵄨󵄨󵄨α ).

46 | 1 Discrete attractor and approximate calculation From Lemma 1.3.2, we have 󵄨 󵄨 󵄨 󵄨 1 + 󵄨󵄨󵄨pn1 󵄨󵄨󵄨α ≤ (1 + 󵄨󵄨󵄨p01 󵄨󵄨󵄨α ) + (η−n − 1)(β(η − 1)−1 − 1), and since β(η − 1)−1 ≤ 1, we derive 󵄨 0󵄨 󵄨 󵄨 󵄨 󵄨󵄨 −n −1 󵄨󵄨(δp)n 󵄨󵄨󵄨α ≤ η󵄨󵄨󵄨(δp)n+1 󵄨󵄨󵄨α + βη (1 + l) ‖ϕ1 − ϕ2 ‖α (1 + 󵄨󵄨󵄨p1 󵄨󵄨󵄨α ). Finally, (1.3.43) follows by induction on n. Now we consider the discrete cone property. Assume u10 , u10 ∈ D(Aα ) and set wn = n u1 − un2 . Our aim is to estimate wn , n ≥ 0, in terms of wn , n. From (1.3.14), R(τ)−1 wn+1 = wn − τ(F(un1 ) − F(un1 )).

(1.3.57)

Calculating the inner product of (1.3.57) with wn+1 , we get 󵄨󵄨 n+1 󵄨󵄨2 󵄨 1 n+1 󵄨2 󵄨 n+1 󵄨 󵄨 n 󵄨 󵄨 α n+1 󵄨 󵄨 n 󵄨 󵄨󵄨w 󵄨󵄨α + τ󵄨󵄨󵄨A 2 w 󵄨󵄨󵄨α ≤ 󵄨󵄨󵄨w 󵄨󵄨󵄨α 󵄨󵄨󵄨w 󵄨󵄨󵄨α + τLF 󵄨󵄨󵄨A w 󵄨󵄨󵄨α 󵄨󵄨󵄨w 󵄨󵄨󵄨α .

(1.3.58)

It follows from γ− 1 󵄨 1 n+1 󵄨 󵄨󵄨 γ n+1 󵄨󵄨 󵄨󵄨A w 󵄨󵄨α ≤ Λ 1 2 󵄨󵄨󵄨A 2 w 󵄨󵄨󵄨α

that γ− 1 󵄨 n 󵄨 󵄨 1 n+1 󵄨 󵄨 1 n+1 󵄨2 󵄨󵄨 n+1 󵄨󵄨2 󵄨󵄨 n+1 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨w 󵄨󵄨α ≤ 󵄨󵄨w 󵄨󵄨α 󵄨󵄨w 󵄨󵄨α + 2(LF Λ 1 2 󵄨󵄨󵄨w 󵄨󵄨󵄨α 󵄨󵄨󵄨A 2 w 󵄨󵄨󵄨α − 󵄨󵄨󵄨A 2 w 󵄨󵄨󵄨α ), 󵄨󵄨 n+1 󵄨󵄨2 󵄨󵄨 n 󵄨󵄨2 2 2γ−1 󵄨󵄨w 󵄨󵄨 ≤ (1 + τLF Λ 1 /2)󵄨󵄨w 󵄨󵄨 , ∀n ≥ 0, 2γ−1

where we have used the inequality x2 − LF Λ 1

2γ−1 2

xy ≥ −L2F Λ 1

n−2 󵄨 0 󵄨󵄨 n n󵄨 2 2γ−1 0󵄨 󵄨󵄨u1 − u2 󵄨󵄨󵄨α (1 + τLF Λ 1 /2) 󵄨󵄨󵄨u1 − u2 󵄨󵄨󵄨α ,

y /4. Hence we have

∀n ≥ 0.

(1.3.59)

Proposition 1.3.2. Given k > 0, assume that N satisfies γ

γ

(1.3.60)

∀n ≥ 0.

(1.3.61)

Λ N+1 − Λ N ≥ 4LF (1 + k −1 )Λ N + (1 + k)Λ N . Then for every τ satisfying (1.3.33), the cone εk given by εk = {v ∈ D(Aα ) : |Pv|α ≥ k|Qv|α }, where P = PN , Q = I − PN , is invariant relative to (1.3.14), i. e., if u01 − u02 ∈ εk ,

then un1 − un2 ∈ εk ,

Moreover, we have the following alternative: either ∃k ≥ 0, or

uk1 − uk2 ∈ εk

󵄨󵄨 n n󵄨 n󵄨 0 0󵄨 󵄨󵄨u1 − u2 󵄨󵄨󵄨α ≤ (1 + 1/k)ρ 󵄨󵄨󵄨u1 − u2 󵄨󵄨󵄨α , ∀n ≥ 0, γ ρ = (1 + τ(1 + k)LF Λ N+1 )(1 + τΛ N+1 )−1 < 1.

(1.3.62)

(1.3.63)

1.3 Inertial manifolds under time-discretization

| 47

Proof. We begin by checking (1.3.61). For given u01 , u02 ∈ D(Aα ), u01 −u02 ∈ εk , i. e., |p0 |α ≥ k|q0 |, where p0 = P(u01 − u02 ), q0 = Q(u01 − u02 ). Our aim is to show that 󵄨 1󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨p 󵄨󵄨α ≥ k 󵄨󵄨󵄨q 󵄨󵄨󵄨α ,

(1.3.64)

where p1 = P(u11 − u12 ), q1 = Q(u11 − u12 ). Using (1.3.14), we have R(τ)−1 p1 = p0 − τP(F(u01 ) − F(u02 )). The inner product of this relation with p0 in D(Aα ) gives 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 (1 + τλ)󵄨󵄨󵄨p1 󵄨󵄨󵄨α 󵄨󵄨󵄨p0 󵄨󵄨󵄨α ≤ 󵄨󵄨󵄨p0 󵄨󵄨󵄨α − τ󵄨󵄨󵄨A−γ P(F(u01 ) − F(u02 ))󵄨󵄨󵄨α 󵄨󵄨󵄨Aγ p0 󵄨󵄨󵄨α , and using (1.3.6) we find 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 (1 + τλ)󵄨󵄨󵄨p1 󵄨󵄨󵄨α 󵄨󵄨󵄨p0 󵄨󵄨󵄨α ≥ 󵄨󵄨󵄨p0 󵄨󵄨󵄨α − τLF λγ 󵄨󵄨󵄨u01 − u02 󵄨󵄨󵄨α 󵄨󵄨󵄨p0 󵄨󵄨󵄨α . Now since |p0 |α ≥ k|q0 |α , it follows that 󵄨󵄨 0 󵄨󵄨 󵄨󵄨p 󵄨󵄨α ≤

1 + τλ

1 − τLF (1 + k1 )λα

󵄨󵄨 1 󵄨󵄨 󵄨 α 1󵄨 󵄨󵄨p 󵄨󵄨α ≡ ηk 󵄨󵄨󵄨A p 󵄨󵄨󵄨.

(1.3.65)

We note that (1.3.33) and (1.3.60) imply that ηk ≤ 4 and, by (1.3.33), R(τ)q1 = q0 − τQ(F(u01 ) − F(u01 )). Calculating inner product of the above equality with q1 in D(Aα ), we derive 1 󵄨󵄨 1 󵄨󵄨2 󵄨 1 1 󵄨2 󵄨 0 󵄨 󵄨 1 󵄨 γ− 1 󵄨 1 1 󵄨 󵄨 1 󵄨 󵄨󵄨q 󵄨󵄨α + τ󵄨󵄨󵄨A 2 q 󵄨󵄨󵄨α ≤ 󵄨󵄨󵄨q 󵄨󵄨󵄨α 󵄨󵄨󵄨q 󵄨󵄨󵄨α + τLF (1 + )ηk Λ 2 󵄨󵄨󵄨A 2 q 󵄨󵄨󵄨α 󵄨󵄨󵄨p 󵄨󵄨󵄨α , k

(1.3.66)

with which one deduces inequality (1.3.64). Indeed, if not, we have |p1 |α ≥ k|q1 |α , then 1 1 󵄨 1󵄨 󵄨󵄨 21 1 󵄨󵄨 −1 󵄨󵄨A q 󵄨󵄨α ≥ Λ 2 󵄨󵄨󵄨q 󵄨󵄨󵄨α ≥ Λ 2 k , 󵄨󵄨 1 󵄨󵄨 −1 γ− 21 󵄨󵄨 1 󵄨󵄨 󵄨󵄨p 󵄨󵄨α ≥ LF (1 + k )ηk Λ 󵄨󵄨p 󵄨󵄨α /2. 1

1

Then we can replace |A 2 q1 |α in (1.3.66) by Λ 2 k −1 , 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (1 + τΛ)󵄨󵄨󵄨q1 󵄨󵄨󵄨α ≤ 󵄨󵄨󵄨q0 󵄨󵄨󵄨α 󵄨󵄨󵄨q1 󵄨󵄨󵄨α + τLF (1 + k −1 )ηk Λγ 󵄨󵄨󵄨p1 󵄨󵄨󵄨α 󵄨󵄨󵄨q1 󵄨󵄨󵄨α . Since k|q0 |α ≤ |p0 |α ≤ |p1 |α , we derive that 󵄨 󵄨 󵄨 󵄨 (1 + τΛ)󵄨󵄨󵄨p1 󵄨󵄨󵄨α ≤ (ηk k −1 + τLF (1 + k −1 )ηk Λα )󵄨󵄨󵄨p1 󵄨󵄨󵄨α . Finally, (1.3.33) and (1.3.60) imply (1.3.64), and the proof of (1.3.61) is achieved.

48 | 1 Discrete attractor and approximate calculation Concerning (1.3.63), we assume that un1 − un2 ∉ εk , ∀n ≥ 0, i. e., 󵄨 n󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨p 󵄨󵄨α ≤ k 󵄨󵄨󵄨q 󵄨󵄨󵄨α ,

pn = P(un1 − un2 ), qn = Q(un1 − un2 ),

∀n ≥ 0.

(1.3.67)

We are going to prove that 󵄨 n󵄨 󵄨󵄨 n+1 󵄨󵄨 󵄨󵄨q 󵄨󵄨α ≤ ρ󵄨󵄨󵄨q 󵄨󵄨󵄨α ,

∀n ≥ 0.

(1.3.68)

The inner product of (1.3.14) with −qn+1 leads to 󵄨 n 󵄨󵄨 n+1 󵄨󵄨2 󵄨󵄨 21 n+1 󵄨󵄨2 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n+1 󵄨󵄨 n 󵄨 󵄨 1 n+1 󵄨 󵄨󵄨q 󵄨󵄨α + 󵄨󵄨A q 󵄨󵄨α ≤ 󵄨󵄨q 󵄨󵄨α 󵄨󵄨q 󵄨󵄨α + τLF 󵄨󵄨󵄨u1 − u2 󵄨󵄨󵄨α 󵄨󵄨󵄨A 2 q 󵄨󵄨󵄨α , 󵄨󵄨 n+1 󵄨󵄨2 󵄨󵄨 21 n+1 󵄨󵄨2 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n+1 󵄨󵄨 γ− 1 󵄨 n 󵄨 󵄨 n+1 󵄨 󵄨󵄨q 󵄨󵄨α + 󵄨󵄨A q 󵄨󵄨α ≤ 󵄨󵄨q 󵄨󵄨α 󵄨󵄨q 󵄨󵄨α + τLF (1 + k)Λ 2 󵄨󵄨󵄨q 󵄨󵄨󵄨α 󵄨󵄨󵄨q 󵄨󵄨󵄨α .

(1.3.69)

If (1.3.66) does not hold, then 1 1 󵄨󵄨 21 n+1 󵄨󵄨2 󵄨 n+1 󵄨 󵄨 n󵄨 γ− 1 󵄨 n 󵄨 󵄨󵄨A q 󵄨󵄨α ≥ Λ 2 󵄨󵄨󵄨q 󵄨󵄨󵄨α ≥ ρΛ 2 󵄨󵄨󵄨q 󵄨󵄨󵄨α ≥ LF (1 + k)Λ 2 󵄨󵄨󵄨q 󵄨󵄨󵄨α /2. 1

1

By replacing |A 2 qn+1 |2α with Λγ− 2 |qn |α in (1.3.69), we obtain 󵄨 󵄨 󵄨 󵄨 (1 + τΛ)󵄨󵄨󵄨qn+1 󵄨󵄨󵄨α ≤ 󵄨󵄨󵄨qn 󵄨󵄨󵄨α (1 + τLF (1 + k)Λα ), which is exactly (1.3.68). Proof of Theorem 1.3.1. It follows from the assumptions on N in the theorem that γ

2γ−1

Λ N+1 ≥ 3LF Λ 1 Λ N+1 − Λ N ≥

/2,

γ 30LF (Λ N

+

(1.3.70)

γ Λ N+1 ).

(1.3.71)

Let us take l = 41 and k = 4, then (1.3.71) implies (1.3.63) and (1.3.60), and the conclusions of Corollary 1.3.1 and Proposition 1.3.2 hold true. Hence we have found a fixed point ϕτ ∈ ε 1 of ετ , and (1.3.62), (1.3.63) are satisfied for ε4 . In order to achieve the 4 proof of Theorem 1.3.1, we have to show the exponential attractivity property. τ n 0 We take u0 ∈ D(Aα ), v0 ∈ Mτ such that dα (u0 , Mτ ) = |u0 − v0 | and set un = (Sϕτ ) u ,

τ n 0 vn = (Sϕτ ) v = Pvn + ϕτ (Pvn ). From (1.3.69), (1.3.71) and the fact τΛ ≤ 1, we derive 2γ−1

/2)(1 + (1 + k)LF Λγ ) ≤ (1 + τΛ),

2γ−1

/2)−1 ln 2], where [x] denotes the integer part of x. Then

(1 + τLF Λ 1 Let n0 = 1 + [2 ln(1 + τLF Λ 1 due to (1.3.69)

2γ−1

(1 + τLF Λ 1

/2)

n0

1

2γ−1

≤ exp(lg 2 2 (1 + τLF Λ 1

k = 4.

/2)) ≤ 4√2/3.

(1.3.72)

(1.3.73)

On the other hand (k = 4), thanks to (1.3.73) and the definition of n0 , 2n0

(1 + (1 + k)τLF Λγ )

2γ−1

(1 + τΛ)−2n0 ≤ (1 + τL2F Λ 1

−2n0

/2)

1 ≤ . 2

(1.3.74)

1.3 Inertial manifolds under time-discretization

| 49

Next we take n ∈ ℕ satisfying n0 ≤ n ≤ 2n0 . Two cases can occur: (i) If un − vn ∈ ε4 , then 󵄨 󵄨 󵄨 󵄨 dα (u0 , Mτ ) ≤ 󵄨󵄨󵄨Q(un − vn )󵄨󵄨󵄨α + 󵄨󵄨󵄨ϕτ (Pun ) − ϕτ (Pvn )󵄨󵄨󵄨α n 2γ−1 󵄨 󵄨 ≤ 󵄨󵄨󵄨un − vn 󵄨󵄨󵄨α (1 + τLF Λ 1 /2) 0 󵄨 󵄨 ≤ 2√2󵄨󵄨󵄨un − vn 󵄨󵄨󵄨α /3. Hence n0 ≤ n ≤ 2n0 , un − vn ∈ ε4 dα (u0 , Mτ ) ≤ (2√2/3)dα (u0 , M2 ).

⇒

(1.3.75)

(ii) If un − vn ∉ ε4 , then, by Proposition 1.3.2, we have un − vn ∉ ε, k = 0, 1, . . . , n0 . It follows from (1.2.63) and uk1 = uk , uk2 = vk that for k = 4 󵄨 󵄨 dα (u0 , Mτ ) ≤ 󵄨󵄨󵄨un − vn 󵄨󵄨󵄨α 5 2n 󵄨 󵄨 ≤ (1 + 5τLF Λγ ) 0 (1 + τΛ)−2n0 󵄨󵄨󵄨u0 − v0 󵄨󵄨󵄨α 4 5󵄨 󵄨 ≤ 󵄨󵄨󵄨u0 − v0 󵄨󵄨󵄨α . 8 We summarize this and (1.2.75) in dα (u0 , Mτ ) ≤ (2√2/3)dα (u0 , Mτ ),

n0 ≤ n ≤ 2n0 .

(1.3.76)

Now if n ≥ 2n0 , we introduce k ∈ ℕ, r ∈ [n0 + 1, 2n0 ] such that n = kn0 + r. According to (1.3.76), we have dα (u0 , Mτ ) ≤ (2√2/3)dα (u0 , M2 ),

n ≥ n0 .

This shows the exponential attractivity of the inertial manifold with σ = [ln( 2√3 2 )]n0 . Next we consider the general case, i. e., the case where C ≠ 0. The discretized equation is (un+1 − un )/τ + (A + C)un+1 + F(un ) = 0,

n ≥ 0.

(1.3.77)

The fractional step method is as follows: 1

1 un+ 2 − un + Aun+ 2 + F(un ) = 0, τ

n+1− 21

u

τ

− un

+C

n+ 21

un+1 + u 2

= 0,

(1.3.78) (1.3.79)

50 | 1 Discrete attractor and approximate calculation As for the case where C = 0, we rewrite (1.3.78) as 1

un+ 2 = R(τ)(un − τF(un )),

(1.3.80)

where R(τ) = (I + τA)−1 . Concerning (1.3.79), we set U(τ)v = ∑ UA (τ)RA v, Λ∈σ(A)

(1.3.81)

where Un (τ) is the unitary operator on RΛ H; UA (τ) = (I + τCΛ /2)−1 = (I − τCΛ /2). The operators U(τ) are unitary on D(As ), s ∈ ℝ. Then (1.3.79) reads 1

un+1 = U(τ)un+ 2 .

(1.3.82)

Hence (1.3.78)–(1.3.79) can be written as un+1 = R(τ)(un − τF(un )),

n ≥ 0,

(1.3.83)

where R(τ) ≡ U(τ)R(τ) = R(τ)U(τ),

∀τ > 0.

In other words, by comparison with (1.3.16), we have replaced R(τ) by R(τ) given above. Thus we have Theorem 1.3.2. We assume that N ≤ 1 is such that γ

2γ−1

Λ N+1 ≥ 3LF Λ 1

/2,

γ

γ

Λ N+1 − Λ N ≥ 30LF (Λ N + Λ N+1 ).

(1.3.84)

Then for every τ > 0 such that τΛ N+1 ≤ 1, the discrete infinite-dimensional dynamical system 1

1 un+ 2 − un + Aun+ 2 + F(un ) = 0, τ 1

1

un+1 − un+ 2 un+1 + un+ 2 +C =0 τ 2 possesses an inertial manifold Mτ , which is the graph of a Lipschitz function, PN H → QN D(Aα ). Moreover, there exist two constants C0 and σ > 0 such that for every u0 ∈ D(Aα ), dα (u0 , Mτ ) ≤ C0 dα (u0 , M2 ),

n ≥ 0.

(1.3.85)

1.3 Inertial manifolds under time-discretization | 51

Assume that N is such that γ

2γ−1

Λ N+1 ≥ 3LF Λ 1

/2,

γ

(1.3.86)

γ

Λ N+1 − Λ N ≥ 30LF (Λ N + Λ N+1 ).

Then we know that the continuous equation (1.3.14) possesses an M-dimensional (M = dim PN H) inertial manifold M, M = Md = {p + ϕ(p), p ∈ PH}. For sufficiently small τ, τΛ N+1 ≤ 1, Theorem 1.3.2 provides an M-dimensional inertial manifold Mτ for the discrete equations (1.3.78)–(1.3.79). A natural question is whether Mτ converges to M as τ → 0. Theorem 1.3.3. We assume that N satisfies (1.3.84) and τΛ N+1 ≤ 1. There exists a constant K which is independent of τ > 0 such that ε

‖ϕ − ϕτ ‖α ≤ Kτξ (1 + |ln τΛ N+1 |) , where ξ = 1,

ε = 1,

ξ = 1 − γ,

s0 ≤ 1,

ε = 0,

γ = 0;

s0 ≤ 1,

−1

ξ = (1 − γ)(2s0 − 1) ,

γ > 0;

ε = 1,

s0 > 1.

We will prove it later. Firstly, we consider the error estimate for the finite-dimensional part. We are given ϕ ∈ Fl , and our aim is to relate the following two dynamical systems: dp + Ap + Cp + PF(p + ϕ(p)) = 0, dt pn+1 = R(τ)(pn − τPF(p + ϕ(p))),

(1.3.87) (1.3.88)

where P = PN , τ satisfies (1.3.23), R(τ) = R(τ)U(τ)(I + τA)−1 (I + τ C2 )−1 (I − τ C2 ), and N is a fixed integer. Assume that ϕ is the graph of an inertial manifold and set en = p(nτ) − pn ,

n ∈ ℤ.

(1.3.89)

Proposition 1.3.3. With the previous hypotheses and notations, let p0 be given in H and let p(t), t ∈ ℝ (resp. pn , n ∈ ℤ) denote the solution to (1.3.87)–(1.3.88) satisfying p(0) = p0 (resp. p0 = p0 ). Then for every negative integer n, we have |en |α ≤

τ2 K(λ) 󵄨 󵄨 (η−n − e−nτλ )(ηe−τλ − 1)(1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α ), 1 − τLF (1 + l)λα

where K(λ) is independent of τ, n, and η = (1 + τΛ)(1 − τLF Λγ (1 + l)) , −1

λ = λ + τLF (1 + l),

λ = ΛN.

(1.3.90)

52 | 1 Discrete attractor and approximate calculation Proof. This is a very classical matter, and we only briefly sketch the main steps of the proof. We introduce the consistency error εn by writing R(τ)−1 p((n + 1)τ) = (pn − τPF(p(nτ) + ϕ(p(nτ)))) + εn .

(1.3.91)

Comparing with (1.3.88), we have R(τ)−1 en+1 = (en − τP(G(pn(τ)) − G(pn ))) + εn ,

(1.3.92)

where G(p) = GF(p + ϕ(p)). The inner product of (1.3.92) with en in D(Aα ) leads then to |en | ≤ η|en+1 |α + |εn |α /(1 − τ(1 − τ(1 + l)LF λα )).

(1.3.93)

This shows that (e0 = 0) −1

|en | ≤ ∑ ηk−n |εk |α (1 − τ(1 + l)LF ) , −1

k=n

∀n ≤ 0.

(1.3.94)

It remains to estimate εk . We introduce the function ϕ(s) by ϕ(s) = R(s)−1 p(σ + s) − p(σ) + sPF(p(σ) + ϕ(p(σ))), where σ is fixed and s ≥ 0. We have |εn |α = ϕ(τ),

σ = nτ,

(1.3.95)

and since ϕ(0) = 0, there exists θ ∈ [0, 1] such that 󵄨 󵄨 |εn |α ≤ τ󵄨󵄨󵄨ϕ󸀠 (θτ)󵄨󵄨󵄨α ,

σ = nτ.

(1.3.96)

Computing ϕ󸀠 (s) leads to ϕ󸀠 (s) = (

dp dp d (σ), R(s)−1 )p(σ + s) − (A + C)p(σ) + R(s)−1 (σ + s) − dt ds ds

which we rewrite as ϕ󸀠 (s) = Ψ1 (s) + Ψ2 (s),

(1.3.97)

where Ψ2 (s) = R(s)−1 (( dp )(σ + s) − ( dp )(σ)). The function Ψ1 is differentiable and, since ds ds Ψ1 (0) = 0, there exists θ1 ∈ [0, 1] such that Ψ1 (s) = sΨ1󸀠 (θ1 s). Concerning Ψ2 , we write dp dp (s) − (σ + s) = (A + C)(p(σ + s) − p(σ)) + P(G(p(σ + s) − G(p(σ))), ds ds

(1.3.98)

1.3 Inertial manifolds under time-discretization | 53

which implies that 󵄨󵄨 dp 󵄨󵄨 dp 󵄨 󵄨 󵄨󵄨 󵄨 (σ + s)󵄨󵄨󵄨 ≤ (λ + λs0 + (1 + l)LF λα )󵄨󵄨󵄨p(σ + s) − p(σ)󵄨󵄨󵄨α . 󵄨󵄨 (s) − 󵄨󵄨α ds 󵄨󵄨 ds Since |R(s)−1 |L (D(Aα )) ≤ (1 + sλ), there exists θ2 ∈ [0, 1] such that 󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 s α 󵄨󵄨 dp 󵄨󵄨Ψ2 (s)󵄨󵄨󵄨α ≤ sΛ + sλ(λ + aλ 0 + (1 + l)LF λ )󵄨󵄨󵄨 (σ + θ2 s)󵄨󵄨󵄨 , 󵄨󵄨α 󵄨󵄨 dt

(1.3.99)

a = |C|L (D(Aα0 ),H) .

(1.3.100)

where

In order to derive from (1.3.98) and (1.3.100) an estimate on εn , we need an estimate on |p(σ)|α , σ ≤ 0. Returning to (1.3.87) and thanks to (1.3.6)–(1.3.7), we find 󵄨󵄨 󵄨 󵄨 0 󵄨 −λσ 󵄨󵄨p(σ)󵄨󵄨󵄨α ≤ (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α )e ,

∀σ ≤ 0.

(1.3.101)

where λ = λ + λγ LF (1 + l). Then using (1.3.87), (1.3.101) and (1.3.99), we derive 󵄨󵄨 󵄨 󵄨 0 󵄨 −λ(s+nt) , 󵄨󵄨Ψ2 (s)󵄨󵄨󵄨α ≤ s(1 + sλ)K1 (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α )e

∀s ≤ 0,

(1.3.102)

where K1 and K2 below are constants which only depend on λ, LF , l, s0 and a. The estimate of (1.3.98) is similar. One obtains 󵄨󵄨 󵄨 󵄨 0 󵄨 −λs s 󵄨󵄨Ψ1 (s)󵄨󵄨󵄨α ≤ s(2λ + (1 + sλ)aλ ) )K2 (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α )e ,

∀s ≥ 0.

(1.3.103)

We sum (1.3.102) and (1.3.103), take σ = nτ ≤ 0, s = τ, and then have 󵄨 󵄨 |εn | ≤ τ2 K(λ)(1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α )e−λnτ ,

∀n ≤ 0.

Finally, (1.3.90) follows from (1.3.94) and (1.3.103). Now we consider the convergence of the mappings Fτ to F as τ → 0. Proposition 1.3.4. The hypotheses are the same as in Proposition 1.3.3. We assume moreover that N is such that γ

Λ N+1 − Λ N ≥ τLF (1 + l)Λ N .

(1.3.104)

Then for every τ > 0, τΛ ≤ 1, and for every p0 ∈ PH, ϕ ∈ Fl we have ε 󵄨󵄨 󵄨 0󵄨 ξ 0 󵄨 󵄨󵄨(Fτ ϕ − ϕ)(p )󵄨󵄨󵄨α ≤ C0 (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α )τ (1 + |ln τΛ N+1 |) ,

(1.3.105)

where the constant C0 depends only on Λ N , Λ N+1 , s0 , γ, but not on τ, ξ , ε ∈ {0, 1} which are as follows: ξ = 1, ξ = 1 − γ, ξ = (1 − γ)(2s0 − 1) , −1

ε = 1,

γ = 0,

ε = 0,

γ ∈ [0, 1],

ε = 1,

γ ∈ [0, 1]

0 < s0 ≤ 1; 0 < s0 ≤ 1; s0 ≤ 1.

54 | 1 Discrete attractor and approximate calculation Proof. Thanks to (1.3.31), we have 0

∞

−∞

k=1

(Fτ ϕ − F ϕ)(p0 ) = ∫ eA+C σQG(p(σ))dσ − τ ∑ R(τ)k QG(p−k ), where G(p) = F(p + ϕ(p)). We split this expression as follows: (Fτ ϕ − F ϕ)(p0 ) = (1.3.107) + (1.3.108) + (1.3.109), ∞

∑ ∫ eA+C σQ(G(p(σ)) − G(p−k ))dσ,

k=1 −kτ−τ ∞

(1.3.106)

−kτ

(1.3.107)

−kτ

∑ ∫ (eA+C σ − R(τ)k )QG(p−k )dσ,

k=1 −kτ−τ

(1.3.108)

0

∫ eA+C QG(p(σ))dσ.

(1.3.109)

−τ

It is easy to find a constant C, which is independent of T satisfying τΛ ≤ 1, such that 󵄨󵄨 γΛ 󵄨 󵄨 0󵄨 1−γ 󵄨󵄨A (1.3.109)󵄨󵄨󵄨α ≤ Cτ (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α ).

(1.3.110)

Now the proof is divided in two parts. First, we estimate (1.3.107) using Proposition 1.3.1. Second, we deal with (1.3.108). (i) An estimate of (1.3.107). We write for k ≥ 0, −kτ − τ < σ ≤ −kτ, p(σ) − p−k = p(σ) − p(−kτ) + p(−kτ) − p−k and deduce from (1.3.87) and (1.3.101) that there exists θ ∈ [0, 1] such that 󵄨󵄨 󵄨󵄨 dp 󵄨 󵄨 󵄨󵄨 󵄨 −k 󵄨 −k 󵄨 󵄨󵄨p(σ) − p 󵄨󵄨󵄨α ≤ τ󵄨󵄨󵄨 (−kτ − θτ)󵄨󵄨󵄨 + 󵄨󵄨󵄨p(−kτ) − p 󵄨󵄨󵄨α , 󵄨󵄨α 󵄨󵄨 dt 󵄨 󵄨󵄨 󵄨 0󵄨 −k 󵄨 −k 󵄨 󵄨󵄨p(σ) − p 󵄨󵄨󵄨α ≤ τK1 (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α ) exp(λ(k + 1)τ) + 󵄨󵄨󵄨p(−kτ) − p 󵄨󵄨󵄨α ,

(1.3.111)

where, and in the sequel, Ki s denote constants which are independent of τ. Now using (1.3.90) with n = −k, we find −1 󵄨󵄨 󵄨 0󵄨 −λτ −k 󵄨 2 k kλτ 󵄨󵄨p(−kτ) − p 󵄨󵄨󵄨α ≤ τ K2 (η − e )(ηe − 1) (1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α ).

We conclude, according to (1.3.111), that for k ≥ 0, 󵄨󵄨 󵄨 0 󵄨 λkτ −k 󵄨 󵄨󵄨p(σ) − p 󵄨󵄨󵄨α ≤ τ(K1 + kτK2 )(1 + 󵄨󵄨󵄨p 󵄨󵄨󵄨α )e , kτ ≤ −σ ≤ (k + 1)τ, k ≥ 0.

(1.3.112)

1.3 Inertial manifolds under time-discretization | 55

Returning to (1.3.107), we set v = A−γ (G(p(σ)) − G(p−k )) and use 󵄨 󵄨 |v|a ≤ LF (1 + l)󵄨󵄨󵄨p(σ) − p−k 󵄨󵄨󵄨α , and then from 󵄨󵄨 γ Aσ 󵄨󵄨 γ −1 γ Λσ 󵄨󵄨A E Qv󵄨󵄨α ≤ (Λ + (τ|γ| ) )e |Qv|a , v ∈ D(Aα ), ∀σ ≤ 0.

(1.3.113)

we have a

󵄨 󵄨 |(1.3.107)|α ≤ τLF (1 + l)(1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α ) ∫ (Λ + (2|σ|−1 ))(K1 + |σ|K2 ) exp(Λ − λ)σdσ. −∞

(1.3.114)

Since γ ∈ [0, 1], the integral above converges, and we can find K3 such that 󵄨 󵄨 |(1.3.107)|α ≤ τK3 (1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α ).

(1.3.115)

(ii) An estimate of (1.3.108). We split this sum into the following terms: ∞

−kτ

∑ ∫ (e(A−C)σ − e(A+C)kτ )QG(p−k )dσ,

k=1 −kτ−τ ∞

τ ∑ (e−(A+C)kτ − R(τ)k )QG(p−k ).

(1.3.116) (1.3.117)

k=1

They will be estimated by the following lemmas. Lemma 1.3.4. There exists a constant C, which is independent of N and τ, τΛ < 1, such that ∞

−kτ

k=1

−kτ−τ

ε 󵄨 󵄨 ∑ eλkτ ∫ 󵄨󵄨󵄨Aγ (e(A+C)σ − e(A+C)kτ )󵄨󵄨󵄨L (D(Aα )) dσ ≤ C1 τξ (1 + |lg τΛ|) 1 ,

(1.3.118)

where ξ1 ∈ [0, 1], ε1 ∈ {0, 1} are given by the following table: γ γ=0 γ=0 0 1, we take δ1 < (1 − γ)/s0 . An analogue of (1.3.123) reads ∞

∑ eλkτ sup τδ1 μγ + δ1 s0 e−μkτ ≤ K10 τ−1+(1−γ)/s0 (1 + lg |τΛ|).

(1.3.125)

μ≥Λ

k=1

Returning to (1.3.118), we have to integrate with respect to σ the estimates (1.3.123), (1.3.124) and (1.3.125). Since these are independent of σ, we have |(1.3.124)|, 󵄨󵄨 󵄨 󵄨󵄨left-hand side of (1.3.118)󵄨󵄨󵄨 ≤ τ(|(1.3.123)| + { |(1.3.125)|,

s0 ≤ 1,

s0 > 1,

),

which is (1.3.118). Lemma 1.3.5. There exists a constant C2 , which is independent of N and τ, satisfying τΛ ≤ 1 and such that ∞

ε 󵄨 󵄨 τ ∑ 󵄨󵄨󵄨Aγ (e−(A+C)kτ − R(τ)k )󵄨󵄨󵄨L (D(Aα )) eλkτ ≤ C2 τξ2 (1 + |lg τΛ|) 2 , k=1

(1.3.126)

where ξ2 ∈ [0, 1], ε2 ∈ {0, 1}, ξ2 = 1 − γ,

ε2 = 0, s0 < 1; −1

ξ2 = (1 − γ)(2s0 − 1) ,

ε2 = 1, γ = 0,

ε2 = 1, s0 ≥ 1.

Proof. Consider μγ (e−(μ+iξ )kτ − (1 + μτ)−k (

k

1 − iτξ /2 ) ), 1 + iτξ /2

(1.3.127)

where μ ≥ Λ, |ξ | ≤ a|μ|s0 , k ≥ 1. We write (1.3.127) = μγ (e−ikτξ − e−ikθ )e−μkτ + μγ (e−μkτ − (1 + τμ)−k )e−ikθ , where θ ≡ 2 arctan(τξ /2). We are going to estimate the two terms above separately. First, we have 󵄨󵄨 ikτξ 󵄨 󵄨 󵄨 − e−ikθ 󵄨󵄨󵄨 = 2󵄨󵄨󵄨 sin(k(θ − τξ ))/2󵄨󵄨󵄨 ≤ 21−δ2 |k|δ2 |θ − τξ |δ2 , 󵄨󵄨e

58 | 1 Discrete attractor and approximate calculation where δ2 ∈ [0, 1] is arbitrary for the moment. Then since for every x ∈ ℝ, |arctan x −x| ≤ x2 , we deduce that 2 󵄨󵄨 −ikτξ 󵄨 − e−ikθ 󵄨󵄨󵄨 ≤ 21−2δ2 τ2δ2 |ξ |2δ2 , 󵄨󵄨e

∀δ2 ∈ [0, 1].

(1.3.128)

Hence using that |ξ | ≤ aμs0 , we obtain 󵄨󵄨 γ −ikτξ 󵄨 − e−ikθ )e−μkτ 󵄨󵄨󵄨 ≤ 21−2δ2 a2δ2 τ2δ2 μ(γ+2δ2 s0 ) k δ2 e−μkτ . 󵄨󵄨μ (e

(1.3.129)

The other term in (1.3.127) is bounded via a first order Taylor formula as follows: 󵄨󵄨 γ −μkτ 󵄨 − (1 + τμ)−k )󵄨󵄨󵄨 ≤ μγ+τ τ2 k(1 + τμ)−k /2. 󵄨󵄨μ (e

(1.3.130)

When k = 1 or k = 2, it is, in fact, better to bound this term as follows: 󵄨󵄨 γ −μkτ 󵄨 − (1 + τμ)−k )󵄨󵄨󵄨 ≤ τ−a , 󵄨󵄨μ (e

k = 1, 2.

(1.3.131)

A careful study of the function of p, which appears in the right-hand side of (1.3.130), shows that ∞

󵄨 󵄨 ∑ (sup 󵄨󵄨󵄨μγ (e−μkτ − (1 + τμ)−k )󵄨󵄨󵄨)eλk ≤ K11 τ−a .

(1.3.132)

k=3

Concerning (1.3.129), we write ∞

󵄨 󵄨 S = ∑ (sup 󵄨󵄨󵄨μγ (e−ikτξ − e−ikθ )e−μkτ 󵄨󵄨󵄨)eλkτ k=1 μ≥Λ

≤ 21−2δ2 a2δ2 τ2δ2 (

k δ2 (

∑

k≤(γ+2δ2 s0 )/2Λ

+

∑

k>(γ+2δ2 s0 )/2Λ

−(γ+2δ2 s0 )

kτ ) γ + 2δ2 s0

eλkτ e−(γ+2δ2 s0 ) )

k δ2 Λγ+δ2 s0 e(λ−Λ)kτ = 21−2δ2 a2δ2 τ2δ2 {S1 + S2 }.

(1.3.133)

Concerning S2 , we notice that for k > (γ + 2δ2 s0 )/2Λ, the function x → x δ2 e(λ−Λ)xτ is decreasing and then γ+2δ2 s0

S2 ≤ Λ

∞

̄

∫ xδ2 e(λ−Λ)xτ dx = K12 Λγ+2δ2 s0 ((λ − Λ)τ)

−(δ2 +1)

.

0

Concerning S1 , we see that if δ2 (I − 2S0 ) − γ > −1, S1 = K13 τ−(γ+2δ2 s0 ) ≤ K14 τ

−(γ+2δ2 s0 )

∑

k≤(γ+2δ2 s0 )/2Λ

(τ−1 )

k δ2 (1−2s0 )−γ

δ2 (1−2s0 )−γ+1

= K14 τ−(γ+2δ2 s0 )−δ2 (1−2s0 )−γ+1 .

1.3 Inertial manifolds under time-discretization | 59

If δ2 (1 − 2s0 ) − γ = −1, we have S1 ≤ K13 τ−(γ+2δ2 s0 ) lg |τΛ|. Suppose first that s0 < 1 and choose δ2 = 1 − γ, then we have δ2 (1 − 2s0 ) − γ < 1. Thus S1 ≤ τ(1−γ)/(2s0 −1)−1 . Therefore, returning to (1.3.127), and using (1.3.131), (1.3.132), and (1.3.133) we find for the left hand side of (1.3.126) that τ1−τ , { { { left-hand side of (1.3.126) ≤ K15 {τ(1 + 󵄨󵄨󵄨󵄨lg(τΛ)󵄨󵄨󵄨󵄨), { { (1−γ)/(2s −1)−1 󵄨 󵄨 0 (1 + 󵄨󵄨󵄨lg(τΛ)󵄨󵄨󵄨), {τ

s0 < 1, γ ∈ [0, 1]; s0 < 1, γ = 0; s0 ≥ 1.

Thanks to these lemmas, we are now able to estimate (1.3.116) and (1.3.117). Indeed, using (1.3.7), (1.3.101) and (1.3.118), we find 󵄨 󵄨 ξ 󵄨 󵄨ε |(1.3.116)|α ≤ K4 (1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α )τ1 (1 + 󵄨󵄨󵄨lg(τΛ)󵄨󵄨󵄨) 1 .

(1.3.134)

In the same way, this time using (1.3.126), we 󵄨 󵄨 ξ 󵄨 󵄨ε |(1.3.117)|α ≤ K5 (1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α )τ2 (1 + 󵄨󵄨󵄨lg(τΛ)󵄨󵄨󵄨) 2 .

(1.3.135)

Combining (1.3.134) and (1.3.135), we deduce 󵄨 󵄨 󵄨 󵄨ε |(1.3.108)|α ≤ K6 (1 + 󵄨󵄨󵄨p0 󵄨󵄨󵄨α )τξ (1 + 󵄨󵄨󵄨lg(τΛ)󵄨󵄨󵄨) ,

ξ = min(ξ1 , ξ2 ),

ε = max(ε1 , ε2 ).

(1.3.136)

Finally, (1.3.106) follows by adding the estimates (1.3.110), (1.3.115) and (1.3.136). Now we are in the position to prove Theorem 1.3.3 using Lemma 1.3.4. Indeed, from Theorem 1.3.1 we know ϕ = F ϕ, ϕ ∈ F1/4 . On the other hand, for τΛ ≤ 1, we have by Theorem 1.3.2 that ϕτ = Fτ ϕτ , ϕτ ∈ F1/4 . Thus ϕ − ϕτ = F ϕ − Fτ ϕτ = F ϕ − Fτ ϕτ + Fτ ϕ − Fτ ϕτ . Now, according to (1.3.34) (with k = 7/10, l = 1/4), we have 󵄨 󵄨ε ‖ϕ − ϕτ ‖α ≤ 10/3‖F ϕ − F ϕτ ‖α ≤ (10C0 /3)τξ (1 + 󵄨󵄨󵄨lg(τΛ)󵄨󵄨󵄨) , which is exactly the desired result of Theorem 1.3.3.

60 | 1 Discrete attractor and approximate calculation In the following, we develop two types of application. Example (Complex amplitude equations). Consider the following family of equations 𝜕u − (1 + iα)Δu + (1 + iβ)f (|u|2 )u = ru, 𝜕t

(1.3.137)

where α, β, γ are real parameters, u = u(x, t) is a complex valued function defined on Ω × ℝ+ , Ω ⊂ ℝd , d = 1 or 2 is a bounded open set. Two typical cases are f (s) = s and f (s) = ±s(1+δs)−1 , δ > 0. In the first case, (1.3.137) is the well-known Ginzburg–Landau PDE which describes the amplitude evolution of instability waves in a very large variety of dissipative systems in fluid mechanics. The second case is a laser equation occurring in the study of propagation of intensive light in a nonlinear medium cavity. Other cases can arise, for example, f (s) = −s + s2 . Various boundary conditions can supplement (1.3.137): (Dirichlet) u(x, t) = 0, x ∈ 𝜕Ω, t ≥ 0; 𝜕u (Neumann) (x, t) = 0, x ∈ 𝜕Ω, t ≥ 0; 𝜕ν (Periodic) u(⋅, t) is Ω-periodic, Ω = [0, L]d ;

(1.3.138) (1.3.139) (1.3.140)

where ν denotes the outward unit normal of 𝜕Ω. Assume that f ∈ ε2 (ℝ+ , ℝ) satisfies one of the two hypotheses: either ∃σ > 0, C1 ≥ 0, R ≥ 0, f > 0, f > 0, such that 󵄨 󵄨 f (s) ≥ fsσ − C1 , 󵄨󵄨󵄨f 󸀠 (s)󵄨󵄨󵄨 ≤ f sσ−1 , ∀s ≥ R,

(1.3.141)

or ∃w ≥ 0, z1 f (|z1 |2 ) − z2 f (|z2 |2 ) ≤ w|z1 − z2 |,

∀zi ∈ ℂ.

(1.3.142)

Notice that the cases f (s) = s, f (s) = −s + s2 satisfy (1.3.141), while the case f (s) = ±s(1+δs)−1 , δ > 0 satisfies (1.3.142). We set H = L2 (Ω)2 and define the bounded operator A as follows: Av = −Δv + v,

v ∈ D(A),

(1.3.143)

where 󵄨 {v ∈ H 2 (Ω), v󵄨󵄨󵄨𝜕Ω = 0} or { { { { { 𝜕v 󵄨󵄨󵄨󵄨 2 { D(A) = {{v ∈ H (Ω), 𝜕ν 󵄨󵄨󵄨󵄨𝜕Ω = 0} or { { { { {{v ∈ H 2 (Ω), v and 𝜕v are Ω-periodic}, 𝜕ν { according to the boundary conditions (1.3.138)–(1.3.140). The operator C is unbounded, C = iα(A − I).

(1.3.144)

1.3 Inertial manifolds under time-discretization

| 61

It commutes with A, is skew-symmetric and continuous, D(As+1 ) → D(As ), ∀s ∈ ℝ, i. e., we have s0 = 1 and F(v) = (iα − γ)v + (1 + iβ)f (|v|2 )v.

(1.3.145)

Consider an abstract equation du + Au + Cu + F(u) = 0, dt u(0) = u0 .

(1.3.146) (1.3.147)

In the case f (s) = s, we cannot expect F to be a globally Lipschitz mapping on D(Aα ), α ∈ ℝ. We first consider the case (1.3.142), namely 1

1

󵄨 󵄨2 ∫ 󵄨󵄨󵄨F(v) − F(w)󵄨󵄨󵄨 dx ≤ ((α2 + γ 2 ) 2 + (1 + β2 ) 2 w) ∫ |v − w|2 dx, Ω

Ω

and (1.3.6), (1.3.7) hold true with α = τ = 0, Lp = (|1 + r| + (1 + β2 )1/2 w). Let us now consider the case (1.3.141). Local in time existence and uniqueness of solutions to (1.3.137), (1.3.138)–(1.3.140) is very classical. Given u0 ∈ H, we denote by S(t)u0 = u(t) the maximal solution u ∈ ε([0, T]; L2 (Ω2 )) ∩ ε([0, T]; H 2 (Ω2 )),

T = Tmax (u0 )

satisfying u(0) = u0 . As we are going to see, a priori estimates on u will prove that Tmax (u0 ) = +∞, ∀u0 ∈ H. In fact, S(t) is bounded on H: Proposition 1.3.5. We assume that f satisfies (1.3.141) with d = 1, σ ≤ 2; d = 2, σ ≤ 1. There exists ρ ≥ 0 depending only on the data α, β, γ, Ω and f such that for every v0 ∈ H, |u0 | ≤ R there exists t0 (R), for which the solution to (1.3.137), (1.3.138)–(1.3.140) with initial data u0 satisfies 󵄨 󵄨2 󵄨 󵄨2 ‖u‖21 ≡ ∫(󵄨󵄨󵄨u(x, t)󵄨󵄨󵄨 + 󵄨󵄨󵄨∇u(x, t)󵄨󵄨󵄨 )dx ≤ ρ2 ,

∀t ≥ t0 (R).

(1.3.148)

Ω

This shows that the semigroup S(t) possesses a bounded absorbing set Bρ = {v ∈ H 1 (Ω)2 , ‖v‖1 ≤ ρ}. Before giving the sketch of the proof of this result, which is postponed to the end of this section, we discuss the applicability of previous abstract framework to (1.3.137), (1.3.138)–(1.3.140). (i) The one-dimensional case. In this case, since the norm in H 1 (Ω) is bounded by the sup-norm, we deduce that S(t) possesses a bounded absorbing set B∞ in L∞ (Ω) : ∃ρ∞ > 0 such that B∞ = {v ∈ L∞ (Ω), supx∈Ω |v(x)| ≤ ρ∞ } is absorbing set. We replace F in (1.3.145) by 2 F(v) = (iα − γ)v + (1 + iβ)θ(|v|2 ρ−2 ∞ )f (|v| )v,

(1.3.149)

62 | 1 Discrete attractor and approximate calculation where θ is a cut-off function: θ ∈ C ∞ (ℝ+ ), θ(s) = 1, s ∈ [0, 2], θ(s) = 0, s ≥ 3. Now it is clear that the new f (s) = θ(sρ−1 ∞ )f (s) satisfies (1.3.142), and we conclude as before that we can take α = γ = 0 in (1.3.6)–(1.3.7). (ii) The two-dimensional case. In this case we take, instead of (1.3.145), 󵄩 󵄩2 F(v) = (iα − γ)v + (1 + iβ)θ(󵄩󵄩󵄩v2 󵄩󵄩󵄩1 ρ−2 )f (|v|2 )v,

(1.3.150)

where θ is as before and ρ was given in Theorem 1.3.5. We claim that there exists a constant w1 , which only depends on r, β, α, ρ, Ω and f such that 󵄩󵄩 󵄩 󵄩󵄩F(v) − F(w)󵄩󵄩󵄩1 ≤ w1 ‖v − w‖1 ,

∀v, w ∈ H 1 (Ω)2 .

(1.3.151)

Let us note that, according to (1.3.151), we see that (1.3.6)–(1.3.7) hold true with α = 21 ,

LF γ = 0, w . Concerning the proof of (1.3.151), we note first that the linear part in (1.3.150) 1 is harmless. Hence we have to show that the nonlinear part satisfies (1.3.151). Since this part is compactly supported in H 1 (Ω)2 , we only have to check that v → θ(‖v‖21 ρ−2 )f (|v|2 ) is a locally Lipschitz function, H 1 (Ω) → H 1 (Ω). But now this follows by the fact that we are in the case where Ω ⊂ ℝ2 and H 1 (Ω) is continuously imbedded in Lp (Ω), ∀p < ∞. We now consider the applications of the results in the one-dimensional case. Setting Ω = [0, L], L > 0, the eigenvalues of A are well known. In the cases (1.3.138) and (1.3.139), the eigenvalues λk = Λ k , k ≥ 1, are distinct, and we have Λ k = 1+(k −1)2 π 2 L2 . In the case (1.3.140), we have Λ k = 1 + 4(k − 1)2 π 2 L−2 with multiplicity mk = 2, m1 = 1. The condition Λ N+1 ≥ 3L2F Λ2γ−1 /2 is satisfied, setting γ = 0, Λ 1 = 1,

N ≥ C0 LF L, γ

where C0 is an absolute constant. The more drastic condition Λ N+1 − Λ N ≥ 30LF (Λ N + γ Λ N+1 ) reads N ≥ C1 LF L2 ,

L ≥ 1.

(1.3.152)

The fractional step scheme reads 󵄨 n 󵄨2 n n+ 1 n n+ 1 n {(u 2 − u )/τ − Δu 2 + (1 + iβ)f (󵄨󵄨󵄨u 󵄨󵄨󵄨 )u = γu , { n+1 n+1 n+ 21 n+ 21 {(u − u )/τ − iαΔ(u + u )/2 = 0,

(1.3.153)

which are supplemented with the boundary conditions (1.3.138)–(1.3.140). Let us first 1 1 deal with the case, where f satisfies (1.3.142). In this case LF = ((α2 + γ 2 ) 2 + (1 + β2 ) 2 w), and summarizing Theorem 1.3.1 we deduce Theorem 1.3.4. We assume that f satisfies (1.3.142). Equations (1.3.137), (1.3.138)– (1.3.140) on Ω = [0, L], L > 0 possess an M-dimensional manifold in L2 (Q)2 , M = Mϕ , where 1

1

M ≤ C3 ((α2 + γ 2 ) 2 + (1 + β2 ) 2 w)L2

(1.3.154)

1.3 Inertial manifolds under time-discretization

1

| 63

1

and C2 is a universal constant. For every τ > 0, satisfying τL2 ((α2 + γ 2 ) 2 + (1 + β2 ) 2 w)2 ≤ C3 , C3 being a universal constant, the discrete iteration (1.3.153) possesses also an Mdimensional inertial manifold, Mτ = Mϕτ . Moreover, we have an error estimate ‖ϕ − ϕτ ‖0 ≤ Kτ (1 + |lg τ|). If f satisfies (1.3.141), we take F given by (1.3.150), i. e., we consider instead of (1.3.137) 𝜕u − (1 + iα)Δu + (1 + iβ)θ × (‖u‖21 ρ−2 )f (|u|2 )u = ru, 𝜕t

(1.3.155)

for which the inertial manifolds are obtained in H 1 (Q)2 and the error estimate is obtained. Now we consider the two dimensional case. As is well known, λk = (4πk/ vol Ω) + 2γ−1 o(k), where vol Ω denotes the area of Ω. It is clear that the condition Λ N+1 ≥ 3lf2 /2Λ 1 is satisfied as soon as N is large. Hence in order to show the spectral gap condition Λ N+1 − Λ N ≥ 60LF for arbitrary LF , we must establish lim sup(Λ N+1 − Λ N ) = +∞.

(1.3.156)

N→+∞

One can answer this question positively in the case where Ω = [0, L1 ] × [0, L2 ] is a rectangle, (1.3.156) follows from ( LL1 )2 being rational. 2

Sketch of proof of Proposition 1.3.5. Let us consider a smooth solution u to (1.3.137), (1.3.138)–(1.3.140). Multiplying by u(−Δu) and integrating on Ω the real part of the resulting identities, we find 1 d ∫ |u|2 dx ∫ |∇u|2 dx + ∫ f (|u|2 )|u|2 dx = γ ∫ |u|2 dx, 2 dt Ω

Ω

Ω

1 d ∫ |∇u|2 dx + ∫ |Δu|2 dx + Re(1 + iβ) 2 dt Ω

(1.3.157)

Ω

Ω

× ∫ ∇(f (|u|2 )u)∇udx = γ ∫ |∇u|2 dx. Ω

(1.3.158)

Ω

According to (1.3.141), there exists C1 such that f (|u|2 ) ≥ f |u|2σ − C1 . Now there exists C2 such that f1 s2σ+2 − (C1 + 1 + γ)s2 ≥ s2 f s2σ+2 /2 − C2 , and we deduce d ∫ |u|2 dx + ∫ |u|2 dx + 2 ∫ |∇u|2 dx + f ∫ |u|2σ+2 dx ≤ C3 . dt Ω

Ω

Ω

(1.3.159)

Ω

This implies 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨u(x, t)󵄨󵄨󵄨 dx ≤ (∫ 󵄨󵄨󵄨u0 (x)󵄨󵄨󵄨 dx)e−t + C3 (1 − e−t ), Ω

Ω

∀t ≥ 0.

(1.3.160)

64 | 1 Discrete attractor and approximate calculation Concerning (1.3.158), we have Re(1 + iβ) ∫(∇f (|u|2 )u)∇udx = ∫ f (|u|2 )|∇u|2 dx Ω

Ω

+ Re(1 + iβ) ∫ f 󸀠 (|u|2 )u∇u Re(u ⋅ ∇u)dx, Ω

and since f (|u|2 ) > −C1 , we conclude by (1.3.141) that 1 d ∫ |∇u|2 dx + ∫ |Δu|2 dx ≤ (r + C1 ) ∫ |∇u|2 dx + C4 ∫ |u|2σ |∇u|2 dx. 2 dt Ω

Ω

Ω

(1.3.161)

Ω

By Hölder inequality, 2σ

2

2σ+2

∫ |u| |∇u| dx ≤ (∫ |u| Ω

σ/(σ+1)

dx)

(∫ |∇u|

Ω

2(σ+1)

dx)

1 σ+1

(1.3.162)

.

Ω

Then due to the following c inequality on two-dimensional domains: σ

∫ |∇u|2(σ+1) dx ≤ Cσ (∫ |∇u|2 dx)(∫ |Δu|2 dx) , Ω

Ω

Ω

where Cσ depends only on σ and Ω, we find 2σ

2

2σ+2

∫ |u| |∇u| dx ≤ C5 (∫ |u| Ω

σ/(σ+1)

dx)

(∫ |∇u| dx)

Ω

≤

2

1 σ+1

σ/(σ+1)

(∫ |Δu|2 dx) Ω

Ω

σ

1 ∫ |Δu|2 dx + C6 (∫ |∇u|2 dx)(∫ |∇u|2σ+2 dx) . 2C4 Ω

Ω

(1.3.163)

Ω

Using this estimate in (1.3.161) then leads to d ∫ |∇u|2 dx + ∫ |Δu|2 dx dt Ω

Ω

σ

≤ 2(r + C1 + C6 (∫ |u|2σ+2 dx) ) ∫ |∇u|2 dx. Ω

(1.3.164)

Ω

This implies that the global solution to (1.3.137), (1.3.157) exists. Integrating (1.3.159) shows that T

∫ ∫(|u|2σ+2 + |∇u|2 )dxdt ≤ C7 (T) < ∞. 0 Ω

(1.3.165)

1.3 Inertial manifolds under time-discretization

| 65

In the two-dimensional case, we have assumed that σ ≤ 1, therefore returning to (1.3.164) and using (1.3.165), it follows that 󵄨2 󵄨 sup ∫ 󵄨󵄨󵄨∇u(x, t)󵄨󵄨󵄨 dx ≤ C8 (T) < ∞.

0≤t≤T

Ω

And now global in time existence follows by classical means. So far we have not proved a uniform in time bound on |u(t)|. It is necessary to make use of a time uniform Gronwall lemma. In the one-dimensional case, the relevant Gagliardo–Nirenberg inequality is 1+ σ2

∫ |∇u|2(σ+1) dx ≤ C6 (∫ |∇u|2 dx) Ω

σ 2

(∫ |Δu|2 dx) .

Ω

Ω

And this leads, instead of (1.3.165), to 2σ

σ+2 d ∫ |∇u|2 dx + ∫ |Δu|2 dx ≤ 2(r + C1 + C7 (∫ |u|2σ+2 dx) ∫ |∇u|2 dx). dt

Ω

Ω

Ω

Ω

In this case σ ≤ 2, so that 2σ/(σ + 2) ≤ 1, and we argue as in the previous case. Example 2. KdV–Burgers equations are given by s

𝜕t u + 𝜕x2p+1 u + κu𝜕x u + (−𝜕x2 ) u = f ,

(1.3.166)

where the unknown function u = u(x, t) is real and 2π-periodic u(x + 2π, t) = u(x, t),

∀x ∈ ℝ, t ≥ 0.

(1.3.167)

The function f and the constant κ are given, p is an integer (usually p = 1 or 2) and (−𝜕x2 )s is the pseudo-differential operator with symbol |k|2s . i. e.,

The Fourier coefficients of a 2π-periodic function v(x) are denoted by vk , k ∈ ℤ, 1

v(x) = (2π)− 2 ∑ vk eikx . k∈ℤ

We denote by HLσ the usual fractional Sobolev space of periodic functions: σ

HLσ = {v ∈ L2 (0, L), ∑ (1 + |k|2 ) |vk |2 < ∞}, k∈ℤ

∘ ∘ ∘ HL0 = L2L = L2 (0, L), HLσ = {v ∈ HLσ , v0 = 0}, we take H = L2 (0, L), (Av)k = |k|2s vk ,

(Cv)k = (−1)p ik 2p+1 vk ,

∘ D(A) = HL2s .

66 | 1 Discrete attractor and approximate calculation It is clear that C is skew-symmetric and maps continuously D(As+s0 ) → D(As0 ), s0 = ∘ (2p+1)/(2s). Assuming that f ∈ L2L , the function F = F(v) = κv𝜕x v −f is locally Lipschitz from HL1 → HL2 . Indeed, assuming that v ∈ HL2 = D(A1/(2s) ), F(v) = 𝜕x (κv2 /2) − f . Since HL2 is an algebra, κv2 /2 ∈ HL2 and then F(v) ∈ L2L . Moreover, we have F(v) − F(w) = κ𝜕x (v2 − w2 )/2, κ κ󵄨 2 󵄨 󵄨󵄨 2󵄨 󵄨󵄨F(v) − F(w)󵄨󵄨󵄨0 ≤ 󵄨󵄨󵄨v − w 󵄨󵄨󵄨0 ≤ |v + w|L∞ |v − w|0 . 2 2 The sup-norm |w|L∞ is bounded by ‖ ⋅ ‖1/2s which is the HL1 (Ω)-norm, therefore this last estimate shows that F is locally Lipschitz, D(As ) → D(Aα−γ ), α = 2S1 . It is clear that this F is not globally Lipschitz. We are going to use here, like in the previous case, the existence of a bounded absorbing set and a truncation method. We assume that u0 ∈ L2L , f ∈ L2L . It is classical that (1.3.166), (1.3.167) (s ≥ 1) possess a unique solution satisfying u ∈ ε(ℝ+ ; L2L ) ∩ L2 ([0, T]; HLs ),

∀T > 0,

and u(0) = u0 . Denoting by S(t) the semigroup u0 → S(t)u0 = u(t), we have Proposition 1.3.6. Consider the bounded sets Bε , ∘ Bε = {v ∈ HL1 , |v|20 ≤ (1 + ε)|f |20 , |vx |20 ≤ Cε (f )}, where Cε (f ) = |f |20 + C0 (1 + ε)4 κ8 |f |10 0 and C0 is a numerical constant. Then Bε are positively invariant sets of S(t), that is, S(t)Bε ⊂ Bε ,

∀ε ≥ 0, ∀t ≥ 0.

Moreover, for every ε > 0, Bε is a bounded absorbing set for S(t) in HL1 , i. e., for every bounded set B in HL1 , there exists Tε (B) such that S(t)B ⊂ Bε ,

∀t ≥ Tε (B), ∀ε > 0.

Proof. According to (1.3.166)–(1.3.167), we have d s ∫ u2 dx + ∫ u(−𝜕x2 ) udx = ∫ ufdx, dt 1 d s ∫(𝜕x u)2 dx + ∫(−𝜕x2 u)(−𝜕x2 ) udx 2 dt = κ ∫ u(𝜕x u)(𝜕xx u)dx + ∫(−𝜕x2 u)fdx. So 1 d 2 |u| + |ux |20 ≤ ∫ ufdx, 2 dt 0

(1.3.168)

1.3 Inertial manifolds under time-discretization

1 d |u |2 + |uxx |20 ≤ κ ∫ uux uxx dx − ∫ uxx fdx. 2 dt x 0

| 67

(1.3.169)

Since |ux |0 ≥ |u|0 , we deduce from (1.3.168) 󵄨 󵄨󵄨 −t −t 󵄨󵄨u(t)󵄨󵄨󵄨0 ≤ |u0 |e + |f |0 (1 − e ).

(1.3.170)

Set 󵄨 󵄨2 ML2∞ = sup 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 ≤ 2|v|0 |vx |∞ ,

∀v ∈ HL1 ,

0≤x≤L

∫ uux uxx dx =

1 1 ∫ u(u2x )x dx = − ∫ u3x dx 2 2 5

1

(1.3.171)

1

≤ |ux |L∞ |ux |20 /2 ≤ 2− 2 |ux |02 |uxx |02 . Since |ux |20 ≤ |u|0 |uxx |0 , we conclude by using Young inequality, 8ab ≤ 7a8/7 + b8 , that κ ∫ uux uxx dx ≤

C κ8 1 |uxx |20 + 0 |u|10 , 4 2

(1.3.172)

where C0 is a numerical constant (77 2−13 ). Returning to (1.3.169) and using (1.3.172), gives 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ uxx fdx󵄨󵄨󵄨 ≤ |f |0 |uxx |0 ≤ |f |2 + |uxx |2 /4, 0 0 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨

|ux |0 ≤ |uxx |0 .

We finally find d |u |2 + |ux |20 ≤ |f |20 + C0 κ8 |u|10 . dt x 0

(1.3.173)

We are now in a position to show that Bε ’s are positively invariant with respect to S(t). We take u0 ∈ Bε , ε ≥ 0. According to (1.3.170), we have 󵄨󵄨 󵄨2 2 󵄨󵄨u(t)󵄨󵄨󵄨0 ≤ (1 + ε)|f |0 ,

∀t ≥ 0.

By (1.3.173) we have d |u |2 + |ux |20 ≤ |f |20 + C0 κ8 (1 + ε)5 |f |50 = Cε (f ). dt x 0 Since |u0x |20 ≤ Cε (f ), we deduce that |ux (t)|20 ≤ Cε (f ), ∀t ≥ 0. Finally, for ε > 0, we consider a bounded set B, B ⊂ {v ∈ HL1 , |v|20 + |vx |20 ≤ R2 } in HL1 . According to (1.3.170), for u0 ∈ B we have 󵄨󵄨 󵄨 2 󵄨󵄨u(t)󵄨󵄨󵄨0 ≤ (1 + ε)|f |0 ,

t ≥ tε (Ω) = lg(ε|f |20 /R).

(1.3.174)

68 | 1 Discrete attractor and approximate calculation On the other hand, thanks to (1.3.170) and (1.3.173), d −t |u |2 + |ux |20 ≤ |f |20 + C0 κ 8 (|u0 |10 e−t + |f |10 0 (1 − e )), dt x 0 which gives after integration 󵄨2 󵄨󵄨 2 8 10 −t 2 8 10 󵄨󵄨ux (t)󵄨󵄨󵄨0 ≤ (R + C0 κ R t)e + |f |0 + C0 k |f |0 ,

∀t ≥ 0.

(1.3.175)

It is clear that there exists Tε (R) ≥ tε (R) such that the right-hand side of (1.3.175) is less than Cε (f ) for t ≥ Tε (R). This and (1.3.174) show that S(t)B ⊂ Bε ,

∀t ≥ Tε (R),

and we complete the proof of Proposition 1.3.6. Now we consider Lipschitz property of F. We introduce the following cut-off function: ξ (x) ∈ C0∞ is such that ξ (x) = 1, x ∈ [0, 2], ξ (x) = 3 − x, x ∈ [2.3, 3], ξ (x) = 0, x ≥ 3. We set F(v) = f − ξ (|vx |20 /2C1 (f ))κv𝜕x v,

(1.3.176)

where Ci (f ) is given in Proposition 1.3.6. According to this result, solutions of (1.3.166)– (1.3.167) are solutions of du + Au + Cu + F(u) = 0. dt We claim that F in (1.3.176) satisfies (1.3.6)–(1.3.7) with α = γ =

(1.3.177) 1 . 2s

We note that

∘ 1 F : HL1 = D(A 2s ) → D(A0 ) = H = L2L , taking v, w ∈ HL1 we have F(v) − F(w) = ξ̃ (v)v𝜕 v − ξ̃ (w)w𝜕 w, x

x

where ξ (x) = κξ (|vx |2 )/2C1 (f ). Using 󵄨󵄨 󵄨 󵄨󵄨ξ (x) − ξ (y)󵄨󵄨󵄨 ≤ |x − y|, 󵄨󵄨 󵄨 󵄨 2 2 󵄨 󵄨󵄨F(v) − F(w)󵄨󵄨󵄨0 ≤ |κ|󵄨󵄨󵄨(v − w )󵄨󵄨󵄨0 /2

+ |κ|(|vx |20 − |wx |20 )|w𝜕x w|0 /2C1 (f ),

we have 󵄨󵄨 󵄨 󵄨 󵄨 2 2 󵄨󵄨F(v) − F(w)󵄨󵄨󵄨0 ≤ 󵄨󵄨󵄨ξ̃ (v)(v − w )x /2󵄨󵄨󵄨0 󵄨 󵄨 + 󵄨󵄨󵄨(ξ̃ (v) − ξ̃ (w))w𝜕x w󵄨󵄨󵄨0 .

1.3 Inertial manifolds under time-discretization

| 69

We claim that 󵄨 󵄨󵄨 4 5 󵄨󵄨F(v) − F(w)󵄨󵄨󵄨0 ≤ C2 |κ|(|f |0 + κ |f |0 )|vx − wx |0 ,

(1.3.178)

where C2 is a numerical constant. Indeed, if |vx |20 and |wx |20 are both larger than 2C1 (f ) then F(u) = F(w) = 0 and (1.3.178) is obviously satisfied. By symmetry we can assume that |vx |20 ≤ 2C1 (f ). Then (1.3.178) follows easily by considering the two cases: (i) |vx − wx |20 ≥ 8C1 (f ) and (ii) |vx − wx |20 ≤ 8C1 (f ). Now we construct inertial manifolds. Thanks to (1.3.178), we see that (1.3.6) holds with α = γ = 2s1 . On the other hand, the eigenvalues of A, Λ k = |k|2s , k ≥ 1, have multiplicity two, and the conditions on N read 2

(N + 1)2s ≥ 3C22 κ2 (|f |0 + κ 4 |f |50 ) /2, 2s

(N + 1) − N

2s

≥

30C22 |κ|(|f |0

+κ

4

|f |50 )(N

(1.3.179) + (N + 1)).

(1.3.180)

For s = 1, (1.3.180) is not satisfied, in general, but for s > 1 it holds. Since (N + 1)2s ≥ N 2s + 2sN 2s−1 , we deduce that there exists a numerical constant C3 (independent of s) such that (1.3.179)–(1.3.180) are satisfied for 2N ≥ C3 (

1/(2s−2)

|κ| (|f |0 + κ4 |f |50 )) s

.

(1.3.181)

The fractional step discretization of (1.3.177) can be explicitly described here as follows. We denote by unk the kth Fourier components of un and denote by unk ⊗ unk = ∑l∈ℤ unk−l unl the kth Fourier component of u2 . Then the fractional step method reads 󵄨 󵄨2 (un+1/2 − unk )γ + |k|2s un+1/2 + ikκξ (󵄨󵄨󵄨unk 󵄨󵄨󵄨0 /2C1 (f ))unk ⊗ unk = 0, { { k { { { {un+1 = (1 + iτ(−1)p k 2p+1 /2)(1 − iτ(−1)p k 2p+1 /2)−1 un+1/2 , k k { { {󵄨󵄨 n 󵄨󵄨 2 󵄨󵄨 n 󵄨󵄨2 { { u = |k| u , k ∈ ℤ. ∑ 󵄨 󵄨 󵄨 󵄨 {󵄨 x 󵄨0 󵄨 k󵄨 k∈ℤ {

(1.3.182)

This scheme was implicit in the physical variables x, t. In the Fourier variables it becomes explicit, and then applying abstract framework theorems we have the following result: ∘ Theorem 1.3.5. Equation (1.3.177) possesses an M-dimensional manifold in HL1 , M = Mϕ , where 1

M ≤ C3 (

2s−2 |κ| + |f |0 + κ 4 |f |50 ) . s

For every τ > 0, satisfying τC42s (|κ|/s)(|f |0 +κ 4 |f |50 )1/(2s−2) , C3 and C4 being numerical constants, the discrete iteration (1.3.182) possesses also an M-dimensional inertial manifold 0

in HL1 , Mτ = Mϕτ . Moreover, we have the error estimate ‖ϕ − ϕτ ‖1/2s ≤ Kτξ (1 + |lg τ|),

70 | 1 Discrete attractor and approximate calculation where ξ = (2s − 1)(4p + 2 − 2s)−1 , ξ = (2s − 1)/(2s),

1 s ∈ [1, p + ], 2

1 s≥p+ . 2

1.4 Landau–Lifschitz equations Guo and Lu [77] consider the discrete attractor problem for the following Landau– Lifschitz equation and its periodic initial boundary valued problem: 𝜕t u = −α1 u × (u × Δu) + α2 (u × Δu), { { { u(x + 2π, t) = u(x, t), { { { {u(x, 0) = u0 (x),

x ∈ Ω, t > 0, x ∈ Ω, t ≥ 0,

(1.4.1)

x ∈ Ω,

where Ω = [0, 2π], × denotes the vector cross-product in ℝ3 , u = (u1 , u2 , u3 ) : Ω × [0, ∞) → ℝ3 is the spin vector, α1 ≥ 0 is a Gilbert damping constant, α2 is another constant. From [102], we know that, in the classical sense, u is a solution of the problem (1.4.1) if and only if u is a solution of the problem 𝜕t u = −α1 (u ⋅ Δu)u + α1 |u|2 Δu + α2 u × Δu.

(1.4.2)

Now we give a spatial discretization of the problem (1.4.1), let J be a nonnegative integer and h = 2π the space step length. The interval Ω is divided into discrete lattice Ωh = J {x1 , x2 , . . . , xj }, where xj = jh. A discrete function u is defined by u = (u1 , u2 , . . . , uJ )T , uj = u(xj ). The spatial difference operators are defined by 1 (u − uj ), h j−1 1 = 2 (uj−1 − 2uj + uj−1 ). h

∇h uj = ujx = Δ h uj = ujxx

Spatial finite-difference discretized version of equation (1.4.1) can be defined by 𝜕t uj = −α1 uj × (uj × Δ h uj ) + α2 (uj × Δ h uj ), { { { uj (x + 2π, t) = uj (x, t), { { { {uj (x, 0) = u0 (xj ).

(1.4.3)

In order to give the following a priori estimates, for the two discrete periodic functions u = {uj | j = 1, 2, . . . , J} and v = {vj | j = 1, 2, . . . , J}, denote the scalar inner product as J

(u, v) = h ∑ uj ⋅ vj , j=1

1.4 Landau–Lifschitz equations | 71

where uj ⋅ vj = u1j vj1 + u2j vj2 + u3j vj3 denotes the inner product in ℝ3 , the discrete norm in L2 (Ωh ) by 1

‖u‖ = (u, u) 2 and the norm in L∞ (Ωh ) by ‖u‖∞ = max |uj |. 1≤j≤J

Lemma 1.4.1. For the smooth solution of problem (1.4.3), we have 󵄨 󵄨 |uj | = 󵄨󵄨󵄨u0 (xj )󵄨󵄨󵄨 and

‖u‖∞ = ‖u0 ‖∞ .

(1.4.4)

Proof. Multiplying the first equation of (1.4.3), we get uj ⋅ 𝜕t uj = 0, So

1 d |u |2 2 dt j

∀1 ≤ j ≤ J, t ∈ ℝ+ .

= 0, and then (1.4.4) is obtained immediately.

Lemma 1.4.2. Assume that the solution of (1.4.3) satisfies u0 ∈ C 1 (Ω), then t

‖∇h u‖ ≤ ‖∇h u0 ‖,

∫ ‖u × Δ h u‖2 dτ ≤ 0

1 ‖∇ u ‖2 . 2α1 h 0

(1.4.5)

Proof. Multiplying the first equation of (1.4.3) by Δ h uj , we get Δ h uj ⋅ 𝜕t uj = −α1 Δ h uj ⋅ (uj × (uj × Δ h uj )) = α1 (uj × Δ h uj ) ⋅ (uj × Δ h uj )

= α1 |uj × Δ h uj |2 .

Taking the inner product in the first equation of (1.4.3) with Δ h uj yields d ‖∇ u‖2 + 2α1 ‖u × Δ h u‖2 = 0. dt h

(1.4.6)

This implies that d ‖∇ u‖2 ≤ 0, dt h so ‖∇h u‖ ≤ ‖∇h u0 ‖. The second inequality of (1.4.5) follows by integrating (1.4.6) with respect to t. Lemma 1.4.3. Assume that u is a discrete function in the interval [0, l], then for any j0 , 1 ≤ j0 ≤ J, we have ‖u‖2 ≤ 2l[|uj0 |2 + l‖∇h u‖2 ], ‖u‖2 ≤

(1.4.7) 󵄨2

󵄨󵄨 J 󵄨 󵄨󵄨 󵄨󵄨 l2 ‖∇h u‖2 + l−1 󵄨󵄨󵄨h ∑ uj 󵄨󵄨󵄨 . 󵄨󵄨 󵄨 2 󵄨 j=1 󵄨󵄨

(1.4.8)

72 | 1 Discrete attractor and approximate calculation Proof. (i) For any j0 , 1 ≤ j0 ≤ J, it is clear that j−1

uj = uj0 + h ∑ ∇h us , s=j0

thus 2

J

|uj |2 ≤ 2[|uj0 |2 + (h ∑ |∇h us |) ], s=1

(1, ∇h u) ≤ ‖1‖‖∇h u‖ = √l‖∇h u‖, yield |uj |2 ≤ 2[|uj0 |2 + l‖∇h u‖2 ]. Summing over j from 1 to J then gives J

‖u‖2 = h ∑ |uj |2 ≤ 2l[|uj0 |2 + l‖∇h u‖2 ]. j=1

(ii) Notice the following identity and its bound: i−1

i−1

s=j

s=j

|ui |2 + |uj |2 − 2ui ⋅ uj = (h ∑ ∇h us ) ⋅ (h ∑ ∇h us ) ≤ l‖∇h u‖2 . Multiplying the above inequality by h2 and then summing over i, j from 1 to J gives J

2l‖u‖2 − 2h2 ∑ ui ⋅ uj ≤ l3 ‖∇h u‖2 .

(1.4.9)

󵄨󵄨 J 󵄨󵄨2 J 󵄨󵄨 󵄨 󵄨󵄨h ∑ uj 󵄨󵄨󵄨 = h2 ∑ uj ⋅ ui , 󵄨󵄨 󵄨󵄨 󵄨󵄨 i=1 󵄨󵄨 i,j=1

(1.4.10)

i,j=1

Since

from (1.4.6) and (1.4.10), inequality (1.4.8) follows. Corollary 1.4.1. (i) Assume that u is a discrete function and there exists an integer j0 such that uj0 = 0. Then ‖u‖ ≤ √2l‖∇h u‖.

(1.4.11)

(ii) Assume that u is an l-periodic discrete function in ℝ. For any integer k > 0, we then have ‖∇hk u‖ ≤ l‖∇hk+1 u‖.

(1.4.12)

1.4 Landau–Lifschitz equations | 73

In the following we always suppose that the magnitude of the spin is finite, i. e., j |u0 | = ∑3j=1 (u0 )2 = 1. Then by Lemma 1.4.1, we have |uj | = 1, ∀t ∈ ℝ, 1 ≤ j ≤ J. Lemma 1.4.4. Assume that |u0 | = 1, then for any j, 1 uj ⋅ Δ h uj = − [∇h uj ⋅ (∇h uj + ∇h uj−1 ) 2 − ∇h uj−1 ⋅ (∇h uj − ∇h uj−1 )].

(1.4.13)

Proof. Using the assumption of the lemma, and applying Lemma 1.4.1, get ui ⋅ uj = 1 for any t ∈ ℝ+ . Acting on uj−1 ⋅ uj−1 = 1 by the operator ∇h and noticing that ∇h (uj−1 ⋅ vj−1 ) = uj ∇h vj−1 + ∇uj−1 ⋅ vj−1 ,

(1.4.14)

∇h uj−1 ⋅ uj + ∇h uj−1 ⋅ uj−1 = 0.

(1.4.15)

we have

Then acting on (1.4.15) by the operator ∇h , we get ∇h uj ⋅ ∇h uj + Δ h uj ⋅ uj + ∇h uj ⋅ ∇h uj−1 + Δ h uj ⋅ uj−1 = 0. Thus (uj + uj−1 ) ⋅ Δ h uj = −(∇h uj ⋅ ∇h uj + ∇h uj ⋅ ∇h uj−1 ), where Δ h uj = h1 (∇h uj − ∇h uj−1 ), so we get 2uj ⋅ Δ h uj = h∇h uj−1 ⋅ Δ h uj − (∇h uj ⋅ ∇h uj + ∇h uj ⋅ ∇h uj−1 ) = ∇h uj−1 (∇h uj − ∇h uj−1 ) − ∇h uj (∇h uj + ∇h uj−1 ),

and the conclusion of the lemma follows. For the norms of the discrete function u and its difference quotients ∇h u of order k ≥ 0, we take the expressions ‖∇hk u‖p

J−k

1 p

󵄨 󵄨p = (h ∑ 󵄨󵄨󵄨∇hk uj 󵄨󵄨󵄨 ) ,

‖∇hk u‖∞ =

j=1

1 ≤ p < +∞,

󵄨 󵄨 max 󵄨󵄨󵄨∇hk uj 󵄨󵄨󵄨,

j=1,2,...,J−k

where ∇hk = ∇h ⋅ ∇h ⋅ ⋅ ⋅ ∇h . Lemma 1.4.5 ([101]). For any discrete function u = {uj | j = 1, . . . , J} on the interval [0, L], 1

1

‖u‖∞ ≤ K1 ‖u‖ 2 (‖∇h u‖ + ‖u‖) 2 ,

(1.4.16)

74 | 1 Discrete attractor and approximate calculation 1

1

‖∇h u‖∞ ≤ K2 ‖u‖ 2 (‖Δ h u‖ + ‖u‖) 2 , ‖∇hk u‖p ≤ K3 ‖u‖1−

k+ 21 − p1 m

(‖∇hm u‖ + ‖u‖)

(1.4.17) k+ 21 − p1 m

,

(1.4.18)

where K1 , K2 and K3 are constants, independent of the discrete function u and step length h, 2 ≤ p ≤ +∞, 0 ≤ k < m. Lemma 1.4.6. Assume that u0 ∈ C 1 , |u0 | = 1. Then ‖∇h u‖2 ≤ ‖∇h u0 ‖2 e−ct + E(1 − e−ct ), where c =

α1 , 4π 2

(1.4.19)

E = 32π 2 K34 (27K34 ‖∇h u0 ‖2 + 1)‖∇h u0 ‖4 .

Proof. By (1.4.2) we have 𝜕t uj = −α1 (uj ⋅ Δ h uj )uj + α1 |uj |2 Δ h uj + α2 uj × Δ h uj .

(1.4.20)

Multiplying (1.4.20) by Δ h u, noticing that (Δ h u, u × Δ h u) = 0 and |u0 | = 1, we have −

1 d ‖∇ u‖2 = α1 ‖u0 ‖2∞ ‖Δ h u‖2 − α1 (Δ h u, (u ⋅ Δ h u)u). 2 dt h

(1.4.21)

By Lemma 1.4.5, we derive 1

3

‖∇h u‖4 ≤ K3 ‖∇h u‖ 4 (‖Δ h u‖ + ‖∇h u‖) 4 . Thus by Lemma 1.4.4, we have 󵄩 󵄩 α1 (Δ h u, (u ⋅ Δ h u)u) ≤ α1 ‖Δ h u‖󵄩󵄩󵄩(u ⋅ Δ h u)u󵄩󵄩󵄩 󵄩 󵄩2 ≤ 2α1 ‖Δ h u‖‖u‖∞ 󵄩󵄩󵄩‖Δ h u‖󵄩󵄩󵄩4 3

≤ 2α1 K32 [(‖Δ h u‖‖∇h u‖) 2 + ‖∇h u0 ‖2 ‖Δ h u‖]

1 27 ‖∇ u ‖4 + ‖∇ u ‖6 ). 4ε h 0 256ε3 h 0

(1.4.22)

1 d 27 1 ‖∇ u‖2 + α1 (1 − 4K32 ε)‖Δ h u‖2 ≤ 2α1 K32 ( ‖∇ u ‖6 + ‖∇h u0 ‖4 ). 2 dt h 4ε 256ε3 h 0

(1.4.23)

≤ 2α1 K32 (2ε‖Δ h u‖2 + Substituting (1.4.22) into (1.4.21), we have

Take ε = (8K32 )−1 , then use (1.4.23) to get α 1 d ‖∇h u‖2 + 1 ‖Δ h u‖2 ≤ D, 2 dt 2 where D = 4α1 K34 (27K34 ‖∇h u0 ‖2 + 1)‖∇h u0 ‖4 . From Corollary 1.4.1 (ii), we get α d ‖∇ u‖2 + 12 ‖∇h u‖2 ≤ 2D. dt h 4π Using Gronwall inequality, we obtain (1.4.19).

1.4 Landau–Lifschitz equations | 75

Lemma 1.4.7. Assume that u0 ∈ C 2 (Ω) and |u0 | = 1. Then there exist constants δ and γ, dependent on ‖u‖Hp1 (Ωh ) , but independent of the step length h, such that ‖Δ h u‖2 ≤ where γ =

α1 , 2π 2

δ (1 − e−γt ) + ‖Δ h u0 ‖2 e−γt , γ

δ = 2 187K412 (2 + 77 K44 +

(1.4.24)

3.217 α28 K44 )‖Δ h u0 ‖10 . 164 α17

Proof. Acting on (1.4.20) by Δ h and then taking scalar product with Δ h u yields (Δ h u, 𝜕t Δ h u) = −α1 (Δ h u, Δ h ((u ⋅ Δ h u)u)) + α1 (Δ h u, Δ2h u) + α2 (Δ h u, Δ h (u × Δ h u)),

(1.4.25)

where 1 d ‖Δ u‖2 , 2 dt h α1 (Δ h u, Δ2h u) = −α1 ‖∇h3 u‖2 , (Δ h u, 𝜕t Δ h u) =

(1.4.26) (1.4.27)

−α1 (Δ h u, Δ h (u ⋅ Δ h u)u) = α1 (∇h Δ h u, ∇h ((u ⋅ Δ h u)u)) 󵄩 󵄩 ≤ α1 ‖∇h3 u‖󵄩󵄩󵄩∇h ((u ⋅ Δ h u)u)󵄩󵄩󵄩.

(1.4.28)

Let 1 Qj = − [∇h uj ⋅ (∇h uj + ∇h uj−1 ) − ∇h uj−1 ⋅ (∇h uj − ∇h uj−1 )]. 2 By Lemma 1.4.4, we get J

󵄩󵄩 󵄩2 󵄨 󵄨2 󵄩󵄩∇h ((u ⋅ Δ h u)u)󵄩󵄩󵄩 ≤ h ∑ 󵄨󵄨󵄨∇h (Qj uj )󵄨󵄨󵄨 j=1

J

󵄨 󵄨2 ≤ 2h ∑(󵄨󵄨󵄨(∇h Qj )uj+1 󵄨󵄨󵄨 + |Qj ∇h uj |2 ) j=1

≤ 4(‖∇h u‖2∞ ‖∇h u‖44 + 2‖∇h u‖2∞ ‖u‖2∞ ‖Δ h u‖2 ). Thus 󵄩󵄩 󵄩2 2 󵄩󵄩∇h ((u ⋅ Δ h u)u)󵄩󵄩󵄩 ≤ 2(‖∇h u‖∞ ‖∇h u‖4 + √2‖∇h u‖∞ ‖u‖∞ ‖Δ h u‖).

(1.4.29)

By Lemma 1.4.4 and (1.4.5), we have 3

1

{ ‖∇h u‖∞ ≤ K3 ‖∇h u0 ‖ 4 (‖∇h3 u‖ + ‖∇h u0 ‖) 4 , { { { { 1 1 ‖Δ h u‖∞ ≤ K3 ‖∇h u0 ‖ 2 (‖∇h3 u‖ + ‖∇h u0 ‖) 2 , { { { { 1 { 7 2 3 4 {‖∇h u‖∞ ≤ K3 ‖∇h u0 ‖ 4 (‖∇h u‖ + ‖∇h u0 ‖) .

(1.4.30)

76 | 1 Discrete attractor and approximate calculation Substituting (1.4.30) into (1.4.29), we get 1 10 󵄩 󵄩󵄩 2 3 󵄩󵄩∇h ((u ⋅ Δ h u)u)󵄩󵄩󵄩 ≤ 2K3 (K3 ‖∇h u0 ‖ 4 (‖∇h u‖ + ‖∇h u0 ‖) 2 3 5 󵄩 󵄩 + √2‖∇h u0 ‖ 4 (󵄩󵄩󵄩∇h3 u󵄩󵄩󵄩 + ‖∇h u0 ‖) 4 ).

(1.4.31)

By (1.4.28) and (1.4.31), it follows that − α1 (Δ h u, Δ h ((u ⋅ Δ h u)u)) ≤ 4ε‖∇h3 u‖2 + C1 ,

(1.4.32)

where C1 =

77 α14 K34 α14 K312 )‖∇h u0 ‖10 + α12 K43 ‖∇h u0 ‖4 /ε(K32 ‖∇h u0 ‖2 + 2). (27 + 256ε4 16ε3

This completes the estimate of the term −α1 (Δ h u, Δ h ((u ⋅ Δ h u)u)) because a2 (Δ h u, Δ h (u × Δ h u)) = −α2 (∇h Δ h u, ∇h (u × Δ h u))

= −α2 (∇h Δ h u, ∇h u × Δ h u + u × ∇h Δ h u)

= −α2 (∇h Δ h u, ∇h u × Δ h u) 󵄩 󵄩 ≤ 3|α2 |󵄩󵄩󵄩∇h3 u󵄩󵄩󵄩‖∇h u‖∞ ‖Δ h u‖.

(1.4.33)

α2 (Δ h u, Δ h (u × Δ h u)) ≤ 2ε‖∇h3 u‖2 + C2 ,

(1.4.34)

By (1.4.30)–(1.4.31), we have

77 ⋅38 α8 K 16

where C2 = 88 ε27 3 ‖∇h u0 ‖10 + (1.4.34) into (1.4.25), we get

9α22 2 K ‖∇ u ‖3 . Substituting (1.4.26), (1.4.27), (1.4.32) and 4ε 3 h 0

1 d 󵄩 󵄩2 ‖Δ u‖2 + (α1 − 6ε)󵄩󵄩󵄩∇h3 u󵄩󵄩󵄩 ≤ C3 , 2 dt h

(1.4.35)

where C3 = C1 + C2 . By Corollary 1.4.1 (ii), from (1.4.35) we have α − 6ε 1 d ‖Δ u‖2 + 1 2 ‖Δ h u‖2 ≤ C3 . 2 dt h 2π Taking ε =

α1 , 12

(1.4.36)

(1.4.36) can then be rewritten as α d ‖Δ u‖2 + 12 ‖Δ h u‖2 ≤ 2C3 , dt h 2π

(1.4.37)

where C3 = 12α1 K34 ‖∇h u0 ‖4 (K32 ‖∇h u0 ‖2 + 2) + + 4 ⋅ 36 α1 K312 (1 + 3 ⋅ 77 K34 +

33 α22 K34 ‖∇h u0 ‖3 α1

39 ⋅ 77 K34 α28 46 α1

By Gronwall lemma, the proof of the lemma is completed.

)‖∇h u0 ‖10 .

1.4 Landau–Lifschitz equations | 77

By the above lemma we can use a common method to prove 2 Theorem 1.4.1. For u0 ∈ Hper (Ωh ) with |u0 | = 1, there exists a unique solution of (1.4.3) which satisfies 2 u ∈ Hper (Ωh ) ∩ {u; |u| = 1, ‖u‖ = √2π} 2 2 and the mapping u0 ∈ Hper (Ωh ) → u(t) ∈ Hper (Ωh ) defines a continuous semigroup S(t).

Theorem 1.4.2. Let Ω = [0, 2π], |u0 | = 1, u0 ∈ H 2 (Ωh ), ‖∇h u0 ‖ ≤ r0 , r0 ∈ ℝ+ . Then the semigroup associated with (1.4.3) possesses a bounded attractor A ⊂ U , where 2 2 U = {u ∈ Hper (Ωh ); |u| = 1, ‖∇h u‖ + ‖Δ h u0 ‖ ≤ r0 }, the attractor A is bounded in Hper (Ωh ) and it is convex and connected. Proof. By Lemmas 1.4.2 and 1.4.6, there exists a bounded attracting set 2

B0 = {u ∈ Hper (Ωh ); ‖u‖Hper 2 (Ω ) ≤ ρ(r0 )}, h 1

where ρ0 (r0 ) = (2π + r02 + δγ ) 2 . In particular, for any ball B(0, R) ⊂ U , as ρ󸀠 ≥ ρ(r0 ), there exists t0 (B, ρ󸀠 ) such that

S(t)B ⊂ B(0, ρ󸀠 ),

t ≥ t0 =

R2 − δ/γ 1 . ln 󸀠 2 γ (ρ ) − δ/γ

By Theorem 1.1 in [91], we have that A = ω(B0 ) is a compact attractor which attracts the bounded set U ; A is convex and connected. Remark 1.4.1. By Theorem 1.4.2, we know that the LL equation possesses a local attractor and the attractor possesses very special properties, which are that the norm is 1 conserved in L2per (Ωh ), the norm is not increasing in Hper (Ωh ), and the norm is decreas2 ing in Hper (Ωh ). Next we give the estimates of dimensions for the attractor. We consider the linear variation equation of (1.4.3) given by d { Uj (t) + L(uj (t))Uj = 0, { { dt { { Uj+rJ = Uj (t), 1 ≤ j ≤ J, r = 0, ±1, ±2, . . . , { { { {Uj (0) = U0 (xj ),

(1.4.38)

where L(u(t)) is defined by L(uj (t))Uj = −(α1 + α2 A(uj ))Δ h Uj − 2α1 (∇h uj ⋅ uj )∇h Uj + (α1 (uj ⋅ Δ h uj ) − α2 A(Δ h uj ))Uj ,

0 ≤ j ≤ J,

(1.4.39)

78 | 1 Discrete attractor and approximate calculation and the 3 × 3 matrix A(uj ) which is derived from uj × Δ h uj has the following form: 0 [ 3 [ uj 2 [−uj

−u3j 0 u1j

u2j ] −u1j ] , 0 ]

u = S(t)u0 is the solution of (1.4.3). Lemma 1.4.8. The skew-symmetry matrix A(u) has the following properties: (1) AT (u) = −A(u); (2) (A(u)v, v) = 0; (3) A(u)u = 0; (4) (v, A(u)w) = −(A(u)v, w); where AT (u) is the transpose of matrix A(u). Lemma 1.4.9. For the skew-symmetry matrix A(u), ∀u ∈ ℝ3 , we have 󵄩󵄩 󵄩 󵄩󵄩A(u)󵄩󵄩󵄩 ≤ 3‖u‖, where ‖A(u)‖ = supv∈ℝ3 \{0}

‖A(u)v‖ . ‖v‖

Since the solution u of the problem (1.4.3) is suitably smooth, we can prove that the linear periodic initial and boundary value problem (1.4.38) possesses a unique global 2 solution U(t) ∈ L∞ ([0, T]; Hper (Ωh )). It can be shown that G(t) is actually the tangent map of S(t) at the point u0 . Proposition 1.4.1. The semigroup S(t) of the problem (1.4.3) is Fréchet differentiable in L2 (Ωh ) with differential L(t, u0 ) : U0 → U(t) ∈ L2 (Ωh ), where U(t) is the solution of (1.4.38). Proof. Set w = S(t)u1 − S(t)u0 − G(t; u0 )U0 = w1 (t) − u(t) − U(t). Thus we have 𝜕t w(t) = F(w1 (t)) − F(u(t)) + L(u(t))U(t) { { { = F(u(t) + U(t) + w(t)) − F(u(t)) + L(u(t))U(t), { { { {w(0) = 0.

(1.4.40)

The first equation of (1.4.2) at uj equals to duj (t) dt

= F(uj (t)).

(1.4.41)

Thus the first equation of (1.4.40) can be written in the form 𝜕t w(t) + L(u(t))w(t) = Λ(u, U, w),

(1.4.42)

1.4 Landau–Lifschitz equations | 79

where Λ(u, U, w) = F(u(t) + U(t) + w(t)) − F(u(t)) + L(u(t))(U(t) + w).

(1.4.43)

Taking the scalar product of (1.4.42) with w, we find that 1 d 󵄩2 󵄩 ‖w‖2 + (L(u(t))w(t), w(t)) ≤ C(‖w‖2 + 󵄩󵄩󵄩Λ(u, U, w)󵄩󵄩󵄩 ). 2 dt

(1.4.44)

Now we estimate every term in (1.4.44). First, − α1 (Δ h w, w) = α1 ‖∇h w‖2 .

(1.4.45)

By Lemma 1.4.8, we have α2 (A(u)Δ h w, w) = −α2 (∇h w, ∇h (A(u)u))

= α2 (∇h w, A(∇h u)w) + α2 (∇h w, A(u)∇h w, ) ≤ C‖w‖2 + ε‖∇h w‖2 , 2

2

2α1 ((∇h u ⋅ u)∇h w, w) ≤ C‖w‖ + ε‖∇h w‖ .

(1.4.46) (1.4.47)

By Lemma 1.4.4, we find − α1 ((u ⋅ Δ h u)w, w) + α2 (A(Δ h u)w, w) ≤ C‖w‖2 + ε‖∇h w‖2 .

(1.4.48)

By (1.4.33) and due to L(u(t)) = −F 󸀠 (u(t)), we have 󵄩2 󵄩󵄩 󵄩 󵄩 2 󵄩󵄩Λ(u, U, w)󵄩󵄩󵄩 ≤ C‖U + w‖ = C 󵄩󵄩󵄩S(t)u1 − S(t)u0 󵄩󵄩󵄩 ≤ C‖u1 − u0 ‖2 .

(1.4.49)

By (1.4.44)–(1.4.49), we find 1 d ‖w‖2 + (α1 − 3ε)‖∇h w‖2 ≤ C‖w‖2 + D‖u1 − u0 ‖4 . 2 dt

(1.4.50)

By Gronwall inequality, we obtain 󵄩󵄩 󵄩 2 󵄩󵄩w(t)󵄩󵄩󵄩 ≤ C‖u1 − u0 ‖ ,

t ∈ [0, T].

(1.4.51)

Therefore, lim

sup

δ→0 u ,u ∈L2 (Ω 0 1 h p ‖u0 −u1 ‖ π2 , x ∈ (0, π2 ], we have

2α m α1 m α m(m + 1)(2m + 1) . ∑ λk ≥ 21 ∑ k 2 = 1 2 k=1 π k=1 3π 2

(1.4.62)

Substituting (1.4.62) into (1.4.61), we find tr(L(S(τ)u0 ) ⋅ Qm (τ)) ≥

3π 2 C1 mα1 [(m + 1)(2m + 1) − ]. α1 3π 2

(1.4.63)

Set m0 = 41 (√1 + 24π 2 C1 /α1 − 3). Then if m ≥ m0 , tr(L(S(τ)u0 ) ⋅ Qm (τ)) ≥ 0.

(1.4.64)

Thus we have Theorem 1.4.3. Under the assumptions of Theorem 1.4.2, the Hausdorff and fractal dimensions of the attractor A of the discrete system (1.4.3) are finite, in fact, dH (A ) ≤ m0 ,

dF (A ) ≤ 2m0 ,

where 1 (√1 + 24π 2 C1 /α1 − 3), 4 9α2 ‖∇ u ‖2 C1 = 2 h 0 + (4α1 + 3|α2 |)‖∇h u0 ‖2∞ + 3|α2 |‖Δ h u0 ‖. α1

m0 =

82 | 1 Discrete attractor and approximate calculation

1.5 Nonlinear Galerkin method The usual computing method is not appropriate in the numerical computation of chaos, inertial manifold and global attractor as t → ∞. This is because it has the error estimate of the form c(h)eT , where c(h) is an appropriate small constant related to h. In theory, it is necessary to establish the error estimate and convergence of the approximate solution. A new numerical method – nonlinear Galerkin method – has come into being, and shown great superiority in long time numerical computations. In this section we will use a class of nonlinear evolution equations as examples to illustrate this method. Suppose that the nonlinear evolution equation has the following form: du = −νAu − R(u), dt

(1.5.1)

R(u) = B(u) + Cu − f ,

(1.5.2)

where

ν > 0 is an viscous parameter, the operator A is a linear unbounded self-adjoint operator in a Hilbert space, A is positive and closed, D(A) is dense in H. Define the powers 1 of A for s ∈ ℝ; D(As ) has the norm |As ⋅ | and is a Hilbert space. Define V = D(A 2 ), 1 ‖ ⋅ ‖ = |A 2 ⋅ |. Because A−1 is compact and self-adjoint, there exists an orthogonal basis {ωj } of H, it is composed of eigenvectors of A: Aωj = λj ωj ,

(1.5.3)

where 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅, λj → ∞, j → ∞. The nonlinear term R(u) satisfies (1.5.2), where B(u) = B(u, u), B(⋅, ⋅) is a bilinear operator from V × V to V; C is a linear operator from V to H, f ∈ H. Let b be the triplet form on V b(u, v, ω) = (B(u, v), ω)V 󸀠 ,V ,

∀u, v, ω ∈ V.

Set b(u, v, ω) = −b(u, ω, v), ∀u, v, ω ∈ V, 1 1 1 1 󵄨󵄨 󵄨 󵄨󵄨b(u, v, ω)󵄨󵄨󵄨 ≤ C1 |u| 2 ⋅ ‖u‖ 2 ⋅ ‖v‖ ⋅ |ω| 2 ‖ω‖ 2 , |Cu| ≤ C2 ‖u‖, ∀u ∈ V,

(1.5.4) ∀u, v, ω ∈ V,

(1.5.5) (1.5.6)

where C1 , C2 and subsequent Ci , i > 2, denote positive constants. Suppose that the mapping B : V × D(A) → B satisfies 1 1 1 1 󵄨󵄨 󵄨 󵄨󵄨B(u, v)󵄨󵄨󵄨 ≤ C3 |u| 2 ⋅ ‖u‖ 2 ⋅ ‖v‖ 2 ⋅ |Au| 2

(1.5.7)

1.5 Nonlinear Galerkin method | 83

1 1 󵄨 󵄨󵄨 󵄨󵄨B(u, v)󵄨󵄨󵄨 ≤ C4 |u| 2 ⋅ |Au| 2 ⋅ ‖v‖,

∀u ∈ V, v ∈ D(A).

(1.5.8)

Finally, νA + C is positive, and then ((νA + C)u, u) ≥ α‖u‖2 ,

∀u ∈ D(A),

(1.5.9)

where α > 0. For (1.5.1), we consider the Cauchy initial value problem with u(0) = u0 ,

u0 ∈ H.

(1.5.10)

We can prove that problem (1.5.1), (1.5.10) has a unique solution u(t), t > 0, and u ∈ ℒ(ℝ+ ; H) ∩ L2 ([0, T]; V),

∀T > 0.

Furthermore, if u0 ∈ V, we have u ∈ ℒ(ℝ+ ; H) ∩ L2 ([0, T]; D(V)),

∀T > 0.

Now we seek to an approximate solution of problem (1.5.1), (1.5.10) of the following form: m

um (t) = ∑ gjm (t)wj , j=1

+

um : ℝ → wm = span{w1 , w2 , . . . , wm }. We seek to the unknown functions um and zm , where 2m

zm (t) = ∑ hjm (t)wj , j=m+1

̃m = span{wm+1 , . . . , w2m }. zm : ℝ+ → w Then (um , zm ) satisfies d (u , v) + ν((um , v)) + (Cum , v) + b(um , um , v) dt m + b(zm , um , v) + b(um , zm , v) = (f , v), ∀v ∈ wm ,

v((zm , ṽ)) + (czm , ṽ) + b(um , um , ṽ) = (f , ṽ),

̃m ∀ṽ ∈ w

(1.5.11) (1.5.12)

and um (0) = Pm u0 , where Pm denotes the orthogonal projection of H onto wn .

(1.5.13)

84 | 1 Discrete attractor and approximate calculation Equations (1.5.11)–(1.5.12) are equivalent to an ordinary differential equation for um . In fact, (1.5.12) is linear in zm and can be written as νAzm + (P2m − Pm )Czm = (P2m − Pm )(f − B(um )).

(1.5.14)

Because of assumption (1.5.9), the operator νA + (P2m − Pm )C is coercive and invertible, and then the explicit solution zm is −1

zm = (νA + (P2m − Pm )C) (P2m − Pm )(f − B(um )).

(1.5.15)

Therefore, (1.5.11)–(1.5.12) are equivalent to the ordinary differential equation dum + νAum + Pm (Cum + B(um ) + B(um , zm )) = Pm f , dt

(1.5.16)

where zm is given by (1.5.15). Obviously, when z0 = 0, this is classical Galerkin method. The existence and uniqueness of the solution um (t) to (1.5.16), (1.5.13) can be obtained by the standard ordinary differential equation theory. In the following a priori estimate, we know Tm = ∞. Therefore we can consider the convergence of approximate solutions as m → ∞. Theorem 1.5.1. Suppose that (1.5.4), (1.5.9) are satisfied, u0 is given in H. Then the solution of problem (1.5.16), (1.5.13) converges to the solution u0 of problem (1.5.1), (1.5.10) as m → ∞, that is, um → u strongly in L2 ([0, T]; V) and Lp ([0, T]; H) for all T > 0, 1 ≤ p < +∞, ∞

um is weakly ∗ convergent to u in L (ℝ+ ; H).

and (1.5.17)

In order to prove Theorem 1.5.1, we need to make an priori estimate of a solution to the problem (1.5.11)–(1.5.12). Setting v = um in (1.5.11) and ṽ = zm in (1.5.12), we add the two equations, and by (1.5.14) we have 1 d |u |2 + ν‖um ‖2 + (Cum , um ) + ν‖zm ‖2 + (Czm , zm ) = (f , um + zm ). 2 dt m

(1.5.18)

From (1.5.9), we get 1 d |u |2 + α(‖um ‖2 + ‖zm ‖2 ) ≤ |f | ⋅ |um + zm |. 2 dt m

(1.5.19)

Since 1 󵄨 1 󵄨 ‖v‖ = 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨 ≥ λ12 |v|,

∀v ∈ V,

equation (1.5.19) implies 1 d −1 |um |2 + α(‖um ‖2 + ‖zm ‖2 ) ≤ λ1 2 |f | ⋅ ‖um + zm ‖ 2 dt

(1.5.20)

1.5 Nonlinear Galerkin method | 85

1 α (‖um ‖2 + ‖zm ‖2 ) + |f |2 , 2 αλ1 d 2 |u |2 + α1 |um |2 ≤ |f |2 . dt m αλ1 ≤

(1.5.21)

Integrating we have 2 󵄨2 −αλ t 2|f | 󵄨2 󵄨 󵄨󵄨 −αλ t 󵄨󵄨um (t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨um (0)󵄨󵄨󵄨 e 1 + 2 2 (1 − e 1 ), λ1 α

∀t ≥ 0.

Therefore, the sequence um is bounded in L∞ (ℝ+ ; H) as m → +∞.

(1.5.22)

Integrating (1.5.21) with respect to t from [0, T], we have ∀T > 0,

um and zm are bounded in L2 ([0, T]; V).

(1.5.23)

Now we will make an a priori estimate of zm . Take ṽ = zm in (1.5.12). Then ν‖zm ‖2 + (Czm , zm ) = −b(um , um , zm ) + (f , zm ). Equations (1.5.9), (1.5.7) imply 󵄨 󵄨 α‖zm ‖2 ≤ 󵄨󵄨󵄨B(um )󵄨󵄨󵄨 ⋅ |zm | + |f | ⋅ |zm | 1

1

≤ C3 |um | 2 ⋅ |Aum | 2 ⋅ |zm | + |f | ⋅ |zm |.

(1.5.24)

̃m , we have Since um ∈ wm , zm ∈ w 1

1

|Aum | ≤ λm2 ‖um ‖, ‖um ‖ ≤ λm2 |um |,

(1.5.25)

|Azm | ≥ λm+1 ‖zm ‖, ‖zm ‖ ≥ λm+1 |zm |.

(1.5.26)

1 2

1 2

Combining these inequalities, by (1.5.24) we have αλm+1 |zm |2 ≤ C3 λm |um |2 ⋅ |zm | + |f | ⋅ |zm |,

αλm+1 |zm |2 ≤ C3 λm |um |2 + |f |.

(1.5.27)

Equation (1.5.22) implies zm is bounded in L∞ (ℝ+ ; H) as m → ∞. It follows from (1.5.25), (1.5.26) and (1.5.24) that 1

αλm−1 |zm | ≤ C3 λm2 |um | ⋅ ‖um ‖ + |f |.

(1.5.28)

86 | 1 Discrete attractor and approximate calculation Since λ1 ≤ λm ≤ λm+1 , we have 1

−1

2 αλm+1 |zm | ≤ C3 |um | ⋅ ‖um ‖ + λ1 2 ⋅ |f |.

From this inequality, (1.5.22) and (1.5.23), we obtain 1

2 zm is bounded in L2 ([0, T]; H) as m → +∞. λm+1

∀T > 0,

Now we will estimate

dum . dt

(1.5.29)

By (1.5.4) and (1.5.5), we have

1 1 1 1 󵄩󵄩 󵄩 󵄩󵄩B(u, v)󵄩󵄩󵄩V 󸀠 ≤ C1 |u| 2 ‖u‖ 2 |v| 2 ‖v‖ 2 ,

∀u, v ∈ V.

Therefore, we deduce from the estimates (1.5.22), (1.5.23) and (1.5.28) that B(um ), B(zm , um ) and B(um , zm ) are bounded in L2 ([0, T]; V 󸀠 ). So from the differential equation (1.5.14) we find that ∀T > 0,

dum is bounded in L2 ([0, T]; V 󸀠 ) as m → ∞. dt

(1.5.30)

Now we will prove the convergence of approximate solution um as m → ∞. Since λm → +∞ as m → ∞, from (1.5.29) we deduce ∀T > 0,

zm strongly ∗ converges to 0 in L2 ([0, T]; H) as m → ∞.

(1.5.31)

Therefore, (1.5.23) and (1.5.28) imply ∀T > 0,

zm weakly converges to 0 in L2 ([0, T]; V) as m → ∞;

and is weakly ∗ convergent in L∞ (ℝ+ , H).

(1.5.32)

Now we consider the convergence of sequence um . We deduce from the estimates (1.5.22), (1.5.23) and (1.5.30) that there exists an element u∗ and a subsequence with m󸀠 → +∞ such that um󸀠 weakly converges to u∗ in L∞ ([0, T]; H), ∞

and is weakly ∗ convergent in L (ℝ+ , H), dum dt

∀T > 0; m󸀠 → +∞.

(1.5.33)

weakly converges to du ∈ L2 ([0, T]; V 󸀠 ), ∀T > 0, as m󸀠 → +∞. From the classical dt compactness principle and (1.5.33), we get ∗

∀T > 0,

um󸀠 → u∗ is strongly convergent in L∞ ([0, T]; H),

m󸀠 → +∞.

(1.5.34)

By (1.5.31)–(1.5.34), we can take the limit in (1.5.11), the only difficulty is the bilinear term. Fix v ∈ wm and m󸀠 ≥ m. Then (1.5.4) implies b(um󸀠 , um󸀠 , v) = −b(um󸀠 , v, um󸀠 ).

1.5 Nonlinear Galerkin method | 87

From (1.5.7), b(⋅, v, ⋅) is the bilinear continuous form from V × H to ℝ. Thus, we deduce from (1.5.33)–(1.5.34) that b(um󸀠 , v, um󸀠 ) strongly converge to b(u∗ , v, u∗ ) in L󸀠 (0, T), ∀T > 0, as m󸀠 → +∞. Therefore, b(um󸀠 , um󸀠 , v) strongly converges to b(u∗ , u∗ , v) in L󸀠 (0, T),

∀t > 0, m󸀠 → +∞.

Similarly, b(zm󸀠 , um󸀠 , v) → b(0, u∗ , v) = 0,

b(um󸀠 , zm󸀠 , v) → b(u∗ , 0, v) = 0,

m󸀠 → +∞, m󸀠 → +∞,

where all sequences are all strongly convergent in L󸀠 (0, T), ∀T > 0. Therefore, we can find the limit function u∗ such that d ∗ (u , v) + ν((u∗ , v)) + (cu∗ , v) + b(u∗ , u∗ , v) = (f , v) dt

(1.5.35)

for all v ∈ wm . By the continuity, this holds ∀v ∈ V. Furthermore, by (1.5.33) we have um󸀠 (0) weakly converges to u∗ (0) in H.

(1.5.36)

Since um󸀠 (0) = Pm󸀠 u0 , from (1.5.36) we deduce that u∗ (0) = u0 .

(1.5.37)

We find that u∗ is the solution to the problem (1.5.1), (1.5.10) through (1.5.35), (1.5.36), so u∗ = u. By (1.5.33) we obtain that the sequence um converges to u. In order to complete the proof of Theorem 1.5.1, we have to verify the strong convergence in (1.5.17). To this end, we introduce the expression T

1󵄨 󵄨2 Xm = 󵄨󵄨󵄨um (T) − u(T)󵄨󵄨󵄨 + ∫{ν‖um − u‖2 2 0

+ (c(um − u), um − u) + ν‖zm ‖2 + (czm , zm )}dt. We have to prove lim Xm = 0.

(1.5.38)

m→+∞

This will complete the proof of Theorem 1.5.1. In fact, (1.5.9), (1.5.38) imply that um strongly converges to u in L2 (0, T; V); moreover, we have um (t) strongly converges to u(t) in H,

∀t ≥ 0.

(1.5.39)

We find that um strongly converges to u in Lp ([0, T]; H), ∀p ∈ [1, ∞) from (1.5.39), (1.5.22) and Lebesgue dominated convergence theorem. Besides the strong convergence result of um in (1.5.17), we note that (1.5.38) also implies zm → 0 strongly in L2 ([0, T]; V),

∀T ≥ 0, m → +∞.

(1.5.40)

88 | 1 Discrete attractor and approximate calculation Now we show that (1.5.38) holds true. Integrating (1.5.18) from 0 to T, we obtain T

1 󵄨󵄨 󵄨2 2 2 󵄨u (T)󵄨󵄨 + ∫{ν‖um ‖ + (cum , um ) + ν‖zm ‖ + (czm , zm )}dt 2󵄨 m 󵄨 0

T

1󵄨 󵄨2 = 󵄨󵄨󵄨um (0)󵄨󵄨󵄨 + ∫(f , um + zm )dt. 2 0

Then Xm can be rewritten as 1󵄨 󵄨2 Xm = −(um (T), u(T)) + 󵄨󵄨󵄨u(T)󵄨󵄨󵄨 + 2 T

1 󵄨󵄨 󵄨2 󵄨u (0)󵄨󵄨 2󵄨 m 󵄨

+ ∫{−2ν((um , v)) + ν‖u‖2 (−cu, um − u) 0

− (cum , u) + (f , um + zm )}dt.

(1.5.41)

Taking advantage of (1.5.31), (1.5.33) and passing to the limit in (1.5.41), we get 1󵄨 󵄨2 1 lim Xm = − 󵄨󵄨󵄨u(T)󵄨󵄨󵄨 + |u0 |2 2 2

m→+∞

T

+ ∫{−ν‖u‖2 − (cu, u) + (f , u)}dt. 0

Using u, u instead of u∗ , v, we find that the limit in of above equality is 0 by (1.5.35). Finally, we obtain (1.5.38). The proof of Theorem 1.5.2 is completed. In order to improve the convergence of the nonlinear Galerkin method in a stronger topology, we need to prove the following theorem: Theorem 1.5.2. Suppose that (1.5.4)–(1.5.9) are satisfied, and, for given u0 ∈ V, the solutions of problem (1.5.16), (1.5.13) converge to the solution of (1.5.1), (1.5.10) as m → +∞ in the following sense: um → u strongly in L2 ([0, T]; D(A)) and in Lp ([0, T]; V), T > 0, 1 ≤ p < ∞, um → u weakly ∗ in L∞ (ℝ+ ; V).

(1.5.42)

Proof. The proof of theorem relies on further a priori estimates of um and zm . We first prove them in L∞ (ℝ+ ; V). Set v = Aum in (1.5.11) and ṽ = Azm in (1.5.12). Adding the corresponding terms, we derive 1 d ‖u ‖2 + ν|Aum |2 + ν|Azm |2 2 dt m = (f , A(um + zm )) − (cum , Aum ) − (Czm , Azm ) − b(um , um , Aum ) − b(zm , um , Aum ) − b(um , zm , Aum ) − b(um , um , Azm ).

(1.5.43)

1.5 Nonlinear Galerkin method | 89

We make the following estimates for every term on the right-hand side of (1.5.43): 6 2 󵄨 ν 󵄨󵄨 2 2 󵄨󵄨(f , A(um + zm ))󵄨󵄨󵄨 ≤ (|Aum | + |Azm | ) + |f | ; 12 ν

(1.5.44)

and equation (1.5.6) implies 󵄨 󵄨󵄨 󵄨󵄨(Cum , Aum )󵄨󵄨󵄨 ≤ C2 ‖um ‖ ⋅ |Aum | 3C 2 ν ≤ |Aum |2 + 2 ‖um ‖2 , 12 ν 󵄨󵄨 󵄨󵄨 󵄨󵄨(Czm , Aum )󵄨󵄨 ≤ C2 ‖zm ‖ ⋅ |Azm | ≤

3C 2 ν |Aum |2 + 2 ‖um ‖2 . 12 ν

(1.5.45)

(1.5.46)

We use (1.5.7) to estimate the bound of the trilinear term, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨b(um , um , Aum )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨B(um , um )󵄨󵄨󵄨 ⋅ |Aum | 1

3

≤ C3 |um | 2 ⋅ ‖um ‖ ⋅ |Aum | 2 C ν ≤ |Aum |2 + 5 |um |2 ⋅ ‖um ‖4 , 12 ν

(1.5.47)

where C5 is an absolute constant. Similarly, we obtain C 󵄨󵄨 󵄨 ν 2 2 2 2 󵄨󵄨b(zm , um , Aum )󵄨󵄨󵄨 ≤ |Aum | + 5 |zm | ⋅ ‖zm ‖ ⋅ ‖um ‖ , 12 ν 1 1 1 1 󵄨󵄨 󵄨 󵄨󵄨b(um , zm , Aum )󵄨󵄨󵄨 ≤ C3 |um | 2 ‖um ‖ 2 ‖zm ‖ 2 |Azm | 2 ⋅ |Azm | C 12 ≤ |Aum |2 + 6 |um | ⋅ ‖um ‖ ⋅ ‖zm ‖ ⋅ |Azm | ν ν C ν ν 2 ≤ |Aum | + |Azm |2 + 27 |um |2 ⋅ ‖um ‖2 ⋅ ‖zm ‖2 . 12 12 ν

(1.5.48)

The estimate of the last item on the right-hand side of (1.5.43) is 1 1 󵄨󵄨 󵄨 󵄨󵄨b(um , um , Azm )󵄨󵄨󵄨 ≤ C3 |um | 2 ⋅ ‖um ‖ ⋅ |Aum | 2 ⋅ |Azm | C ν ν ≤ |Aum |2 + |Azm |2 + 27 |um |2 ⋅ ‖um ‖4 . 12 12 ν

(1.5.49)

Combining with the above inequality, (1.5.43) implies d ‖u ‖2 + ν|Aum |2 + ν|Azm |2 dt m 6C 2 12 ≤ |f |2 + 2 ‖zm ‖ ν ν + C8 ‖um ‖2 ⋅ (1 + |um |2 ⋅ ‖um ‖2 + |zm |2 ⋅ ‖zm ‖2 + |um |2 ⋅ ‖zm ‖2 ),

(1.5.50)

90 | 1 Discrete attractor and approximate calculation where C8 = C8 (ν) depends on ν. Equation (1.5.50) can be rewritten as the following differential inequality dym ≤ gm ym + hm , dt

(1.5.51)

where 󵄩2 󵄩 ym (t) = 󵄩󵄩󵄩um (t)󵄩󵄩󵄩 ,

2 12 2 6C2 |f | + ‖zm ‖2 , ν ν 󵄨 󵄨2 󵄩 󵄩2 gm (t) = C8 (1 + 󵄨󵄨󵄨um (t)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩um (t)󵄩󵄩󵄩 󵄨 󵄨2 󵄩 󵄩2 󵄨 󵄨2 󵄩 󵄩2 + 󵄨󵄨󵄨zm (t)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩zm (t)󵄩󵄩󵄩 + 󵄨󵄨󵄨um (t)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩zm (t)󵄩󵄩󵄩 ).

hm (t) =

(1.5.52)

Integrating (1.5.51), we have t

ym (t) ≤ ym (0) exp(∫ gm (s)ds) t

0

t

+ ∫ hm (s) exp(∫ gm (σ)dσ)ds,

∀t ≥ 0.

(1.5.53)

s

0

Combining this inequality with (1.5.22)–(1.5.23) and (1.5.28), we find that um is bounded in L∞ ([0, T]; V) for all T > 0. However, the boundedness of ‖um ‖2 in ℝ+ can be obtained by the following uniform Gronwall inequality: Lemma 1.5.1. Suppose that g(t), h(t), y(t) are three locally integrable functions in (t0 , +∞), which satisfy dy ∈ L1loc ([t0 , ∞]), dt

dy ≤ gy + h, dt

t+1

t+1

∫ g(s)ds ≤ a1 ,

∫ h(s)ds ≤ a2 ,

t

t

t+1

∫ y(s)ds ≤ a3 ,

t ≥ t0 , (1.5.54)

t ≥ t0 ,

t

where ai are positive constants (i = 1, 2, 3). Then y(t) ≤ (a3 + a2 ) exp(a1 ),

∀t ≥ t0 + 1.

(1.5.55)

Now we consider (1.5.51). Because of the previous a priori estimate, the assumptions of Lemma 1.5.1 hold. Since um , zm are bounded in L∞ (ℝ+ ; H), integrating (1.5.21)

1.5 Nonlinear Galerkin method | 91

with respect to t from t to t + 1, we then have t+1

∫ (‖um ‖2 + ‖zm ‖2 )ds ≤ C8󸀠 ,

(1.5.56)

t

where the constant C8󸀠 does not depend on m. The functions ym , gm , hm satisfy (1.5.54), where the constants a1 , a2 , a3 are all independent on m. Therefore, (1.5.55) implies 󵄩 󵄩2 ym (t) = 󵄩󵄩󵄩um (t)󵄩󵄩󵄩 ≤ C9 ,

∀t ≥ 1,

(1.5.57)

where C9 = C9 (ν) does not depend on m. Therefore, (1.5.57) implies a uniform bound of ‖um (t)‖ for t ≥ 1. However, (1.5.53) implies a uniform bound of ‖um (t)‖ for t ∈ [0, 1]. Hence, um (t) is bounded in L∞ (ℝ+ ; H),

m → +∞.

(1.5.58)

Integrating (1.5.50), we obtain ∀T > 0,

um (t) and zm are bounded in L2 ([0, T]; D(A)),

m → +∞.

(1.5.59)

Next we estimate zm , and hope to obtain an estimate similar to (1.5.59). Choosing ṽ = Azm in (1.5.12), we get ν|Azm |2 = −(Czm , Azm ) − b(um , um , Azm ) + (f , Azm ) 1

1

≤ C2 ‖zm ‖ ⋅ |Azm | + C3 |um | 2 ⋅ ‖um ‖ ⋅ |Aum | 2 |Azm | + |f | ⋅ |Azm |, 1

1

1

ν|Azm | ≤ C2 |um | 2 + C3 |um | 2 ⋅ ‖um ‖|Aum | 2 + |f |, then (1.5.25)–(1.5.26) imply 1

1

3

1

2 νλm+1 ‖zm ‖ ≤ C2 ‖zm ‖ + C3 λm4 |um | 2 ‖um ‖ 2 + |f |.

For any m, which is sufficiently large, we have ‖zm ‖ ≤

1 2

1

νλm+1 − C2

1

1

3

(C3 λm4 |um | 2 ‖um ‖ 2 + |f |).

Combining this inequality with (1.5.58), we obtain zm → 0 strongly in L(ℝ+ ; H),

m → +∞.

(1.5.60)

du

At last we need to estimate dtm . Equations (1.5.7) and (1.5.58)–(1.5.60) imply that B(um ), B(zm , um ) and B(um , zm ) are uniformly bounded in L4 ([0, T]; H). However, Aum is uniformly bounded in L2 ([0, T]; H). Therefore, (1.5.16) implies ∀T > 0,

dum is bounded in L2 ([0, T]; H), dt

m → +∞.

(1.5.61)

92 | 1 Discrete attractor and approximate calculation The convergence result comes from the estimates (1.5.58)–(1.5.61). First, we combine previous convergence result (1.5.17) to obtain um → u weakly in L2 ([0, T]; D(A)),

∀T > 0,

(1.5.62)

∞

um → u weakly ∗ in L (ℝ+ ; V), dum du → weakly in L2 ([0, T]; H), ∀T > 0, dt dt zm → 0 weakly in L2 ([0, T]; D(A)), ∀T > 0,

(1.5.63) (1.5.64) (1.5.65)

as m → +∞. This implies the weak convergence of (1.5.47). In order to obtain the strong convergence result of (1.5.52), we introduce the expression T

1󵄩 󵄩2 Ym = 󵄩󵄩󵄩um (T) − u(T)󵄩󵄩󵄩 + ν ∫(|Aum − Au|2 + |Azm |2 )dt. 2 0

We only need to prove that lim Ym = 0.

m→+∞

In fact, the strong convergence in Lp ([0, T]; V) follows from the estimate (1.5.58) and Lebesgue dominated convergence theorem. Integrating (1.5.43) from 0 to T, we have T

1 󵄩󵄩 1 󵄩2 2 2 2 󵄩u(T)󵄩󵄩󵄩 + ν ∫(|Aum | + |Azm | )ds = zm + ‖u0m ‖ , 2󵄩 2 T

0

zm = ∫{(f , A(um + zm )) − (Cum , Aum ) − (Czm , Azm ) 0

− b(um , um , Aum ) − b(zm , um , Aum ) − b(u, zm , Aum )

− b(um , um , Azm )}ds. Therefore, Ym can be rewritten as

1󵄨 󵄨2 1 Ym = −(um (T), u(T)) + 󵄨󵄨󵄨u(T)󵄨󵄨󵄨 + ‖u0m ‖2 2 2 T

+ ν ∫(−2(Aum , Au) + |Au|2 )ds + zm . 0

And (1.5.62) and (1.5.64) imply T

T

0

0

󵄩 󵄩2 lim {−(um (T), u(T))} − 2ν ∫(Aum , Au)ds = −󵄩󵄩󵄩u(T)󵄩󵄩󵄩 − 2ν ∫ |Au|2 ds.

m→+∞

(1.5.66)

1.5 Nonlinear Galerkin method | 93

Taking advantage of the weak convergence of um , zm in L2 ([0, T]; D(A)), the strong convergence in L2 ([0, T]; V) and the boundedness in L∞ ([0, T]; V), we can obtain limm→+∞ zm . In particular, we can obtain the trilinear results in (1.5.66) similarly. Here we just consider the first term. We have T

T

∫ b(um , um , Aum )dt − ∫ b(u, u, Au)dt 0

T

0

T

T

= ∫ b(um − u, um , Aum )ds + ∫ b(u, um − u, Aum )ds + ∫ b(u, u, A(um − u))ds. 0

0

(1.5.67)

0

For the first term on the right-hand side of (1.5.67), equations (1.5.8), (1.5.58) and Hölder inequality imply T 󵄨󵄨 T 󵄨󵄨 1 1 󵄨󵄨 󵄨 󵄨󵄨 ∫ b(um − u, um , Aum )ds󵄨󵄨󵄨 ≤ C4 ∫ |um − u| 2 ‖um ‖ ⋅ 󵄨󵄨󵄨A(um − u)󵄨󵄨󵄨 2 ⋅ |Aum |ds 󵄨 󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨0 󵄨 0 T

1 󵄨 󵄨1 ≤ C ∫ |um − u| 2 󵄨󵄨󵄨A(um − u)󵄨󵄨󵄨 2 ⋅ |Aum |ds

0

T

1 4

T

󵄨 󵄨2 ≤ C(∫ |um − u| ds) (∫ 󵄨󵄨󵄨A(um − u)󵄨󵄨󵄨 ds) 2

0

0

1 2

T

1 4

⋅ (∫ |Aum |2 ds) . 0

By (1.5.17) and (1.5.59), we find that these terms tend to 0 as m → ∞. For the second term we take advantage of (1.5.7) to find T 󵄨󵄨 T 󵄨󵄨 1 1 1 1 󵄨󵄨 󵄨 󵄨󵄨 ∫ b(um , um − u, Aum )ds󵄨󵄨󵄨 ≤ C3 ∫ |u| 2 ‖u‖ 2 ‖um − u‖ 2 ⋅ 󵄨󵄨󵄨A(um − u)󵄨󵄨󵄨 2 ⋅ |Aum |ds 󵄨 󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨0 󵄨 0 T

1 󵄨 󵄨1 ≤ C ∫ ‖um − u‖ 2 󵄨󵄨󵄨A(um − u)󵄨󵄨󵄨 2 ⋅ |Aum |ds

0

T

1 4

T

󵄨 󵄨2 ≤ C(∫ ‖um − u‖ ds) (∫ 󵄨󵄨󵄨A(um − u)󵄨󵄨󵄨 ds) 0

T

2

1 2

⋅ (∫ |Aum |2 ds) . 0

0

1 4

94 | 1 Discrete attractor and approximate calculation Sequences in (1.5.17) and (1.5.59) tend to 0 as m → +∞. Finally, the last term in (1.5.67) is linear with respect to um . Hence, we prove that T

T

∫ b(um , um , Aum )ds → ∫ b(u, u, Au)ds, 0

m → +∞.

0

Similarly, we can deal with the rest of (1.5.66) and obtain T

1󵄩 󵄩2 1 lim Y = − 󵄩󵄩󵄩u(T)󵄩󵄩󵄩 + ‖u0 ‖2 − ν ∫ |Au|2 ds m→+∞ m 2 2 0

T

+ ∫{(f , Au) − (Cu, Au) − b(u, u, Au)}ds = 0. 0

This completes the proof of strong convergence in (1.5.42). Thus the proof of Theorem 1.5.2 is completed. Now we consider (1.5.42), (1.5.10) with another nonlinear Galerkin method. For a given u0 ∈ V, we select the eigenvectors of the operator as the basis of V. If the approximate solution um has the form m

um (t) = ∑ gim (t)wj , j=1

um : ℝ+ → span{w1 , w2 , . . . , wm } = Wm . Introduce the unknown function zm of the form 2m

zm (t) = ∑ hjm (t)wj , j=m+1

̃m = span{wm+1 , . . . , w2m }. zm : ℝ+ → w We need that um , zm satisfy the following equations: d (u , v) + ν((um , v)) + (Cum , v) + b(um , um , v) dt m + b(zm , um , v) + b(um , zm , v) + b(zm , zm , z) = (f , v),

ν((zm , ṽ)) + ((zm , ṽ)) + b(um , um , ṽ)

+ b(zm , um , ṽ) + b(um , zm , ṽ) = (f , ṽ),

um (0) = Pm u0 .

∀v ∈ Wm ,

̃m , ∀ṽ ∈ w

Equation (1.5.69) can be rewritten as νAzm + (P2m − Pm )Czm + (P2m − Pm )(B(zm , um ))

(1.5.68) (1.5.69) (1.5.70)

1.5 Nonlinear Galerkin method | 95

+ B(um , zm ) = (P2m − Pm )(f − B(um )).

(1.5.71)

We use D(um ) to denote the linear operator acting on zm on the left-hand side in (1.5.69). In order to prove the existence of a solution {um , zm } to problem (1.5.68)–(1.5.70) on a ̃m . We have small scale, we must prove that D(um ) is invertible on w (D(um )ṽ, ṽ) = ν‖ṽ‖2 + (C ṽ, ṽ) + b(ṽ, um , ṽ) ≥ α‖ṽ‖2 − C1 |ṽ| ⋅ ‖ṽ‖ ⋅ ‖um ‖ −1

2 )‖um ‖. ≥ ‖ṽ‖2 (α − C1 λm+1

(1.5.72)

We choose sufficiently large m such that −1

2 α − C1 λm+1 ‖u0 ‖ ≥

α , 2

(1.5.73)

then by the existence theorem of ordinary differential equations, equations (1.5.68)– (1.5.70) admit a solution (um (t), zm (t)), t ∈ [0, Tm ]. On this interval, equations (1.5.68)– (1.5.69) equal to the ordinary differential equation of um : dum + νAum + Pm (Cum + B(um + zm )) = Pm f , dt zm = D(um )−1 {(P2m − Pm )(f − B(um ))}.

(1.5.74)

Condition (1.5.73) is satisfied. Due to the fact that λm → ∞ as m → +∞, in the following, we will prove that (at least for sufficiently large m) Tm = +∞, namely the solution to (1.5.74) is defined in ℝ+ . Furthermore, we prove that the solution to (1.5.74) tends to the solution of problem (1.5.1) as m → +∞. We have the following theorem: Theorem 1.5.3. Suppose that (1.5.4)–(1.5.9) hold. If u0 is given in V, then we have (i) There exists a constant k = k(u0 ) such that if m satisfies −1

2 α − C1 kλm+1 ≥

α , 2

(1.5.75)

then (1.5.74), (1.5.70) admit a solution um which is defined in ℝ+ ; (ii) The solution um of (1.5.74), (1.5.70) converges to the solution u of (1.5.1), (1.5.10) as m → +∞: um → u strongly in L2 ([0, T]; D(A)) and Lp ([0, T]; V),

∀T > 0, 1 ≤ p < +∞, and the sequence is weakly ∗ convergent in L∞ (ℝ+ ; V). (1.5.76) Proof. The constant k in (1.5.75) will be determined later. Now if (1.5.73) holds, (1.5.68)– (1.5.70) admit a solution {um , zm } in the certain interval (0, Tm ). We obtain some a priori estimates of {um , zm } in (0, Tm ). This is similar to Theorems 1.5.1 and 1.5.2.

96 | 1 Discrete attractor and approximate calculation (i) The a priori estimate (I). Taking v = um in (1.5.68), ṽ = zm in (1.5.69), and adding the corresponding equalities, we derive the following equality from (1.5.4): 1 d |u |2 + ν‖um ‖2 + (cum , um ) + ν‖zm ‖2 2 dt m + (czm , zm ) = (f , um + zm ).

(1.5.77)

Therefore, similar to (1.5.22), (1.5.23) obtained by (1.5.18), we have um is uniformly bounded in L∞ ([0, T]; H).

(1.5.78)

Functions um , zm are uniformly bounded in L2 ([0, T]; H), ∀T ∈ (0, Tm ); if Tm < +∞,

T = Tm .

(1.5.79)

(ii) The a priori estimate (II). Taking v = Aum in (1.5.68), ṽ = Azm in (1.5.69) and adding the corresponding equalities, we have 1 d ‖u ‖2 + ν|Aum |2 + ν|Azm |2 2 dt m = (f , A(um + zm )) − (cum , Aum ) − (Czm , Azm ) − b(um , um , Aum ) − b(zm , um , Aum ) − b(um , zm , Aum ) − b(zm , zm , Aum )

− b(um , um , Azm ) − b(zm , um , Azm ) − b(um , zm , Azm ).

(1.5.80)

Some terms on the right-hand side of (1.5.80) are bounded, which is proved in Theorem 1.5.2. Then (1.5.7), (1.5.25) and (1.5.26) imply 1 1 3 󵄨󵄨 󵄨 󵄨 1 󵄨󵄨b(zm , um , Aum )󵄨󵄨󵄨 ≤ C3 󵄨󵄨󵄨zm | 2 ⋅ ‖zm ‖ 2 ‖um ‖ 2 |Aum | 2 3

4 λ ≤ C3 ( m ) ⋅ ‖um ‖2 ⋅ |Azm | ≤ C3 ‖um ‖2 ⋅ |Azm | λm+1 C ν ≤ |Azm |2 + 10 ‖um ‖4 , 24 ν

(1.5.81)

󵄨󵄨 󵄨 󵄨󵄨b(zm , zm , Aum )󵄨󵄨󵄨 ≤ C3 |zm | ⋅ ‖zm ‖ ⋅ ‖Azm ‖ |Aum | 1 2

1 2

3

4 λ ≤ C3 ( m ) ⋅ ‖um ‖2 ⋅ |Azm | ≤ C3 ‖um ‖2 ⋅ |Azm | λm+1 C ν ≤ |Azm |2 + 10 ‖um ‖2 ‖zm ‖2 , 24 ν

(1.5.82)

1 1 1 󵄨󵄨 󵄨 󵄨󵄨b(zm , um , Azm )󵄨󵄨󵄨 ≤ C3 |zm | 2 ⋅ ‖zm ‖ 2 ⋅ ‖Aum ‖ 2 |Azm | 1

4 λ ≤ C3 ( m ) ⋅ ‖um ‖ ⋅ ‖zm ‖‖Azm ‖ λm+1 C ν ≤ |Azm |2 + 10 ‖um ‖2 ‖zm ‖2 . 24 ν

(1.5.83)

1.5 Nonlinear Galerkin method | 97

Finally, the estimate of the last term on the right-hand side of (1.5.80) is 1 1 1 1 󵄨 󵄨󵄨 󵄨󵄨b(um , zm , Azm )󵄨󵄨󵄨 ≤ C3 |zm | 2 ⋅ ‖um ‖ 2 ⋅ ‖zm ‖ 2 |Azm | 2 C ν ≤ |Azm |2 + 11 |um |2 ⋅ ‖um ‖2 ⋅ ‖zm ‖2 . 24 ν

Combining with (1.5.44), (1.5.49), (1.5.81) and (1.5.83), we obtain the differential inequality 6C 2 d 12 ‖um ‖2 + ν|Aum |2 + ν|Azm |2 ≤ |f |2 + 2 ‖zm ‖2 dt ν ν + C1 2‖um ‖2 ⋅ (1 + ‖um ‖2 + |um |2 ⋅ ‖um ‖2 + ‖zm ‖2 + |um |2 ⋅ ‖zm ‖2 ),

(1.5.84)

where C12 = C12 (ν) depends on ν. Bound (1.5.84) implies 6C 2 d 12 ‖um ‖2 ≤ |f |2 + 2 ‖zm ‖2 + C12 ‖um ‖2 ⋅ (1 dt ν ν 2 + ‖um ‖ + |um |2 ⋅ ‖um ‖2 + ‖zm ‖2 + |um |2 ⋅ ‖zm ‖2 ).

(1.5.85)

Integrating (1.5.85) from 0 to T and using (1.5.78)–(1.5.79), we obtain um is uniformly bounded in L∞ ([0, T]; V), 0 < T < Tm ,

if Tm < +∞, T = Tm .

(1.5.86)

Therefore, if Tm < +∞, (1.5.85) implies an upper bound of ‖um (t)‖, 0 ≤ t ≤ Tm . Similarly, we obtain an upper bound of ‖um (t)‖ as Tm = +∞. In fact, (1.5.77) implies t+1

t+1

t

t

∫ ‖um ‖2 ds + ∫ ‖zm ‖2 ds ≤ C13 ,

∀t ≥ 0,

(1.5.87)

where the constant C13 is independent of m. Inequality (1.5.85) satisfies the conditions of Lemma 1.5.1, which follows from (1.5.87), (1.5.87). Inequality (1.5.55) implies a bound of ‖um (t)‖, t ≥ 1 which is independent of m. As (1.5.55) gives a bound of ‖um (t)‖, 0 ≤ t ≤ 1, we have um is uniformly bounded in L∞ ([0, Tm ]; V) with respect to m.

(1.5.88)

Integrating (1.5.84) with respect to t from 0 to T, we have um , zm are uniformly bounded in L2 ([0, T]; D(A)), 0 < T < Tm ,

if Tm < +∞, T = Tm .

(iii) The a priori estimate (III). Taking ṽ = Azm in (1.5.69), we have ν|Azm |2 = −(Czm , Azm ) − b(um , um , Azm )

− b(zm , um , Azm ) − b(um , zm , Am ) + (f , Azm )

(1.5.89)

98 | 1 Discrete attractor and approximate calculation 1

1

≤ C2 ‖zm ‖|Azm | + C3 |um | 2 ‖um ‖|Aum | 2 ⋅ |Azm | 1

3

+ C4 |zm | 2 ⋅ |Aum | 2 ⋅ ‖um ‖ 1

1

1

3

+ C3 |um | 2 ‖um ‖ 2 ‖zm ‖ 2 |Azm | 2 + |f | ⋅ |Azm |. Dividing the above inequality by |Azm | and using (1.5.28), (1.5.29) and (1.5.88), we have 1

−1

−1

2 2 ν|Azm | ≤ C2 λm+1 |Azm | + C3 λm4 + C4 λm+1 |Azm |

−1

2 |Azm | + |f |, + C3 λm+1

t ∈ [0, Tm ).

So −1

1

−1

−1

4 2 2 }|Azm | ≤ C3 λm4 + |f |. {ν − C2 λm+1 − C4 λm+1 − C3 λm+1

As m is large enough, we have 1 ν |Azm | ≤ C3 λm4 + |f |. 2

(1.5.90)

1

2 Since |Azm | ≥ λm+1 ‖zm ‖, (1.5.90) implies

zm → z weakly ∗ in L∞ ([0, Tm ]; V),

m → +∞.

(1.5.91)

(iv) Taking the limit. First, we verify that the solution to (1.5.74) is defined in ℝ+ for sufficiently large m. In fact, (1.5.88) implies that there exists a constant k (which does not depend on m) such that 󵄩󵄩 󵄩 󵄩󵄩um (t)󵄩󵄩󵄩 ≤ k,

0 ≤ t < Tm .

(1.5.92)

So if m satisfies −1

2 ≥ α − C1 kλm+1

α , 2

then by (1.5.72), we have (D(um )ṽ, ṽ) ≥

α 2 ‖ṽ‖ , 2

̃m . t ∈ [0, Tm ), ṽ ∈ w

Therefore, the operator D(um ) is uniformly coercive in (0, Tm ). This implies Tm = +∞. So we complete the proof of the (i) part in Theorem 1.5.3. Assume that (1.5.75) holds, where k is given by (1.5.92). Then the estimates (1.5.89), (1.5.91) hold as Tm = +∞. These estimates are similar to (1.5.58)–(1.5.60). Therefore we can take the limit m → +∞. Then um converges to the solution u of problem (1.5.1), (1.5.10) due to (1.5.17), (1.5.42). The proof of Theorem 1.5.3 is completed.

1.5 Nonlinear Galerkin method | 99

Now we consider a numerical computation scheme for some nonlinear evolution equation. Among such methods, there are the spectral, pseudo-spectral, finite element and finite difference methods for spatial discretization. Time discretization can have two formats, semiexplicit and explicit schemes. Assume that Vh denotes a finite-dimensional vector space, and it has two scalar products and respective norms: ((⋅, ⋅))h , ‖ ⋅ ‖h and (⋅, ⋅)h , | ⋅ |h . Space Vh is an approximation of a classical Sobolev space, ‖ ⋅ ‖h is discrete Sobolev norm, | ⋅ |h is discrete L2 -norm, Vh ∈ {Vl }, l ∈ ℋ, {Vl } approaches infinite-dimensional space V as h tends to 0. Assume that Ci are positive absolute constants which do not depend on h, they tend to 0 as h → 0, in general. Assume that |uh |h ≤ C1 ‖uh ‖h ,

S1 (h)‖uh ‖h ≤ |uh |h ,

∀uh ∈ Vh .

(1.5.93)

The bilinear continuous form ah (⋅, ⋅) in Vh satisfies 󵄨󵄨 󵄨 󵄨󵄨ah (uh , vh )󵄨󵄨󵄨 ≤ C2 ‖uh ‖h ‖vh ‖h ,

∀uh , vh ∈ Vh .

(1.5.94)

The trilinear continuous form bh (⋅, ⋅, ⋅) in Vh satisfies bh (uh , vh , vh ) = 0, ∀uh , vh ∈ Vh , 1 1 1 1 󵄨󵄨 󵄨 󵄨󵄨bh (uh , vh , wh )󵄨󵄨󵄨 ≤ C3 |uh |h2 ‖uh ‖h2 ‖vh ‖h |wh |h2 ‖wh ‖h2 .

(1.5.95) (1.5.96)

The bilinear continuous form dh (⋅, ⋅) in Vh satisfies 󵄨󵄨 󵄨 󵄨󵄨dk (uh , vh )󵄨󵄨󵄨 ≤ C4 ‖uh ‖h ⋅ |vh |h , ∀uh , vh ∈ Vh , ah (uh , uh ) + dh (uh , uh ) ≥ C5 ‖uh ‖2h , ∀uh ∈ Vh .

(1.5.97) (1.5.98)

Now we consider the following problem: Find a function uh : ℝ+ → Vh such that d (u , v ) + ah (uh , vh ) + bh (uh , uh , vh ) + dh (uh , vh ) = (fh , vh ), dt h h h uh (0) = u0h ,

∀vh ∈ Vh ,

(1.5.99) (1.5.100)

where u0h is given in Vh , fh ∈ L∞ (ℝ+ ; Vh ). Since Vh is finite-dimensional, (1.5.94)– (1.5.98) imply that there exists a unique solution to the initial problem (1.5.99)– (1.5.100), uh ∈ L∞ (ℝ+ ; Vh )

(1.5.101)

and uh is uniformly bounded with respect to h. Many equations which are significant for physics can be rewritten in the form of (1.5.99). The conditions (1.5.94)–(1.5.98) are satisfied. Assume that Vh = Vh2 ⊕ Wh ,

(1.5.102)

100 | 1 Discrete attractor and approximate calculation where Vh2 ⊂ Vh = Vh1 , the elements of Vh2 are represented as yh , ỹh , . . . , the elements of Wh are represented as zh , z̃h , . . . , and any uh ∈ Vh can be uniquely represented as uh = yh + zh ,

yh ∈ Vh2 , zh ∈ Wh .

(1.5.103)

Now we state the following assumption on decomposition formula (1.5.102): 󵄨 󵄨󵄨 󵄨󵄨(yh , zh )󵄨󵄨󵄨 ≤ (1 − δ)‖yh ‖h ⋅ ‖zh ‖h , ∀yh ∈ Vh2 , ∀zh ∈ Wh ,

(1.5.104)

where δ ∈ (0, 1), which does not depend on h. Moreover, we assume that |zh |h ≤ S2 (h)‖zh ‖h ,

∀zk ∈ Wh ,

(1.5.105)

where S2 (h) → 0, as h → 0. Now we consider three important situations of the decomposition (1.5.102): (i) Spectral scheme. Space Vh is a subspace of the Hilbert space V and its inner product and norm are ((⋅, ⋅)) and ‖ ⋅ ‖, respectively. The constraint on Vh for the form ah (⋅, ⋅) is that it is bilinear continuous symmetric coercive form on V, and V is continuously imbedded and dense in another Hilbert space H with inner product (⋅, ⋅) and norm | ⋅ |. Then ‖uh ‖h = ‖uh ‖,

|uh |h = |uh |,

∀uh ∈ Vh .

The unbounded self-adjoint operator A, which is connected with a, V and H possesses an orthogonal basis {wj }, j ∈ ℕ, D(A) ⊂ V in H and V, which satisfies Awj = λj wj ,

0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅, λj → ∞, j → ∞.

(1.5.106)

Given m = m1 ∈ ℕ, h = m1 , Vh = span{w1 , . . . , wm1 }, (1.5.99)–(1.5.100) can be seen as a 1 Galerkin approximation of an infinite-dimensional problem in V. Consider the decomposition (1.5.102). We assume that another integer m2 ∈ ℕ is chosen, m2 < m1 . Set Vh2 = span{w1 , . . . , wm2 },

wh = span{wm2 +1 , . . . , wm1 },

Vh2 and Wh are orthogonal in Vh (with scalar product ((⋅, ⋅))h = ((⋅, ⋅))), (1.5.104) is satisfied with δ = 1. For all zh ∈ Wh ,

m1

zh = ∑ ξj wj , j=m2 +1

we have |zh |2h

󵄩󵄩 m1 󵄩󵄩2 m1 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∑ ξj wj 󵄩󵄩󵄩 = ∑ |ξj |2 λj 󵄩󵄩 󵄩󵄩 j=m2 +1 󵄩 j=m2 +1 󵄩

1.5 Nonlinear Galerkin method | 101 m1

≥ λm2 +1 ∑ |ξj |2 = λm2 +1 |zh |2h . j=m2 +1

Therefore 1

S2 = (λm2 +1 )− 2 .

(1.5.107)

Also a and V are related to the boundary value problem for elliptic equations with space boundary condition, (1.5.99) is spectral or pseudo-spectral approximation for the continuous problem. (ii) Finite elements. We just consider the simplest case of a 1D piecewise linear element. The space Vh is a subspace of V = H01 (Ω), Ω = (0, L), L > 0. Introduce the inner product L

((uh , vh ))h = ((uh , vh )) = ∫ 0

duh dvh dx. dx dx

2

The scalar product in L ([0, L]) is T

(uh , vh )h = (uh , vh ) = ∫ uh vh dx. 0 1 Denote h1 = h = 2N , h2 = N1 , n ∈ ℕ. Then Vh is a space of real continuous functions in (0, L) which are zero at 0 and L, and which are linear in the intervals (jh, (j + 1)h], j = 0, 1, . . . , 2N − 1. The space Vh = V2h is defined in the same way. However, functions of V2h are linear on the intervals [2jh, 2(j + 1)h], j = 0, 1, . . . , N − 1. The node basis of Vh is composed of the functions wj,h of Vh ,

wj,h (jh) = 1;

wj,h (ih) = 0,

i, j = 1, 2, . . . , 2N − 1, i ≠ j.

Similarly, the node basis of V2h is composed of the functions wj,2h of V2h , wj,2h (2jh) = 1,

w2j,h (2ih) = 0,

i, j = 1, 2, . . . , N − 1, i ≠ j.

The basis of Vh consists of the basis of V2h and the basis of Wh , namely functions wj,h , j = 2i + 1, i = 0, 1, . . . , N − 1. And then uh ∈ Vh can be expanded as 2N−1

uh = ∑ uh (jh)wj,h j=1

(1.5.108)

which can be decomposed into N−1

N−1

j=1

i=0

uh = ∑ uh (2jh)wj,2h + ∑ uh ((2i + 1)h)w2i+1,h ,

(1.5.109)

102 | 1 Discrete attractor and approximate calculation where uh ((2i + 1)h) is an interpolation of uh : 1 uh ((2i + 1)h) = uh ((2i + 1)h) − (uh (2ih) + uh (2i + 2)h). 2

(1.5.110)

We note that the first sum of (1.5.109) corresponds to yh ∈ Vh2 ⊂ Vh , while the second sum corresponds to zh ∈ Wh . If uh (jh) = u(jh), j = 0, 1, . . . , 2N, then uh ((2i + 1)h) =

h2 󸀠󸀠 u ((2i + 1)h) + o(h3 ), 2

yh and uh have the same order, thus zh has the factor h2 . It is easy to verify that every function yh ∈ Vh2 is orthogonal to any function zh ∈ Wh in V. Thus (1.5.104) holds with δ = 1. Inequality (1.5.105) is easy to be satisfied, it yields S2 (h) = h/√3.

(1.5.111)

It is easy to use a common method to verify (1.5.93). For example, for S1 (h), set ξi = uh (ih) and L

∫( 0

2

duh 1 2N−1 ) dx = ∑ (ξi+1 − ξi )2 . dx 2 i=0

Similarly, L

∫(uh )2 dx = 0

Therefore Sl (h) =

h 2N−1 2 2 + ξi ξi+1 ). ∑ (ξ + ξi+1 3 i=0 i

h . 2√3

(iii) Finite difference method. Assume V = H01 ([0, L]); h = L/2N, N ∈ ℕ. Let Vh be the space of step functions which are constant in [jh, (j + 1)h), j = 0, 1, . . . , 2N − 1, they are 0 on [0, h] and [L − h, L], i. e., Vh = span{wj,h , j = 1, 2, . . . , 2N − 2}, 1,

[jh, (j + 1)h);

0,

otherwise,

wj,h = {

2N−2

uh = ∑ uh (jh)wj,h , j=1

∀uh ∈ Vh .

The set {wj,h } is a natural basis of Vh . Define scalar products L−h

((uh , vh ))h = ∫ ∇h uh ∇h vh dx, 0

1.5 Nonlinear Galerkin method |

103

L

(uh , vh )h = ∫ uh vh dx, 0

where ∇h is the forward difference operator given by (∇h φ)(x) =

φ(x + h) − φ(x) . h

Set h2 = 2h, and similarly define Vh2 = V2h , its basis consists of functions wj,2h , j = 1, . . . , N − 2, V2h ⊂ Vh . Define Wh = span{w2i+1,h , i = 0, 1, . . . , N − 2} in the decomposition (1.5.102). Any function uh ∈ Vh can be written as uh = yh + zh , N−2

yh ∈ V2h ,

yh = ∑ uh (2jh)wj,2h ,

(1.5.112)

j=1

N−2

zh = ∑ zh ((2i + 1)h)w2i+1,h + zh ((2N − 2)h)w2N−2,h . i=0

When i = 0, 1, . . . , N − 2, zh ((2i + 1)h) = ū h ((2i + 1)h) = uh ((2i + 1)h) − uh (2ih),

zh ((2N − 2)h) = uh ((2N − 2)h).

(1.5.113) (1.5.114)

Obviously, if uh is the constraint of a smooth function u ∈ H01 ([0, L]) in Vh , then yh and uh have the same order, zh possesses the factor h. We use the following lemmas to verify (1.5.104) and (1.5.105): Lemma 1.5.2. The following Cauchy–Schwarz inequality holds: 2 󵄨󵄨 󵄨 󵄨󵄨((yh , zh ))h 󵄨󵄨󵄨 ≤ √ ‖yh ‖h ‖zh ‖h , 3

∀y ∈ V2h , ∀zh ∈ Wh .

(1.5.115)

Proof. We must prove that L−h

L−h

1 2

Lh

1 2

2 ∫ ∇h yh ∇h zh dx ≤ √ ( ∫ |∇h yh |2 dx) ⋅ (∫ |∇h zh |2 dx) . 3 0

0

(1.5.116)

0

Hence we only need to fully prove that (1.5.116) holds as the interval (0, L) is substituted with a rough grid having intervals (2jh, 2(j + 1)h), j = 1, 2, . . . , N − 3. Set m1 , yh = { m2 ,

x ∈ [2jh, 2(j + 1)h), x ∈ [2(j + 1)h, (2j + 3)h),

104 | 1 Discrete attractor and approximate calculation 0, { { { { { {P1 , zh = { { 0, { { { { {P2 ,

x ∈ [2jh, (2j + 1)h), x ∈ [(2j + 1)h, 2(j + 1)h), x ∈ [2(j + 1)h, (2j + 3)h), x ∈ [(2j + 3)h, 2(j + 2)h).

In the interval (2jh, 2(j + 1)h), ∇h yh = 0,

∇h zh =

P1 . h

In the interval [2(j + 1)h, 2(j + 1)h), ∇h yh = 2(j+1)h

m2 − m1 , h

∇h zh = −

P1 , h

1 ∫ ∇h yh ∇h zh dx = − (m2 − m1 )P1 h

2jh

1 2

2(j+1)h

2(j+1)h

1 ≤ ( ∫ |∇h yh |2 dx) ⋅ ( ∫ |∇h zh |2 dx) √2 2jh

1 2

2jh

1 = |m2 − m1 | ⋅ |P1 |. h

(1.5.117)

Now we first consider the end points j = 0, j = N − 2, N − 1 of the interval. The only difference is m1 = 0 for j = 0. Therefore, (1.5.117) still holds. Next, we consider the interval [L − 4h, L − 2h) and (L − 2h, L − h). Function zh is not 0 on [L − 2h, L − h] (for example, zh = P2 ), but zh = 0 in [L − h, L). Therefore L−h

1 ∫ ∇h yh ∇h zh dx = − m1 (P2 − P1 ) h

L−4h

≤

1 1 2 √ |m1 |((P2 − P1 )2 + P12 + P22 ) 2 h 3

L−h

1 2

L−h

1 2

2 = √ ( ∫ |∇h yh |2 dx) ⋅ ( ∫ |∇h zh |2 dx) . 3 L−4h

Since

1 √2

L−4h

< √ 32 , we obtain (1.5.115).

Lemma 1.5.3. For the functions in Wh , there exists the following strong discrete Poincaré inequality |zh |h ≤ S2 (h)‖zh ‖h ,

∀zh ∈ Wh , S2 (h) = h.

(1.5.118)

1.5 Nonlinear Galerkin method | 105

Proof. Similar to Lemma 1.5.2, we prove that similar inequalities hold in the interval [2jh, 2(j + 1)h), j = 0, 1, . . . , N − 1, namely 2(j+1)h

∫

zh2 dx

≤

S22 (h)

2jh

2(j+1)h

∫ |∇h zh |2 dx.

(1.5.119)

2jh

As in Lemma 1.5.2, the integral on the right-hand side of (1.5.119) equals to 2P12 /h, j = 0, 1, . . . , N − 3, while the left-hand side equals to 2P12 h. Therefore, S2 (h) = h/√2 in (1.5.119), the integral of zh2 equals to h(P12 + P22 ) in [L − 4h, L), the integral of |∇h zh |2 equals to h1 ((P1 − P2 )2 + P12 + P22 ) in [L − 4h, L). Thus we derive (1.5.119), S2 (h) = h. Hence, (1.5.118) holds. and

Now we prove the second inequality in (1.5.93). Set ξj = uh (jh), uh = ∑2N−2 j=1 ξj Wj,h , 2N−2

|uh |2h = h ∑ ξj2 , j=1

‖uh ‖2h = =

2N−2

1 2 2N−2 2 ∑ (ξj+1 − ξj )2 ≤ ∑ (ξ + ξ 2 ) h j=1 h j=1 j+1 j 4 2N−2 2 4 ∑ ξ (as ξ0 = ξ2N−1 = 0) = 2 |uh |2h . h j=1 j h

Therefore, S1 (h) = h/2.

(1.5.120)

Lemma 1.5.4. The following equalities hold: |yh |h = |yh |2h ,

‖yh ‖2h = 2‖yh ‖22h ,

∀yk ∈ V2h = Vh2 .

(1.5.121)

Proof. Obviously the first equality holds, since | ⋅ |h and | ⋅ |2h are L2 -norms. Similar to Lemma 1.5.2, it follows from ∇2h yh = (m2 − m1 )/2h, [2jh, 2(j + 1)h) and {0, ∇h yh = { m2 − m1 , { h

[2jh, (2j + 1)h), [(2j + 1)h, 2(j + 1)h),

that 2(j+1)h

∫ |∇2h yh |2 dx =

2jh

(m2 − m1 )2 1 = 2h 2

2(j+1)h

∫ |∇h yh |2 dx.

2jh

Therefore (1.5.121) can be obtained by the sums of j = 0, 1, . . . , N − 2.

106 | 1 Discrete attractor and approximate calculation Remark 1.5.1. The space V2h plays the same role as Vh . Hence we can obtain the corresponding conclusions which are similar to (1.5.93)–(1.5.98). In particular, the second assumption in (1.5.93) reads S1 (2h)‖yh ‖2h ≤ |yh |2h ,

∀yh ∈ Vh2 = V2h .

(1.5.122)

It follows from (1.5.120) and (1.5.121) that (1.5.122) leads to S1 (h)‖yh ‖h ≤ |yh |h ,

∀yh2 = Vh = V2h ,

S(h) = S1 (2h)/2.

(1.5.123)

For the other forms of discretization scheme (spectral and finite element), ‖yh ‖h2 = ‖yh ‖h , |yh |h2 = |yh |h , ∀yh ∈ Vh . Therefore (1.5.123) still holds, but S1 (h) = S(h2 ). Now we consider time discretization. Scheme I. The initial value u0h in (1.5.101) is decomposed into u0h = yh2 + zh2 ,

yh0 ∈ Vh2 , zh0 ∈ Wh .

We determine yhn ∈ Vh2 , zhn ∈ Wh in a recurrent sequence as follows: Assume that yhn , zhn are known. Define yhn+1 ∈ V2h and zhn+1 ∈ Wh as follows: 1 n+1 (y − yhn , ŷh )h + ah (yhn+1 + zhn+1 , ŷh ) + dh (yhn+1 + zhn+1 , ŷh ) + bh (yhn , yhn , ŷhn ) k h + bh (yhn , zhn , ŷh ) + bh (zhn , yhn , ŷh ) = (fhn , ŷh )h ; ∀ŷh ∈ Vh2 , (1.5.124) 1 n+1 (z − zhn , ẑh )h + ah (yhn+1 + zhn+1 , ẑh ) k h

+ dh (yhn+1 + zhn+1 , ŷh ) + bh (yhn , yhn , ŷhn ) = (fhn , ẑh )h ;

∀ẑh ∈ Wh ,

(1.5.125)

where k = Δt is the time step, fhn denotes the average of fh : fhn

1 = k

(n+1)k

∫ f (t)dt.

(1.5.126)

nk

If f is smooth, we can take fhn = fh (nk). Equations (1.5.124)–(1.5.125) are linear equations in yhn+1 , zhn+1 . By (1.5.98), there exists a unique yhn+1 , zhn+1 from Lax–Milgram theorem. Scheme I󸀠 . It is a little different from the Scheme I. Equation (1.5.124) yields 1 n+1 (y − yhn , ŷh )h + ah (yhn+1 + zhn+1 , ŷh ) + dh (yhn+1 + zhn+1 , ŷh ) + bh (yhn , yhn , ŷh ) k h + bh (yhn , zhn , ŷh ) + bh (zhn+1 , zhn+1 , ŷh ) = (fhn , ŷh )h ;

∀ŷh ∈ Vh2 .

(1.5.127)

1.5 Nonlinear Galerkin method | 107

Scheme II. Let yhn , zhn be known. Define yhn+1 , zhn+1 as follows: 1 n+1 (y − yhn , ŷh )h + ah (yhn + zhn+1 , ŷh ) k h + dh (yhn + zhn+1 , ŷh ) + bh (yhn , yhn , ŷh )

+ bh (yhn , zhn+1 , ŷh ) + bh (zhn+1 , zhn , ŷh ) = (fhn , ŷh )h ;

∀ŷh ∈ Vh2 ,

1 n+1 (z − zhn , ẑh )h + ah (yhn + zhn+1 , ẑh ) + dh (yhn + zhn+1 , ẑh ) k h + bh (yhn , yh , ẑhn ) = (fhn , ẑh )h ; ∀ẑh ∈ Wh .

(1.5.128)

(1.5.129)

In fact, due to (1.5.129), (1.5.127), we obtain zhn+1 , yhn+1 , respectively. Scheme III. It is different from Scheme II, in which we neglect the term zhn+1 − zhn . The computation of zhn+1 can be determined by the following: ah (yhn + zhn+1 , ẑh ) + dh (yhn + zhn+1 , ẑh ) + bh (yhn , yhn , ẑh ) = (fhn , ẑh ),

∀ẑh ∈ Wh ,

(1.5.130)

the solvability of zhn+1 in (1.5.130) can be obtained by (1.5.94), (1.5.98) and Lax–Milgram theorem. Now we give two examples. Example 1 (Burgers equation). Assume Ω = (0, L), L > 0. Set V = H01 ([0, L]), H = L2 ([0, l]). For v > 0, f is given, the equation is ut − νuxx + uux = f ,

u(0, t) = u(L, t) = 0,

Ω × ℝ+ , u(x, 0) = u0 (x).

In order to find u : ℝ+ → H01 (Ω) = V, we rewrite the equation as d (u, v) + ν((u, v)) + b(u, u, v) = (f , v), dt

∀v ∈ V,

(1.5.131)

where L

(φ, ψ) = ∫ φψdx,

L

((φ, ψ)) = ∫

0

0

dφ dψ dx, dx dx

∀φ, ψ,

b is an antisymmetric nonlinear term, L

1 b(φ, ψ, θ) = ∫ φ(φx θ − ψθx )dx. 3 0

We utilize the spectral, finite element and finite difference methods to discretize the equation and get equation (1.5.99), d = 0. Equations (1.5.94)–(1.5.98) are satisfied, C2 = C5 = ν, C4 = 0, C3 is an appropriate constant.

108 | 1 Discrete attractor and approximate calculation Example 2 (Navier–Stokes equation). Set Ω = ℝ2 , 1 𝜕u − νΔu + (u ⋅ ∇)u + (div u)u = f , 𝜕t 2 u = 0, 𝜕Ω, u(x, 0) = u0 (x).

x ∈ Ω, t ≥ 0,

Translating the equation into the form of (1.5.130), u : ℝ+ → V = H01 (Ω), we have 2

(φ, ψ) = ∫ φψdx, ((φ, ψ)) = ∑ ∫ i,j=1 Ω

Ω

b(φ, ψ, θ) =

𝜕φi 𝜕ψi dx, 𝜕xj 𝜕xj

2

𝜕φj 𝜕θj 1 θ − ψj )dx. ∑ ∫ φi ( 2 i,j=1 𝜕xi j 𝜕xi Ω

Utilizing the spectral, 2D finite element and 2D finite difference methods to discretize the equation, we have equation (1.5.99), d = 0. Equations (1.5.94)–(1.5.98) are satisfied, C2 = C5 = ν, C4 = 0, C3 is an appropriate constant.

1.6 Stability analysis and numerical results Consider the stability of nonlinear Galerkin method for the following 2D Navier– Stokes equation: 𝜕u − νΔu + (u ⋅ ∇)u + ∇p = f, 𝜕t ∇u = 0, Ω × ℝ,

Ω × ℝ+ ,

u is Ω-periodic,

(1.6.1) (1.6.2) (1.6.3)

where u = (u1 , u2 ) is a velocity vector, p = p(x, t) is pressure, x = (x1 , x2 ), f is external force, Ω = (0, L1 ) × (0, L2 ). As usual, from the space of projection (1.6.1) on ∇ ⋅ u = 0, we obtain 𝜕u − νΔu + B(u, u) = f. 𝜕t

(1.6.4)

Take L1 = L2 = 2π. The inner product (⋅, ⋅) and norm ‖ ⋅ ‖ in L2 (Ω) are (u, v) = ∫ u(x)v(x)dx,

|u|2 = (u, u).

(1.6.5)

Ω

For fixed m, we consider an approximate solution u2m of nonlinear Galerkin method, 2m

u2m (x, t) = ∑ gk,m (t)ωk (x), k=1

(1.6.6)

1.6 Stability analysis and numerical results | 109

where ωk (x) is the eigenfunction of Stokes operator. We decompose u2m into u2m = ym + zm , m

ym (x, t) = ∑ gk,m (t)ωk (x) = Pm u2m ,

(1.6.7)

k=1

2m

zm (x, t) = ∑ gk,m (t)ωk (x) = (P2m − Pm )u2m = Pm u2m .

(1.6.8)

k=m+1

Choose the following approximation such that (ym , zm ) satisfies the equations: 𝜕ym − νΔym + Pm (B(ym , ym ) + B(ym , zm ) + B(zm , ym )) = Pm f , 𝜕t 𝜕zm − νΔzm + Rm B(ym ) = Rm f . 𝜕t

(1.6.9) (1.6.10)

Obviously, if we take zm = 0 in (1.6.9), then it corresponds to a general Galerkin spectral method of order m. Now we take the time-discretization of (1.6.9)–(1.6.10): yn+1 − yn − νAyn+1 + P(B(yn ) + B(yn , z n+1 ) + B(z n+1 , yn )) = ρf n , k z n+1 − z n + νAz n+1 + RB(yn ) = Rf n , k

(1.6.11) (1.6.12)

where P = PN , R = P2N − PN , A = −Δ. Since |ϕ| ≤ C1 ‖ϕ‖,

∀ϕ ∈ V,

|y|L∞ ≤ C2 LN ‖y‖,

(1.6.13)

∀y ∈ Pn V,

(1.6.14)

where 1

LN = (1 + lg λN /λ1 ) 2 ,

0 < λ1 < λ2 < ⋅ ⋅ ⋅ < λj → +∞,

j → ∞,

{λj } is the eigenvalue sequence of Stokes operator, ‖ϕ‖ ≤ SN |ϕ|,

1

∀ϕ ∈ PN V, SN = λN2 .

Performing the operation ((1.6.11), yn+1 ) + ((1.6.12), z n+1 ) yields 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 (󵄨󵄨󵄨yn+1 󵄨󵄨󵄨 + 󵄨󵄨󵄨z n+1 󵄨󵄨󵄨 ) − (󵄨󵄨󵄨yn 󵄨󵄨󵄨 + 󵄨󵄨󵄨z n 󵄨󵄨󵄨 ) + 󵄨󵄨󵄨yn+1 − yn 󵄨󵄨󵄨 󵄨 󵄨2 󵄩 󵄩2 󵄩 󵄩2 + 󵄨󵄨󵄨z n+1 − z n 󵄨󵄨󵄨 + 2kν(󵄩󵄩󵄩yn+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩 ) ≤ 2k(f n , yn+1 + z n+1 ) − 2k(b(yn , yn , yn+1 − yn ))

+ b(yn , z n+1 , yn+1 − yn ) + b(z n+1 , yn , yn+1 − yn ),

(1.6.15)

110 | 1 Discrete attractor and approximate calculation 󵄨 n 󵄨 󵄩 n 󵄩󵄨 n+1 󵄨󵄨 n n n+1 n󵄨 n 󵄨 󵄨󵄨b(y , y , y − y )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨y 󵄨󵄨󵄨L∞ 󵄩󵄩󵄩y 󵄩󵄩󵄩󵄨󵄨󵄨y − y 󵄨󵄨󵄨, 󵄨 n 󵄨 󵄩 n+1 󵄩󵄨 n+1 󵄨󵄨 n n+1 n+1 n󵄨 n 󵄨 󵄨󵄨b(y , z , y − y )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨y 󵄨󵄨󵄨L∞ 󵄩󵄩󵄩z 󵄩󵄩󵄩󵄨󵄨󵄨y − y 󵄨󵄨󵄨, 󵄨 n 󵄨 󵄩 n+1 󵄩󵄨 n+1 󵄨󵄨 n+1 n n+1 n󵄩 n 󵄨 󵄨 n+1 󵄨󵄩 n+1 n 󵄨 󵄨󵄨b(z , y , y − y )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨y 󵄨󵄨󵄨L∞ (󵄩󵄩󵄩z 󵄩󵄩󵄩󵄨󵄨󵄨y − y 󵄨󵄨󵄨 + 󵄨󵄨󵄨z 󵄨󵄨󵄨󵄩󵄩󵄩y − y 󵄩󵄩󵄩) 󵄨 󵄩󵄨 󵄨 󵄨 󵄩 ≤ 󵄨󵄨󵄨yn 󵄨󵄨󵄨L∞ 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩󵄨󵄨󵄨yn+1 − yn 󵄨󵄨󵄨(1 + SN /SN+1 ) 󵄨 󵄩󵄨 󵄨 󵄨 󵄩 ≤ 2󵄨󵄨󵄨yn 󵄨󵄨󵄨L∞ 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩󵄨󵄨󵄨yn+1 − yn 󵄨󵄨󵄨, and then 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 (󵄨󵄨󵄨yn+1 󵄨󵄨󵄨 + 󵄨󵄨󵄨z n+1 󵄨󵄨󵄨 ) − (󵄨󵄨󵄨yn 󵄨󵄨󵄨 + 󵄨󵄨󵄨z n 󵄨󵄨󵄨 ) + 󵄨󵄨󵄨yn+1 − yn 󵄨󵄨󵄨 󵄨 󵄨2 󵄩 󵄩2 󵄩 󵄩2 + 󵄨󵄨󵄨z n+1 − z n 󵄨󵄨󵄨 + 2kν(󵄩󵄩󵄩yn+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩 ) 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩󵄨 󵄨 ≤ 2k 󵄨󵄨󵄨f n 󵄨󵄨󵄨(󵄨󵄨󵄨yn+1 󵄨󵄨󵄨 + 󵄨󵄨󵄨z n+1 󵄨󵄨󵄨) + 2k 󵄨󵄨󵄨yn 󵄨󵄨󵄨L∞ (󵄩󵄩󵄩yn 󵄩󵄩󵄩 + 3󵄩󵄩󵄩z n+1 󵄩󵄩󵄩)󵄨󵄨󵄨yn+1 − yn 󵄨󵄨󵄨.

(1.6.16)

If |yj |L∞ ≤ M1 , ∀j, we derive 1󵄨 󵄨2 󵄩 󵄩2 (1.6.16) ≤ 󵄨󵄨󵄨yn+1 + yn 󵄨󵄨󵄨 − 4k 2 {M12 󵄩󵄩󵄩yn 󵄩󵄩󵄩 2 󵄩 󵄩2 󵄩 󵄩 󵄩 󵄩󵄨 󵄨 + 9M12 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩 } + 2kC1 {󵄩󵄩󵄩yn+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩}󵄨󵄨󵄨f n 󵄨󵄨󵄨. Therefore, 1 󵄨 n+1 󵄨󵄨 n+1 󵄨󵄨2 󵄨󵄨 n+1 󵄨󵄨2 󵄨󵄨 n 󵄨󵄨2 󵄨󵄨 n 󵄨󵄨2 n 󵄨2 󵄨 n+1 n 󵄨2 󵄨󵄨y 󵄨󵄨 + 󵄨󵄨z 󵄨󵄨 − (󵄨󵄨y 󵄨󵄨 + 󵄨󵄨z 󵄨󵄨 ) + 󵄨󵄨󵄨y − y 󵄨󵄨󵄨 + 󵄨󵄨󵄨z − z 󵄨󵄨󵄨 2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 + 2kν󵄩󵄩󵄩yn+1 󵄩󵄩󵄩 − 4k 2 M12 󵄩󵄩󵄩yn 󵄩󵄩󵄩 + 2k(ν − 18kM12 )󵄩󵄩󵄩z n+1 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄨 󵄨 ≤ 2kC1 {󵄩󵄩󵄩yn+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩z n+1 󵄩󵄩󵄩}󵄨󵄨󵄨f n 󵄨󵄨󵄨.

(1.6.17)

If the constant k satisfies the following nonlinear stability condition: k< we obtain

ν 4M12

and k
0 is Rayleigh constant. Equations (1.7.1)–(1.7.3) can be

1.7 Two-dimensional Newton–Boussinesq equation

| 117

rewritten as R 𝜕θ 𝜕 ΔΨ + J(Ψ, ΔΨ) = Δ2 Ψ − a , 𝜕t Pr 𝜕x 𝜕θ 1 + J(Ψ, θ) = Δθ, 𝜕t Pr

(1.7.4) (1.7.5)

where J(u, v) = uy vx − ux vy . Consider the periodic initial value problem of (1.7.4)–(1.7.5), namely Ψ(x + 2D, y, t) = Ψ(x, y, t), θ(x + 2D, y, t) = θ(x, y, t), Ψ(x, y, 0) = Ψ0 (x, y),

Ψ(x, y + 2D, t) = Ψ(x, y, t), θ(x, y + 2D, t) = θ(x, y, t),

(1.7.6)

θ(x, y, 0) = θ0 (x, y),

(1.7.7)

where Ψ0 (x, y) and θ0 (x, y) are given 2D-periodic functions Suppose that {wj (x, y)}, j = 1, 2, . . . are the periodic eigenvectors of operator A = −Δ and satisfy − Δωj = λj ωj ,

j = 1, 2, . . . , λ1 < λ2 < ⋅ ⋅ ⋅ .

(1.7.8)

For every integer m, we seek an approximate solution of the following form: um (x, y, t) = Ψm (x, y, t) + ξm (x, y, t),

θm (x, y, t) = θm (x, y, t) + ηm (x, y, t), m

m

j=1

j=1

Ψm (x, y, t) = ∑ αjm (t)ωj , θm (x, y, t) = ∑ βjm (t)ωj , 2m

2m

j=m+1

j=m−1

(1.7.9)

ξm (x, y, t) = ∑ δjm (t)ωj , ηm (x, y, t) = ∑ γjm (t)ωj .

(1.7.10)

Ψm (t), θm (t) : ℝ+ → Wm , Wm is the subspace span{w1 , w2 , . . . , wm }, ξm (t), ηm (t) : ℝ+ → ̃m , W ̃m is the subspace span{wm+1 , wm+2 , . . . , w2m }. Then (Ψm , ξm ) and (θm , ηm ) satisfy W the following equations: d (−ΔΨm , v) + r(Ψ, ΔΨm , v) + r(ξm , ΔΨm , v) + r(Ψm , Δξm , v) dt R + (ΔΨm , Δv) + a (θmx , v) = 0, ∀v ∈ Wm , Pr R ̃m , (Δξm , Δv) + r(Ψm , ΔΨm , v) + a (θmx , v) = 0, ∀v ∈ W Pr d 1 (θ , v) + r(Ψm , θm , v) + r(ξm , θm , v) + r(Ψm , ηm , v) = (Δθm , v), dt m Pr

(1.7.11) (1.7.12) v ∈ Wm ,

(1.7.13)

118 | 1 Discrete attractor and approximate calculation 1 (Δηm , v) + r(Ψm , θm , v) = 0, Pr Ψm (0, x, y) = Pm Ψ0 (x, y), −

̃m , v∈W

(1.7.14) (1.7.15)

θm (0, x, y) = Pm θ0 (x, y),

(1.7.16)

where Pm denotes the orthogonal projection from L2 to Wm , r(u, v, w) = ∬(uy vx − ux vy )wdxdy,

Ω = [0, 2D] × [0, 2D].

Now we give uniform a priori estimates of approximate solutions. Lemma 1.7.1. Let ∇Ψ0 (x, y) ∈ L2 (Ω), θ0 (x, y) ∈ L2 (Ω). Then for the solution {Ψm , ξm } and {θm , ηm } of problem (1.7.11)–(1.7.16), the following estimates hold: 1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∇Ψm (⋅, ⋅)󵄩󵄩󵄩 ≤ 2󵄩󵄩󵄩∇Ψm (⋅, 0)󵄩󵄩󵄩 exp(− t) C + C02 ( α

0

2

Ra 󵄩󵄩 󵄩2 ) 󵄩θ (⋅, 0)󵄩󵄩󵄩 ≤ E0 , Pr 󵄩 m

(1.7.17)

T

󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 ∫(󵄩󵄩󵄩∇θm (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇ηm (t)󵄩󵄩󵄩 )dt + ∫ 󵄩󵄩󵄩ΔΨm (t)󵄩󵄩󵄩 dt ≤ E1 ,

(1.7.18)

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩θm (⋅, t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩θm (⋅, 0)󵄩󵄩󵄩,

(1.7.19)

0

t ≥ 0,

0

where E0 and E1 are constants independent of M, C0 is the minimum constant satisfying the following Poincaré inequality: ‖u‖ ≤ C0 ‖∇u‖,

(1.7.20)

where ∬ udxdy = 0. Proof. Taking v = Ψm in (1.7.11), v = ξm in (1.7.13), v = θm in (1.7.13) and v = ηm in (1.7.14), we have (−ΔΨm , Ψm ) + r(Ψm , ΔΨm , Ψm ) + r(ξm , ΔΨm , Ψm ) R + r(Ψm , Δξm , Ψm ) + (ΔΨm , ΔΨm ) + a (θmr , Ψm ) = 0, Pr Ra (Δξm , Δξm ) + r(Ψm , θm , ξm ) + (θ , Ψ ) = 0, Pr mx m (θmt , θm ) + r(Ψm , θm , θm ) + r(ξm , θm , θm ) 1 + r(Ψm , ηm , θm ) = (Δθm , θm ), Pr 1 − (Δηm , ηm ) + r(Ψm , θm , ηm ) = 0, Pr

(1.7.21) (1.7.22) (1.7.23) (1.7.24)

1.7 Two-dimensional Newton–Boussinesq equation

| 119

where r(Ψm , ΔΨm , Ψm ) = ∬[Ψmy (ΔΨm )x − Ψmx (ΔΨm )y ]Ψmdxdy =

1 ∬[(Ψm2 )y (ΔΨm )x − (Ψm2 )x (ΔΨm )y ]mdxdy = 0, 2

r(ηm , ΔΨm , Ψm ) = ∬[ξmy (ΔΨm )x − ξmx (ΔΨm )y ]Ψm dxdy = − ∬ ξm [(ΔΨm )xy Ψm + (ΔΨm )x Ψmy ]dxdy + ∬ ξm [(ΔΨm )xy Ψm + (ΔΨm )y Ψmx ]dxdy = − ∬ ξm [Ψmy (ΔΨm )x − Ψmx (ΔΨm )y ]dxdy = −r(Ψm , ΔΨm , ξm ). Similarly, we have r(Ψm , Δξ , Ψm ) = 0,

r(Ψm , θm , θm ) = 0,

r(Ψm , ηm , θm ) = −r(Ψm , θm , ηm ).

r(ξm , θm , θm ) = 0,

Adding (1.7.21) to (1.7.22) and (1.7.23) to (1.7.24), respectively, gives (−ΔΨm , Ψm ) + (ΔΨm , ΔΨm ) + (θm , θm ) +

Ra (θ , Ψ + ξ ) + (Δξm , Δξm ) = 0, Pr mx m m

1 [(∇θm , ∇θm ) + (∇ηm , ∇ηm )] = 0. Pr

(1.7.25) (1.7.26)

It follows from (1.7.26) that 1 d 󵄩󵄩 󵄩2 1 󵄩 󵄩2 󵄩 󵄩2 󵄩θ (t)󵄩󵄩 + (󵄩󵄩󵄩∇θm (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇ηm (t)󵄩󵄩󵄩 ) = 0, 2 dt 󵄩 m 󵄩 Pr 󵄩󵄩󵄩θm (⋅, t)󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩θm (⋅, 0)󵄩󵄩󵄩2 , 󵄩 󵄩 󵄩 󵄩

(1.7.27) (1.7.28)

∞

P 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 ∫ (󵄩󵄩󵄩∇θm (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇ηm (t)󵄩󵄩󵄩 )dt ≤ r 󵄩󵄩󵄩θm (⋅, 0)󵄩󵄩󵄩 < ∞. 2

(1.7.29)

0

It follows from (1.7.25) that 1 d ‖∇Ψm ‖2 + ‖ΔΨm ‖2 + ‖Δξm ‖2 2 dt ≤

2

R 1 (‖Ψmx ‖2 + ‖ξmx ‖) + C0 ( a ) ‖θm ‖2 2C0 Pr 2

R 1 ≤ (‖ΔΨm ‖2 + ‖Δξm ‖2 ) + C0 ( a ) ‖θm ‖2 , 2 Pr

(1.7.30)

120 | 1 Discrete attractor and approximate calculation where ‖Ψmx ‖2 ≤ ‖∇Ψm ‖2 ≤ C0 ‖ΔΨm ‖2 ,

∬ ∇Ψm dxdy = 0.

From (1.7.30) we have 2

R 1 d 1 󵄩2 󵄩 ‖∇Ψm ‖2 + ‖∇Ψm ‖2 ≤ C0 ( a ) 󵄩󵄩󵄩θm (⋅, 0)󵄩󵄩󵄩 . 2 dt 2C0 Pr

(1.7.31)

By Gronwall inequality, 1 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩∇Ψm (⋅, t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩∇Ψm (⋅, 0)󵄩󵄩󵄩 exp(− t) C 0

+ C02 (1 − exp(−

2

R 1 󵄩 󵄩2 t))( a ) 󵄩󵄩󵄩θm (⋅, 0)󵄩󵄩󵄩 . C0 Pr

Lemma 1.7.2 (Uniform Gronwall lemma, ([91])). Suppose that g, h, y are three positive locally integrable functions on [t0 , ∞), y󸀠 is also locally integrable function on [t0 , ∞) and g, h, y satisfy the inequalities

t+r

dy ≤ gy + h, dt

t ≥ t0 ,

(1.7.32)

t+r

∫ g(s)ds ≤ a1 ,

∫ h(s)ds ≤ a2 ,

t

t

∫ y(s)ds ≤ a3 ,

t ≥ t0 ,

t+r

(1.7.33)

t

where a1 , a2 , a3 are positive constants. Then y(t + r) ≤ (

a3 + a2 ) exp(a1 ), r

t ≥ t0 .

(1.7.34)

Lemma 1.7.3. Suppose that the conditions of Lemma 1.7.1 are satisfied, and ΔΨm (⋅, 0) ∈ L2 (Ω), ∇θm (⋅, 0) ∈ L2 (Ω). Then for the solution {Ψm , ξm } and {θm , ηm } of problem (1.7.11)– (1.7.16), we have 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩∇θm (⋅, t)󵄩󵄩󵄩 + 󵄩󵄩󵄩ΔΨm (⋅, t)󵄩󵄩󵄩 ≤ E2 ,

t ≥ 0,

(1.7.35)

T

󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 ∫(󵄩󵄩󵄩∇ΔΨm (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇Δξm (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩Δθm (t)󵄩󵄩󵄩 )dt ≤ E3 , 0

where the constants E2 and E3 are all independent of m.

t ≥ 0,

(1.7.36)

1.7 Two-dimensional Newton–Boussinesq equation

| 121

Proof. Take v = −ΔΨm in (1.7.11), v = −Δξm in (1.7.12), v = −Δθm in (1.7.13), and v = −Δηm in (1.7.14), then

(−ΔΨmt , −ΔΨm ) + r(Ψm , ΔΨm , −ΔΨm ) + r(ξm , ΔΨm , −ΔΨm ) R + r(Ψm , Δξm , −ΔΨm ) + (ΔΨm , −Δ2 Ψm ) + a (θmx , −ΔΨm ) = 0, Pr Ra 2 (Δξm , −Δ ξm ) + r(Ψm , ΔΨm , −Δξm ) + (θ , −Δξm ) = 0, Pr mx (θmt , −Δθm ) + r(Ψm , θm , −Δθm ) + r(ξm , θm , −Δθm ) + 1 + r(Ψm , ηm , −Δθm ) = (Δθm , −Δθm ), Pr 1 − (Δηm , −Δηm ) + r(Ψm , θm , −Δηm ) = 0, Pr

(1.7.37) (1.7.38) (1.7.39) (1.7.40)

where 1 d ‖ΔΨm ‖2 , 2 dt (ΔΨm , −Δ2 Ψm ) = ‖ΔΔΨm ‖2 , 1 d ‖∇θm ‖2 , (θmt , −Δθm ) = 2 dt 1 1 − (Δηm , −Δηm ) = ‖Δηm ‖2 , Pr Pr r(Ψm , ΔΨm , −ΔΨm ) = 0, r(ξm , ΔΨm , −ΔΨm ) = 0, (−ΔΨmt , −ΔΨm ) =

r(Ψm , Δξm , −ΔΨm ) = r(Ψm , ΔΨm , Δξm ) 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , θm , −Δθm )󵄨󵄨󵄨 ≤ (‖Ψmx ‖∞ + ‖Ψmy ‖∞ )‖∇θm ‖‖Δθm ‖, 1

1

1

1

1

1

‖Ψmx ‖∞ ≤ C‖Ψmx ‖ 2 ‖ΔΨmx ‖ 2 ≤ C1 ‖∇ΔΨm ‖ 2 , ‖Ψmy ‖∞ ≤ C‖Ψmy ‖ 2 ‖ΔΨmy ‖ 2 ≤ C1 ‖∇ΔΨm ‖ 2 , 1 󵄨󵄨 󵄨 ‖Δθm ‖2 + 2Pr C12 ‖∇ΔΨm ‖‖∇θm ‖2 󵄨󵄨r(Ψm , θm , −Δθm )󵄨󵄨󵄨 ≤ 4Pr 1 1 ≤ ‖Δθm ‖2 + ‖∇ΔΨm ‖2 + 3Pr2 C14 ‖∇θm ‖2 , 4Pr 3 󵄨󵄨󵄨r(ξm , θm , −Δθm )󵄨󵄨󵄨 ≤ 1 ‖Δθm ‖2 + 1 ‖∇Δξm ‖2 + 3P 2 C 4 ‖∇θm ‖2 , r 1 󵄨 󵄨 4P 3 r 󵄨󵄨 󵄨󵄨 󵄨󵄨r(Ψm , ηm , −Δθm )󵄨󵄨 ≤ (‖Ψmy ‖∞ + ‖Ψmx ‖∞ )‖∇ηm ‖‖Δθm ‖

(1.7.41) (1.7.42)

1

≤ 2C1 ‖∇ΔΨm ‖ 2 ‖∇ηm ‖‖Δθm ‖ 1 1 ≤ ‖Δθm ‖2 + ‖∇ΔΨm ‖2 + 3Pr2 C12 ‖∇ηm ‖4 4Pr 3 1 1 ≤ ‖Δθm ‖2 + ‖∇ΔΨm ‖2 + 3Pr2 C12 ‖ηm ‖2 ‖∇ηm ‖2 . 4Pr 3

(1.7.43)

122 | 1 Discrete attractor and approximate calculation In order to estimate r(Ψm , ηm , −Δθm ), we need to estimate ‖ηm ‖2 . To this end, (1.7.24) implies 1 ‖∇ηm ‖2 + r(Ψm , ηm , θm ) = 0, Pr where 󵄨 󵄨󵄨 󵄨󵄨r(Ψm , θm , ηm )󵄨󵄨󵄨 ≤ (‖Ψmx ‖∞ + ‖Ψmy ‖∞ )‖∇θm ‖‖ηm ‖ 1

1

1

≤ 2C1 ‖∇ΔΨm ‖ 2 ‖∇θm ‖‖ηm ‖ ≤ 2C1 λm2 ‖Ψm ‖λm2 ‖θm ‖‖ηm ‖

≤ 2C1 λm E0 ‖∇θ(⋅, 0)‖‖ηm ‖. If

‖∇ηm ‖2 ≥ λm+1 ‖ηm ‖2 , we have λ 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩ηm (t)󵄩󵄩󵄩 ≤ 2C1 E0 ( m )󵄩󵄩󵄩ηm (⋅, 0)󵄩󵄩󵄩 ≤ 2C1 E0 󵄩󵄩󵄩ηm (⋅, 0)󵄩󵄩󵄩. λ m+1

(1.7.44)

Substituting (1.7.44) into (1.7.43) then leads to 1 󵄨󵄨 󵄨 ‖Δθm ‖2 + 󵄨󵄨r(Ψm , ηm , −Δθm )󵄨󵄨󵄨 ≤ 4Pr 1 ‖Δθm ‖2 + ≤ 4Pr

1 󵄩 󵄩2 ‖∇ΔΨm ‖2 + 12Pr2 C14 E02 󵄩󵄩󵄩θ(⋅, 0)󵄩󵄩󵄩 ‖Δηm ‖2 3 C 1 (1.7.45) ‖∇ΔΨm ‖2 + 2 ‖Δηm ‖2 . 3 Pr

On the other hand, 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , ηm , −Δηm )󵄨󵄨󵄨 ≤ (‖Ψmx ‖∞ + ‖Ψmy ‖∞ )‖∇θm ‖‖Δηm ‖ 1 1 ‖Δηm ‖2 + ‖∇ΔΨm ‖2 ≤ 2Pr (C2 + 1) 4 + Pr2 (C2 + 1)2 C14 ‖∇θm ‖4 .

Therefore, it follows from (1.7.37)–(1.7.40), (1.7.45) and (1.7.46) that 1 d 1 ‖ΔΨm ‖2 + ‖∇ΔΨm ‖2 + ‖∇Δηm ‖2 2 dt 2 (C + 1) 1 1 d + ‖Δθm ‖2 + ‖∇θm ‖2 + 2 ‖Δηm ‖2 Pr 2 dt Pr R 3 ≤ a ‖∇θm ‖‖ΔΨm ‖ + ‖Δθm ‖2 Pr 4Pr C 1 + ‖Δηm ‖2 + Pr2 (C2 + 1)3 C14 ‖∇θm ‖4 + 2 ‖Δηm ‖2 2Pr Pr

(1.7.46)

1.7 Two-dimensional Newton–Boussinesq equation

+

11 ‖∇ΔΨm ‖2 + (6Pr2 C14 + Pr2 (C2 + 1)2 C14 )‖∇θm ‖4 + C, 12

| 123

(1.7.47)

that is, 1 d 1 1 d ‖ΔΨm ‖2 + ‖∇θm ‖2 + ‖∇ΔΨm ‖2 2 dt 2 dt 12 1 1 1 + ‖∇Δξm ‖2 + ‖Δηm ‖2 + ‖Δθm ‖2 2 2Pr 4Pr R ≤ a ‖∇θm ‖‖ΔΨm ‖2 + C3 ‖∇θm ‖4 + C4 , Pr

(1.7.48)

where C3 = Pr2 (C2 + 1)3 C14 + 6Pr2 C14 + Pr2 (C2 + 1)2 C14 . Equation (1.7.48) can be written in the following form: dym ≤ gm ym + hm , dt

(1.7.49)

1 ym (t) = (‖ΔΨm ‖2 + ‖∇θm ‖2 ), 2 gm (t) = C3 ‖∇θm ‖2 , R hm (t) = a [‖∇θm ‖2 + ‖ΔΨm ‖2 ] + C4 . 2Pr

(1.7.50)

where

t+1

Integrating (1.7.27) and (1.7.31) with respect to s ∈ (t, t + 1), we find that ∫t t+1

‖∇θm ‖2 ds

and ∫t ‖ΔΨm ‖2 ds are bounded ∀t ≥ 0 and independent of m. Hence, ym (t), hm (t), gm (t) are given by (1.7.51) which satisfy (1.7.33), and the constants ai , i = 1, 2, 3 are independent of m. Equation (1.7.34) implies 1 󵄩 󵄩2 󵄩 󵄩2 ym (t) = (󵄩󵄩󵄩ΔΨm (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇θm (t)󵄩󵄩󵄩 ) ≤ C5 , 2

t ≥ 1,

(1.7.51)

where the constant C3 is independent of m. By the usual Gronwall inequality, ym (t), 0 ≤ t ≤ 1 is uniformly bounded, which follows from (1.7.49). Thus the proof of (1.7.35) is completed. Equation (1.7.36) holds by integrating (1.7.48) with respect to t ∈ [0, T]. Lemma 1.7.4. Under the conditions of Lemma 1.7.3, we have 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩 ΔΨ 󵄩󵄩 ≤ E4 , 󵄩󵄩 ΔΨm 󵄩󵄩󵄩 󵄩󵄩 dt 󵄩󵄩L2 ([0,T];H −1 (Ω)) 󵄩󵄩󵄩 dt m 󵄩󵄩󵄩L∞ (ℝ+ ;H −2 (Ω)) 󵄩󵄩 dθ 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 ≤ E5 , 󵄩󵄩 󵄩󵄩 󵄩󵄩 dt 󵄩󵄩L∞ (ℝ+ ;H −1 (Ω)) where the constants E4 and E5 are independent of m.

(1.7.52)

124 | 1 Discrete attractor and approximate calculation Proof. It follows from (1.7.11) that 󵄨󵄨 d 󵄨󵄨 󵄨 R 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 (ΔΨm , v)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨r(Ψm , ΔΨm , v)󵄨󵄨󵄨 + 󵄨󵄨󵄨(ΔΨm , Δv)󵄨󵄨󵄨 + a 󵄨󵄨󵄨(θmx , v)󵄨󵄨󵄨 󵄨󵄨 dt 󵄨󵄨 Pr 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨r(ξm , ΔΨm , v)󵄨󵄨󵄨 + 󵄨󵄨󵄨r(Ψm , Δξm , v)󵄨󵄨󵄨, v ∈ Wm ,

(1.7.53)

where 󵄩 󵄨 󵄩 󵄨󵄨 󵄨󵄨(ΔΨm , Δv)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩∇(ΔΨm )󵄩󵄩󵄩‖∇v‖, 󵄨󵄨 R 󵄨󵄨 R 󵄨󵄨 a 󵄨 󵄨󵄨 (θmx , v)󵄨󵄨󵄨 ≤ a ‖∇θm ‖‖v‖, 󵄨󵄨 Pr 󵄨󵄨 Pr 󵄨󵄨 󵄨󵄨 󵄨 󵄨 r(Ψm , ΔΨm , v) = 󵄨󵄨󵄨 ∬(Ψmy vx − Ψmx vy )ΔΨm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(1.7.54) (1.7.55)

≤ (‖Ψmy ‖L4 + ‖Ψmx ‖L4 )‖ΔΨm ‖L4 ‖∇v‖, 1

1

1

‖Ψmy ‖L4 ≤ C‖Ψmy ‖ 2 ‖Ψmy ‖H2 1 ≤ C‖ΔΨm ‖ 2 ≤ const,

‖Ψmx ‖L4 ≤ const,

1 1 󵄩 󵄩1 ‖ΔΨm ‖L4 ≤ C‖ΔΨm ‖ 2 ‖ΔΨm ‖H2 1 ≤ C 󵄩󵄩󵄩∇(ΔΨm )󵄩󵄩󵄩 2 , 󵄨󵄨 󵄨 󵄩 󵄩1 󵄨󵄨r(Ψm , ΔΨm , v)󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩∇(ΔΨm )󵄩󵄩󵄩 2 ‖v‖H 1 .

(1.7.56)

The estimates of the forth and fifth terms on the right-hand side of (1.7.53) are as follows: 󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨r(ξm , ΔΨm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(ξmy vx − ξmx vy )ΔΨm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (‖ξmy ‖∞ + ‖ξmx ‖∞ )‖ΔΨm ‖‖v‖H 1 1

≤ C‖∇Δξm ‖ 2 ‖v‖H 1 , that is, r(ξm , ΔΨm , v) ∈ L4 ([0, T]), ‖v‖H 1 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , Δξm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy vx − Ψmx vy )Δξm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(1.7.57)

≤ (‖Ψmy ‖L4 + ‖Ψmx ‖L4 )‖Δξm ‖L4 ‖v‖H 1 1

≤ C‖∇Δξm ‖ 2 ‖v‖H 1 .

(1.7.58)

Since v ∈ Wm is dense in v ∈ H 1 , by (1.7.53)–(1.7.58) we have ‖ΔΨm ‖L2 ([0,T];H −1 (Ω)) ≤ E4 ,

(1.7.59)

where the constant E4 is independent of m. On the other hand, it follows from (1.7.53) that 󵄨󵄨 󵄨 󵄨󵄨(ΔΨm , Δv)󵄨󵄨󵄨 ≤ ‖ΔΨm ‖‖Δv‖ ≤ C‖v‖H 2 ,

(1.7.60)

1.7 Two-dimensional Newton–Boussinesq equation

󵄨󵄨 R 󵄨󵄨 R 󵄨 󵄨󵄨 a 󵄨󵄨 (θmx , Δv)󵄨󵄨󵄨 ≤ a ‖∇θm ‖‖Δv‖ ≤ C‖v‖H 2 , 󵄨󵄨 Pr 󵄨󵄨 Pr 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨r(Ψm , ΔΨm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy vx − Ψmx vy )ΔΨm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

| 125

(1.7.61)

≤ (‖Ψmy ‖L4 ‖vx ‖L4 + ‖Ψmx ‖L4 ‖vy ‖L4 )‖ΔΨm ‖ ≤ C‖v‖H 2 , 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨r(ξm , ΔΨm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy vx − Ψmx vy )ΔΨm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (‖ξmy ‖L4 ‖vx ‖L4 + ‖ξmx ‖L4 ‖vy ‖L4 )‖ΔΨm ‖

(1.7.62)

≤ C‖∇ξm ‖ 2 ‖Δξm ‖ 2 ‖v‖H 2 , 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨r(ξm , Δξm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy vx − Ψmx vy )Δξm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(1.7.63)

1

1

≤ (‖Ψmy ‖L4 ‖vx ‖L4 + ‖Ψmx ‖L4 ‖vy ‖L4 )‖Δξm ‖

≤ C‖Δξm ‖‖v‖H 2 .

(1.7.64)

Now we estimate ‖∇ξm ‖ and ‖Δξm ‖. By (1.7.22) we get ‖Δξm ‖2 + r(Ψm , ΔΨm , ξm ) = 0, where 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , Δξm , ξm )󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy (ΔΨm )x − Ψmx (ΔΨm )y )ξm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

3

≤ C(‖Ψmy ‖∞ + ‖Ψmx ‖∞ )‖∇ΔΨm ‖‖ξm ‖ ≤ Cλm4 ‖ξm ‖, 3

2 λm+1 ‖ξm ‖2 ≤ ‖Δξm ‖2 ≤ Cλm4 ‖ξm ‖, 3

4 λ λm+1 ‖ξm ‖ ≤ C( m ) ≤ const. λm+1 5 4

(1.7.65)

On the other hand, 󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨r(Ψ, ΔΨm , ξm )󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy ΔΨmx − Ψmx ΔΨmy )ξm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨 ∬(Ψmy ξmx − Ψmx ξmy )ΔΨm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (‖Ψmy ‖∞ + ‖Ψmx ‖∞ )‖∇ξm ‖‖ΔΨm ‖ 1

1

≤ C‖∇ΔΨm ‖ 2 ‖∇ξm ‖ ≤ Cλm4 ‖∇ξm ‖, 3

4 λm+1 ‖∇ξm ‖ ≤ C(

1

λm 4 ) ≤ const, λm+1 1

λm+1 ‖∇ξm ‖2 ≤ ‖Δξm ‖2 ≤ Cλm4 ‖∇ξm ‖2 , 3

4 λm+1 ‖∇ξm ‖ ≤ C(

1 4

λm ) ≤ const. λm+1

(1.7.66) (1.7.67) (1.7.68)

126 | 1 Discrete attractor and approximate calculation It follows from (1.7.68) that ‖∇Δξm ‖2 + r(Ψm , ΔΨm , −Δξm ) +

Ra (θ , Δξm ) = 0, Pr mx

(1.7.69)

where 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨r(Ψm , ΔΨm , −Δξm )󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy ΔΨmx − Ψmx ΔΨmy )Δξm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (‖Ψmy ‖∞ + ‖Ψmx ‖∞ )‖∇ΔΨm ‖‖Δξm ‖ 3

3

≤ C‖∇ΔΨm ‖ 2 ‖Δξm ‖ ≤ Cλm4 ‖Δξm ‖, 3

λm+1 ‖Δξm ‖2 ≤ ‖∇Δξm ‖2 ≤ C(λm4 + 1)‖Δξm ‖, 3

1 4

λm ‖Δξm ‖ ≤ C(

λm4 + 1 3

λm4

) ≤ const.

(1.7.70)

Therefore, by (1.7.66), (1.7.70), (1.7.63) and (1.7.64), we have 󵄨󵄨 󵄨 󵄨󵄨r(ξm , ΔΨm , v)󵄨󵄨󵄨 ≤ C‖v‖H 2 , 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , Δξm , v)󵄨󵄨󵄨 ≤ C‖v‖H 2 .

(1.7.71) (1.7.72)

From (1.7.53), (1.7.60)–(1.7.72), we get ‖ΔΨM ‖L∞ (ℝ+ ;H −2 (Ω)) ≤ E4 ,

(1.7.73)

where the constant E3 is independent of m. It follows from (1.7.13) that 󵄨󵄨 󵄨󵄨 d 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 (θm , v)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨r(Ψm , θm , v)󵄨󵄨󵄨 + 󵄨󵄨󵄨r(ξm , θm , v)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 dt 󵄨 󵄨 󵄨 1󵄨 + 󵄨󵄨󵄨r(Ψm , ηm , v)󵄨󵄨󵄨 + 󵄨󵄨󵄨(Δθm , v)󵄨󵄨󵄨, Pr

(1.7.74)

where 1 󵄨󵄨 󵄨 1 󵄨(Δθ , v)󵄨󵄨 ≤ ‖∇θm ‖‖∇v‖ ≤ C‖v‖H 1 , Pr 󵄨 m 󵄨 Pr 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , θm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy θmx − Ψmx θmy )vdxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

≤ (‖Ψmy ‖L4 ‖θmx ‖ + ‖Ψmx ‖L4 ‖θmy ‖)‖v‖L4

≤ C‖ΔΨm ‖‖∇θm ‖‖v‖H 1 ≤ C‖v‖H 1 , 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨r(ξm , θm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(ξmy θmx − ξmx θmy )vdxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

≤ (‖ξmy ‖L4 ‖∇θm ‖ + ‖Ψmx ‖L4 ‖∇θm ‖)‖v‖L4

(1.7.75)

(1.7.76)

1.7 Two-dimensional Newton–Boussinesq equation

1

1

≤ C‖∇ξm ‖ 2 ‖Δξm ‖ 2 ‖∇θm ‖‖v‖H 1 ≤ C‖v‖H 1 , 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨r(Ψm , ηm , v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy ηmx − Ψmx ηmy )vdxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (‖Ψmy ‖L4 ‖∇ηm ‖ + ‖Ψmx ‖L4 ‖∇ηm ‖)‖v‖L4 ≤ ‖∇ηm ‖H 1 ‖v‖H 1 .

| 127

(1.7.77)

(1.7.78)

Now we need to estimate ‖∇ηm ‖. From (1.7.40) we have 1 ‖Δηm ‖2 + r(Ψm , θm , −Δηm ) = 0, Pr

(1.7.79)

where 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , ηm , −Δηm )󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∬(Ψmy θmx − Ψmx θmy )Δηm dxdy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ (‖Ψmy ‖∞ + ‖Ψmx ‖∞ )‖∇θm ‖‖Δηm ‖ 1

1

≤ C‖∇ΔΨm ‖ 2 ‖Δηm ‖ ≤ Cλm4 ‖Δηm ‖. Equation (1.7.79) implies 1

‖Δηm ‖2 ≤ CPr λm4 ‖Δηm ‖, which in turn gives 1

1

2 λm+1 ‖∇ηm ‖ ≤ ‖Δηm ‖ ≤ CPr λm4 , 1

4 λm+1 ‖∇ηm ‖ ≤ CPr (

λm ) ≤ const. λm+1

(1.7.80)

By (1.7.78) we have 󵄨󵄨 󵄨 󵄨󵄨r(Ψm , ηm , v)󵄨󵄨󵄨 ≤ C‖v‖H 1 .

(1.7.81)

It follows from (1.7.74), (1.7.76), (1.7.77) and (1.7.81) that 󵄩󵄩 dθ 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 ≤ E4 , 󵄩 󵄩󵄩 󵄩󵄩 dt 󵄩󵄩󵄩L∞ (ℝ+ ;H −1 (Ω))

(1.7.82)

where the constant E4 is independent of m. The proof of Lemma 1.7.4 is completed. Definition 1.7.1. The unknown functions Ψ(x, y, t), θ(x, y, t) are called the global generalized solution, if they satisfy: (1) Ψ(x, y, t) ∈ L∞ (ℝ+ ; H 2 (Ω)),

ΔΨ(x, y, t) ∈ L2 ([0, T]; H 1 (Ω)),

128 | 1 Discrete attractor and approximate calculation ΔΨt (x, y, t) ∈ L∞ (ℝ+ ; H −2 (Ω)) ∩ L2 ([0, T]; H −1 (Ω)),

θ(x, y, t) ∈ L∞ (ℝ+ ; H 1 (Ω)), Δθ(x, y, t) ∈ L2 ([0, T]; L2 (Ω)), θt (x, y, t) ∈ L∞ (ℝ+ ; H −1 (Ω)),

Ω = [0, 2D] × [0, 2D].

(2) The following equalities hold in L1 ([0, T]), ∀T > 0: d (ΔΨ, v) + r(Ψ, ΔΨ, v) + (ΔΨ, Δv) dt R + a (θx , v) = 0, v ∈ H 2 (Ω), Pr d 1 (θ, v1 ) + r(Ψ, θ, v1 ) − (∇θ, ∇v1 ) = 0, dt Pr

(1.7.83) v1 ∈ H 1 (Ω),

(1.7.84)

where r(u, v, w) = ∬(uy vx − ux vy )wdxdy. (3) Ψ(x + 2D, y, t) = Ψ(x, y, t), θ(x + 2D, y, t) = θ(x, y, t), 2

(x, y) ∈ ℝ ,

Ψ(x, y + 2D, t) = Ψ(x, y, t), θ(x, y + 2D, t) = θ(x, y, t),

t ≥ 0;

(1.7.85)

(4) Ψ(x, y, 0) = Ψ0 (x, y),

θ(x, y, 0) = θ0 (x, y),

(1.7.86)

where Ψ0 (x, y) ∈ H 2 (Ω), θ0 (x, y) ∈ H 1 (Ω), and they are 2D-periodic functions of the variables x, y. By the previous uniform a priori estimates, and usual compactness principle, we obtain Theorem 1.7.1. Suppose that Ψ0 (x, y) ∈ H 2 (Ω), θ0 (x, y) ∈ H 1 (Ω) and they are 2D-periodic functions of x, y. Then the approximate solution {Ψm , θm } of (1.7.12)–(1.7.17) converges to the generalized solution {Ψ, θ} of (1.7.4)–(1.7.7) as m → ∞, where Ω = (0, 2D) × (0, 2D). Lemma 1.7.5. Suppose that the conditions of Theorem 1.7.1 are satisfied, then the generalized solution of (1.7.4)–(1.7.8) θ(x, y, t) ∈ L∞ (ℝ+ ; L∞ (Ω)). Theorem 1.7.2. Under the conditions of Theorem 1.7.1, the solution of periodic initial problem (1.7.4)–(1.7.8) for Newton–Boussinesq equation is unique.

1.7 Two-dimensional Newton–Boussinesq equation

| 129

Now consider the following practical example: R { 𝜕t ΔΨ − Δ2 Ψ + a 𝜕x θ + J(Ψ, ΔΨ) = 0, { { Pr { { { { { 1 { { { {𝜕t θ − Pr Δθ + 𝜕x Ψ + J(Ψ, θ) = 0, (E1 ) { { {Ψ(x + L1 , y, t) = Ψ(x, y, t), Ψ(x, 0, t) = Ψ(x, 1, t) = 0, { { { { { { θ(x + L1 , y, t) = θ(x, y, t), θ(x, 0, t) = θ(x, 1, t) = 0, { { { {Ψ(x, y, 0) = Ψ0 (x, y), θ(x, y, 0) = θ0 (x, y),

(1.7.87)

where Ψ is the flow function; θ = θ −1+y; θ is temperature; Ra > 0 is Rayleigh number; Pr is Prandtl number; J(u, v) = uy vx − ux vy . Suppose that {ωj (x, y)}j∈ℕ is an orthogonal basis which consists of eigenvectors for the operator A = −Δ and satisfies −Δωj = λj ωj , j = 1, 2, . . . , λ1 < λ2 < ⋅ ⋅ ⋅. For every given integer m, an approximate solution has the following form: m

Ψm (t) = ∑ αjm (t)ωj , j=1

2m

ξm (t) = ∑ δjm (t)ωj , j=m+1

m

θm (t) = ∑ βjm (t)ωj , j=1

2m

ηm (t) = ∑ γjm (t)ωj . j=m+1

The time-discretized nonlinear Galerkin method is R (n) { (ΔΨm(n+1) , v) − (ΔΨm(n) , v) − Δt(ΔΨm(n+1) , Δv) + Δt a (θm , v) { { Pr { { { { { { + Δtr(Ψm(n) , ΔΨm(n) , v) + Δtr(Ψm(n) , Δξm(n+1) , v) { { { { (n+1) (n) { { { { + Δtr(ξm , ΔΨm , v) = 0, ∀v ∈ Wm , { { { R (n) { { ̃m , { −(Δξm(n+1) , v) + a (θmx , v) + r(Ψm(n) , Δξn(n) , v) = 0, ∀v ∈ W { { Pr (A1 ) { { Δt (n+1) (n) (n+1) (n) (n) { { (θm , v) − (θm , v) + (∇θm , ∇v) + Δt(Ψmx , v) + Δtr(Ψm(n) , θm , v) { { P { r { { { { + Δtr(Ψ , η(n+1) , v) + Δtr(ξ (n+1) , θ(n) , v) = 0, { { m m m m { { { {1 { (n+1) (n) (n) (n) { ̃ { { P (∇ηm , ∇v) + (Ψmx , v) + r(Ψm , θm , v) = 0, ∀v ∈ Wm , { { r { { (0) (0) {Ψm = Pm Ψ0 , θm = Pm θ0 .

(1.7.88)

Set θ0 = 0, Ψ0 = (sin πx, sin πy)τ , L1 = 1, Pr = 10 and consider sample point (x0 , y0 ) = ( 41 , 41 ). From the numerical results we can see that the nonlinear Galerkin method is valid. However, comparing with the classical Galerkin method, the nonlinear Galerkin method cannot save more time in numerical computation and sometimes gives rise to a lower level of accuracy.

130 | 1 Discrete attractor and approximate calculation

1.8 Numerical computation and analysis of cubic Ginzburg–Landau equation Consider the following cubic Ginzburg–Landau equation: ut = c0 u + (c0 + i)uxx − (c0 − i)|u|2 u,

(1.8.1)

where u(x, t) is an unknown complex-valued function; c0 is a real parameter. It corresponds to Newell–Whitchenel equation when c0 → ∞, but it is an integrable cubic nonlinear Schrödinger equation as c0 = 0. Equation (1.8.1) has a traveling wave solution us (x, t) = u0 ei(k0 x+w0 t) ,

(1.8.2)

w0 = 2|u0 |2 − 1, { 2 k0 = 1 − |u0 |2 .

(1.8.3)

where

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ Setting u(x, t) = u(x, t) r i(k0 x+w0 t) , we derive the equation which u(x, t) satisfies. At this moment we still denote it by u(x, t), and so ut = αu + μux + βuxx − γ|u|2 u,

(1.8.4)

where α = c0 − iw − k02 (c0 + i), μ = 2k0 (c0 + i), β = c0 + i, γ = c0 − i. In order to study the linear stability of equation (1.8.4), we consider the linear equation related to (1.8.4) by a small perturbation method. Let u(x, t) = u0 + δu(x, t),

(1.8.5)

then the linearized equation of (1.8.4) is [

L1 L3

L2 𝜕u ][ ] = 0, L4 𝜕u∗

(1.8.6)

where L1 = 𝜕t − α − μ𝜕x − β𝜕xx + L(|u0 |2 ), L2 = h(uc ),

L3 = h∗ (u0 ),

L + 4 = 𝜕t − α∗ − μ∗ 𝜕x − β∗ 𝜕xx + L∗ (|u0 |2 ), L(|u0 |2 ) = 2γ|u0 |2 ,

h(u0 ) = γu20 ,

and ∗ denotes the complex conjugate. In order to facilitate more specific analysis, let u(x, t) = u0 + δu+ ei(qx−Ωt) + δu− e−i(qx+Ω t) , ∗

(1.8.7)

where δu+ and δu− are small quantities in comparison with u0 , q is a real number.

1.8 Numerical computation and analysis of cubic Ginzburg–Landau equation

| 131

Substituting (1.8.7) into (1.8.6), we get Ω2 − 2 Re AΩi = |γ|2 |u0 |4 − |A|2 = 0,

(1.8.8)

where {

Re A = c0 (q2 + 2k0 q + |u0 |2 ),

(1.8.9)

Im A = q2 + 2k0 q − |u0 |2 .

Set Ω = Ωr + iρ, then ρ ± (c0 , |u0 |, q) = c0 (q2 + 2k0 q + |u0 |2 ) 2

± √(c0 + 1)|u0 |4 − (q2 + 2k0 q − |u0 |2 ) .

(1.8.10)

Obviously, in order to study the linear instability of equation (1.8.4), we just care about ρ(c0 , |u0 |, q). In fact, when q < q∗ = √1 + √1 + c02 |u0 |2 − √1 − |u0 |2 , ρ+ (c0 , |u0 |, q) > 0 is

unstable, qmax can be obtained such that ρ− (c0 , |u0 |, qmax ) = 0. Thus ρ− ≤ 0, and we find that q0 is the minimum of ρ− (c0 , |u0 |, q0 ). That is, q0 = min ρ ± (c0 , |u0 |, q) = ρ− (c0 , |u0 |, q0 ) = −|u0 |2 (1 − c0 )2 . 0≤q≤qmax

(1.8.11)

This is shown in Figure 1.6.

Figure 1.6: The relation of q and ρ− (|u0 | = 1, c0 = 0.25).

We can obtain the following conclusions on the phase plane (|u(x, t), 𝜕t |u(x, t)|): Proposition 1.8.1. On the phase plane (|u(x, t)|, 𝜕t |u(x, t)|), one has: (1) If c0 = 0, equation (1.8.1) reduces to the integrable cubic nonlinear Schrödinger equation.

132 | 1 Discrete attractor and approximate calculation (i) (|us |, 0) is a hyperbolic point for the traveling wave solution us as u0 ≠ 0. (ii) (0, 0) is an elliptic point for the traveling wave solution 0. (2) When c0 > 0, (i) if 0 < q < qmax , 0 < c0 ≤ 1, (|us |, 0) is a hyperbolic point for the traveling wave solution us . (ii) if q > qmax , 0 < c0 ≤ 1, (|us |, 0) is an asymptotic stable point for the traveling wave solution us . Next we use the pseudo-spectral method to study equation (1.8.1): n+1

n−1

̃N − u ̃N u { c j j { { ̃ n+1 ̃ n−1 = 0 (u { Nj + u Nj ) { 2Δt 2 { { { { (c + i) 2 2 n+1 { { { ̃N + u ̃ n−1 − 0 j q0 (u { Nj ) j { 2 { { { 2 { 󵄨 ̃ n 󵄨󵄨 n { { − (c0 − i){󵄨󵄨󵄨u { N 󵄨󵄨 uN }j , { 1 0 { ̃N − u ̃N { u j j { { ̃ 0N − (c0 + i)j2 q02 u ̃ 0N = c0 u { { j j { Δt { { { 2 0 󵄨 󵄨 0 { 󵄨 󵄨 { ̃ N 󵄨󵄨 uN }j , − (c0 − i){󵄨󵄨u { { { { { N−1 { { { { ̃ 0N = h ∑ u0 (xm ) exp(ijxm ), u { j m=0 {

(1.8.12)

̃ nN denotes the jth discrete Fourier coefwhere u0 (x) is a given initial value function; u j n ficient of uN ∈ SN . SN = span−N/2≤j≤N/2−1 {exp(ijq0 x)}. Let xm = mh, N−1

̃ nN exp(ijxm ). unN (xm ) = ∑ u j j=0

One can prove that Theorem 1.8.1. Suppose that the solution u of the periodic initial value problem for equation (1.8.1) belongs to C 3 ([0, T]; Hps (C)). Then there exist Cj , j = 1, 2, 3, which do not depend on Δt, and division point number N such that 󵄩󵄩 n n󵄩 2 −s 󵄩󵄩uN − u 󵄩󵄩󵄩 ≤ C3 (Δt + N ) as long as Δt ≤ C1 , N > C3 . For the numerical results, fix an initial value u0 (x, 0) = ux (x, 0) + εeiθ cos(qx),

1.9 One-dimensional Kuramoto–Sivashinsky equation

| 133

Figure 1.7: (a) subharmonic orbit (|u0 | = 1, c0 = 0); (b) a trajectory (|u0 | = 1, c0 = 0.0025) on phase plane (|u(0, t)|, |u(u(0, t))|t); (c) a trajectory (|u0 | = 1, c0 = 0.025, T > 104 ) on phase plane (|u(0, t)|, |u(u(0, t))|t).

where θ = arctan w0 , θ = arctan w0 + π correspond to the unstable manifold of the saddle point (|u0 |, 0), ε = 0.02, u0 = 1, k0 = 0, w0 = 1. The numerical results of the equation are shown in Figures 1.7–1.13.

1.9 One-dimensional Kuramoto–Sivashinsky equation Consider the following 1D Kuramoto–Sivashinsky (KS) equation 𝜕u 𝜕4 u 𝜕2 u 𝜕u { { + + +u = 0, { { 𝜕x { 𝜕t 𝜕x4 𝜕x2 { {u(x, 0) = u0 (x), { { { {u(x, t) ∈ u(x + L, t),

(x, t) ∈ ℝ × ℝ+ , x ∈ ℝ, L > 0.

Let the initial value u0 (x) be an odd function, namely u0 (x) = −u0 (L − x). In [81], the following estimates were established: for any ρ > 0 there exists T ∗ (ρ) > 0 such that 5 󵄩󵄩 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩L2 ≤ ρ0 = C0 L 2 ,

(1.9.1)

134 | 1 Discrete attractor and approximate calculation

Figure 1.8: The image for c0 = 0.25 (a) power spectrum; (b) trajectory of phase plane; (c) Poincaré cross-section. 7 󵄩󵄩 󵄩 󵄩󵄩∇u(t)󵄩󵄩󵄩L2 ≤ ρ1 = C1 L 2 ,

∀t ≥ T ∗ ,

(1.9.2)

as ‖u0 ‖L2 ≤ ρ, where C0 , C1 are absolute constants, which are independent of L and the initial value u0 . We can rewrite (KS) equation in the form of a functional differential equation 1 du + Au − A 2 u + B(u, u) = 0, dt

where A =

𝜕4 , 𝜕x4

u ∈ H,

and its domain is 4 D(A) = Hper ((0, L)).

The bilinear operator B has the following form: 𝜕v 1 , ∀u, v ∈ Hper ((0, L)), 𝜕x 󵄨 H = {u ∈ L2 ((0, L)) 󵄨󵄨󵄨 u(x, t) = u(x + L, t),

B(u, v) = u

u(x, t) = −u(L − x, t), x ∈ ℝ}.

The inner product in H is (⋅, ⋅) with the corresponding norm | ⋅ |. Set 󵄨 󵄨 󵄩 1 󵄩 󵄨󵄨 𝜕u 󵄨󵄨 ‖u‖ = 󵄩󵄩󵄩A 2 u󵄩󵄩󵄩 = 󵄨󵄨󵄨 󵄨󵄨󵄨. 󵄨󵄨 𝜕x 󵄨󵄨

(1.9.3)

1.9 One-dimensional Kuramoto–Sivashinsky equation

| 135

Figure 1.9: The image for c0 = 0.58 (a) amplitude (0 < t < 2 780); (b) power spectrum (0 ≤ t ≤ ); (d) Poincaré cross-section. 2 048); (c) trajectory of phase plane (|u(L/2, t)|, 𝜕|u(L/2,t)| 𝜕t

Since A is a self-adjoint positive operator, its inverse operator A−1 : L2 ((0, L)) → L2 ((0, L)) is compact. Therefore, there exists a complete orthogonal basis of H, which consists of eigenvectors {ωj }∞ j=1 of A, where ωj (x) = sin(2πx/L) and the corresponding eigenvalues are λj = (2πx/L)4 , j = 1, 2, . . . . Use P = Pm to denote the orthogonal projection of H onto the subspace span{ω1 , . . . , ωm }, Q = Qm = I − P. Since P and Q commute with A and its powers, equation (1.9.3) can be divided into 1 dp + Ap − A 2 p + pB(u, u) = 0, dt 1 dq + Aq − A 2 q + QB(u, u) = 0, dt

P = Pu,

(1.9.4)

q = Qu.

(1.9.5)

By Agmon inequality 1

1

‖u‖∞ ≤ √2|u| 2 ‖u‖ 2 , thus 1 1 󵄨󵄨 󵄨 󵄨󵄨(B(u, v), u)󵄨󵄨󵄨√2|u| 2 ‖u‖ 2 ‖v‖|ω|.

(1.9.6)

136 | 1 Discrete attractor and approximate calculation

Figure 1.10: The image for c0 = 0.57 (a) power spectrum; (b) Poincaré cross section; (c) wave number spectrum.

It is easy to verify (B(u, u), u) = 0,

1 ∀u ∈ Hper ((0, L)).

(1.9.7)

First we discuss the analyticity of solution in time t for the KS equation. Consider the complex form of KS equation 1 du + Au − A 2 u + B(u, u) = 0, dζ u ∈ H, ζ ∈ ℂ, Re ζ > 0,

where H, D(A), A and B are the corresponding spaces and complex extensions of operators. Theorem 1.9.1. Let ρ > 0, and let ℒ denote the union of all the disks whose center is δ + τ, τ ≥ 0, and radius is δ, where 1

δ = δ(ρ) =

λ4

1

8ρ + 2λ 4

.

(1.9.8)

1.9 One-dimensional Kuramoto–Sivashinsky equation

| 137

Figure 1.11: The image for c0 = 0.525 (a) power spectrum; (b) trajectory of phase plane; (d) Poincaré cross section.

Suppose that u(s) is the solution on a set containing the positive real axis region such that 󵄨󵄨 󵄨 󵄨󵄨u(t)󵄨󵄨󵄨 ≤ ρ,

∀t ≥ 0,

(1.9.9)

then there exists a solution u(ζ ) in ℒ, and the mapping u : ℒ → D(A) is analytic and satisfies 󵄨󵄨 󵄨 󵄨󵄨u(ζ )󵄨󵄨󵄨 ≤ 2ρ, ∀ζ ∈ S, 󵄨󵄨 du(t) 󵄨󵄨 4ρ 1 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 dt 󵄨󵄨󵄨 ≤ δ , ∀t > 2 δ, 󵄨 󵄨 13 󵄨󵄨󵄨Au(t)󵄨󵄨󵄨 ≤ K1 (ρ) = 32ρ + 4ρ + 10ρ 5 , 󵄨 󵄨 δ

(1.9.10) (1.9.11) 1 ∀t ≥ δ. 2

(1.9.12)

Proof. Compute the inner product of (1.9.8) with u and multiply it by eiθ . Then taking the real part, we have d 󵄨󵄨 󵄨 1 󵄨2 󵄨 1 󵄨2 󵄨 󵄨 iθ 󵄨2 󵄨u(se )󵄨󵄨󵄨 = 2 cos θ(−󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + 󵄨󵄨󵄨A 4 u󵄨󵄨󵄨 ) + 2󵄨󵄨󵄨B(u, u), u󵄨󵄨󵄨. ds 󵄨

(1.9.13)

138 | 1 Discrete attractor and approximate calculation

Figure 1.12: Doubly periodic Poincaré cross-section (a) c0 = 0.35; (b) c0 = 0.275; (c) c0 = 0.26 (d) c0 = 0.0255.

Using interpolation and Young inequalities, 󵄨󵄨 41 󵄨󵄨2 󵄨 1 󵄨 1 2 󵄨󵄨A u󵄨󵄨 ≤ |u|󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≤ |u| + 2

1 󵄨󵄨 21 󵄨󵄨2 󵄨A u󵄨󵄨 . 2󵄨

(1.9.14)

It follows from (1.9.6) that 3 3 −1 󵄨󵄨 󵄨 2 󵄨󵄨B(u, u), u󵄨󵄨󵄨 ≤ √2|u| 2 ‖u‖ 2 ≤ √2λ1 8 |u|‖u‖ −1 󵄨 1 󵄨 ≤ √2λ1 8 |u|󵄨󵄨󵄨A 2 u󵄨󵄨󵄨.

By Young inequality, then 󵄨󵄨 󵄨 󵄨󵄨B(u, u), u󵄨󵄨󵄨 ≤

1 4

|u|4

λ1 cos θ

+

1 󵄨 1 󵄨2 cos θ󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 . 2

From (1.9.13), (1.9.14) and (1.9.15), we get d 2 |u| ≤ ds

1 4

2

λ1 cos θ

|u|4 + cos θ|u|2 .

(1.9.15)

1.9 One-dimensional Kuramoto–Sivashinsky equation

Figure 1.13: The trajectory on phase plane (|u(L/2, t)|, 0.26 (d) c0 = 0.0255.

𝜕|u(L/2,t)| ) 𝜕t

| 139

(a) c0 = 0.35; (b) c0 = 0.275; (c) c0 =

Set y(s) = |u(seiθ )|2 , y0 = |u0 |2 , then we have dy ≤ αy(y + β), dt where α =

1 λ14

2 cos θ

1

, β = 21 λ14 cos2 θ. Integrating directly leads to y y ≤ s cos θ 0 . y+β y0 + β

We seek an upper bound of s such that y ≤ 2y0 .

(1.9.16)

Note that the function z(z + β)−1 is an increasing function, z ∈ (0, ∞). Equation (1.9.16) holds as long as s cos θ

y0 2y0 ≤ , y0 + β 2y0 + β

140 | 1 Discrete attractor and approximate calculation or s≤ holds. Because of

2(y + β) β 1 1 lg( 0 )= lg(1 + ) cos θ 2y0 + β cos θ 2y0 + β

1 lg(1 + z) ≥ z, 2

z ∈ (0, 1),

we have 1

λ4 β 1 lg(1 + ) ≥ cos θ( 1 ) ≥ √2δ cos θ. cos θ 2y0 + β Therefore, (1.9.16) holds true on the domain iθ 󵄨 ℒ = {ζ = se 󵄨󵄨󵄨 s = √2δ cos θ, −π/2 ≤ θ ≤ π/2},

thus the solution of (1.9.3) is analytic on ℒ. By assumption (1.9.9), we obtain that the solution of (1.9.3) is analytic on ℒ + t0 for the initial value u0 (t), t0 > 0. Now we give a proof of (1.9.11). Employing Cauchy integral formula for a circle Γ the center of which lies at any point of the real axis t > 21 δ, radius is 21 δ, we derive from (1.9.9) that 󵄨󵄨 4ρ 󵄨󵄨 du 󵄨󵄨 󵄨󵄨 1 u(ζ ) 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 dζ 󵄨󵄨󵄨 ≤ . ∫ 󵄨󵄨 󵄨󵄨 = 󵄨󵄨 󵄨󵄨 dt 󵄨󵄨 󵄨󵄨 2πi (t − ζ ) 󵄨󵄨 δ Γ

In order to prove (1.9.12), by (1.9.3) we get 󵄨󵄨 du 󵄨󵄨 󵄨 󵄨 󵄨 1 󵄨 󵄨 󵄨 |Au| ≤ 󵄨󵄨󵄨 󵄨󵄨󵄨 + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + 󵄨󵄨󵄨B(u, u)󵄨󵄨󵄨. 󵄨󵄨 dt 󵄨󵄨 From (1.9.6) and Cauchy–Schwarz inequality, we have 󵄨󵄨 du 󵄨󵄨 1 1 3 󵄨 1 󵄨1 󵄨 󵄨 |Au| ≤ 󵄨󵄨󵄨 󵄨󵄨󵄨 + |u| 2 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 2 + √2|u| 2 ‖u‖ 2 . 󵄨󵄨 dt 󵄨󵄨 It follows from interpolation inequality 3

1

‖u‖ ≤ |u| 4 |Au| 4 , Young inequality and (1.9.17) that 󵄨󵄨 du 󵄨󵄨 1 13 3 1 󵄨 󵄨 |Au| ≤ 󵄨󵄨󵄨 󵄨󵄨󵄨 + |u| + √2|u| 8 |Au| 8 . 󵄨 󵄨 2 󵄨 dt 󵄨 2 Again by Young inequality, we obtain 󵄨󵄨 du 󵄨󵄨 1 4 13 1 5 3 󵄨 󵄨 |Au| ≤ 󵄨󵄨󵄨 󵄨󵄨󵄨 + |u| + × 2 5 |u| 5 + |Au|. 󵄨󵄨 dt 󵄨󵄨 2 2 8 8 From the assumptions (1.9.9) and (1.9.11), (1.9.12) holds.

(1.9.17)

1.9 One-dimensional Kuramoto–Sivashinsky equation

| 141

dq(t) . dt

Now we give upper bound estimates of q(t) and

Theorem 1.9.2. Suppose that m is sufficiently large such that 1

−1

λm+1 ≥ max{16, 8(2ρ0 ρ31 ) 2 λ1 4 },

(1.9.18)

where ρ0 , ρ1 are given by (1.9.1) and (1.9.2). Then for any ρ > 0, any solution u(t) = p(t) + q(t), |u(0)| ≤ ρ of (1.9.3) has the following estimate: 1 󵄨󵄨 󵄨2 󵄨 ∗ ∗ 󵄨2 󵄨󵄨q(t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨q(T )󵄨󵄨󵄨 exp{− λm−1 (t − T )} 2 −2 + 4ρ0 ρ31 λm−1 ,

∀t ≥ T ∗ (ρ),

(1.9.19)

where T ∗ is given by (1.9.1) and (1.9.2). Furthermore, 󵄨󵄨 dq(t) 󵄨󵄨 󵄨󵄨 󵄨󵄨 −1 󵄨󵄨 󵄨 ≤ K2 λm−1 , 󵄨󵄨 dt 󵄨󵄨󵄨

∀t ≥ T ∗∗ (ρ),

(1.9.20)

where 󵄨 1 −1 󵄨󵄨 3 2 T ∗∗ (ρ) = max{T ∗ (ρ) + λm−1 󵄨󵄨lg(4ρ1 /λm−1 ρ0 )󵄨󵄨󵄨, }, 8 1

K2 = 32(2ρ0 ρ31 ) 2 . Proof. Let t > T ∗ (ρ), take the inner product of (1.9.5) with q. Then 1 d 2 󵄨󵄨 41 󵄨󵄨2 󵄨󵄨 41 󵄨󵄨2 󵄨󵄨 󵄨 |q| + 󵄨󵄨A q󵄨󵄨 ≤ 󵄨󵄨A q󵄨󵄨 + 󵄨󵄨B(u, u), q󵄨󵄨󵄨 2 dt − 21 󵄨 1 󵄨2 󵄨 󵄨󵄨A 4 q󵄨󵄨 + 󵄨󵄨B(u, u), q󵄨󵄨󵄨. ≤ λm+1 󵄨 󵄨 󵄨 󵄨 It follows from (1.9.1), (1.9.2) and (1.9.6) that 1 3 1 3 − 21 󵄨 1 󵄨 󵄨󵄨 󵄨 󵄨󵄨A 2 q󵄨󵄨 󵄨󵄨B(u, u), q󵄨󵄨󵄨 ≤ √2|u| 2 ‖u‖ 2 |q| ≤ √2ρ02 ρ12 λm+1 󵄨 󵄨 1 1 2 󵄨 󵄨 −1 ≤ ρ0 ρ31 λm+1 + 󵄨󵄨󵄨A 2 q󵄨󵄨󵄨 . 2

From (1.9.21) we have d 2 1 − 21 − 21 󵄨 1 󵄨2 |q| + 2( − λm+1 )󵄨󵄨󵄨A 2 q󵄨󵄨󵄨 ≤ 2ρ0 ρ31 λm+1 . dt 2 By (1.9.18), we obtain d 2 1 − 21 . |q| + λm+1 |q|2 ≤ 2ρ0 ρ31 λm+1 dt 2

(1.9.21)

142 | 1 Discrete attractor and approximate calculation Then (1.9.19) follows from the standard Gronwall inequality. Setting ρ 1 1 − 21 2 (8ρ0 ρ31 ) 2 λm+1 λm+1 ,

=

we derive the following inequality from (1.9.19): 󵄨 󵄨󵄨 󵄨󵄨q(t)󵄨󵄨󵄨 ≤ ρ,

∀t ≥ T ∗∗ (ρ).

Similar to (1.9.11), we deduce that 󵄨󵄨 dq(t) 󵄨󵄨 4ρ 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨≤ 󵄨󵄨 dt 󵄨󵄨󵄨 δ(ρ)

∀t ≥ T ∗∗ ,

where δ(ρ) is taken as in Theorem 1.9.1. Inequality δ(ρ) ≥ fore (1.9.20) holds.

(1.9.22) 1 4

follows from (1.9.18). There-

Corollary 1.9.1. Suppose that m is sufficiently large and satisfies (1.9.18). For every solution u(t) = p(t) + q(t), we have 3 󵄨󵄨 󵄨2 4ρ0 ρ 󵄨󵄨q(t)󵄨󵄨󵄨 ≤ 2 1 , λm+1

∀t ∈ ℝ

and 󵄨󵄨 dq(t) 󵄨󵄨 󵄨󵄨 󵄨󵄨 −1 󵄨󵄨 ≤ K2 λm+1 󵄨󵄨 󵄨󵄨 dt 󵄨󵄨 on the global attractor. We first consider a variety of approximate inertial manifolds. Set B⊥ = {q ∈ QH : ‖q‖ ≤ 2ρ1 }.

B = {p ∈ PH : ‖p‖ ≤ 2ρ1 },

We need to prove that there exists a mapping Φs : B → QH such that 1

AΦs (p) − A 2 Φs (p) + QB(p + Φs (p), p + Φs (p)) = 0,

p ∈ B.

(1.9.23)

We denote the graph of function Φs by μs = graph Φs , which contains the steady state of (1.9.3), thus μs is called the stationary approximate inertial manifold. Theorem 1.9.3. Suppose that m is sufficiently large such that λm+1 ≥ max(4r22 , r12 /4ρ21 ),

(1.9.24)

where 1

1

− −4 , r1 = 2ρ1 + 8√2ρ21 λ1 8 λm+1

−1 −1

4 r2 = 1 + 8ρ1 λ1 8 λm+1 ,

and ρ0 , ρ1 are defined by (1.9.1) and (1.9.2). Thus there exists a unique mapping Φs : B → QH which satisfies (1.9.17), and its graph is an analytic manifold of C. Furthermore, 󵄩󵄩 s 󵄩󵄩 −1 󵄩󵄩Φ (p)󵄩󵄩 ≤ λm+1 r1 ,

∀p ∈ B.

1.9 One-dimensional Kuramoto–Sivashinsky equation

| 143

Proof. For ∀p ∈ B, define 1

Tp (q) = A− 2 q − A−1 QB(p + q, p + q),

∀q ∈ B⊥ .

(1.9.25)

Firstly we show Tp : B⊥ → B⊥ . It follows from (1.9.19) that 󵄩󵄩 󵄩 󵄨 −1 󵄨 󵄨 −3 󵄨 󵄩󵄩Tp (q)󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨A 4 q󵄨󵄨󵄨 + 󵄨󵄨󵄨A 4 QB(p + q, p + q)󵄨󵄨󵄨. By (1.9.6), we get − 1 − 43 − 21 󵄩󵄩 󵄩 + 8√2ρ21 λ1 8 λm+1 󵄩󵄩Tp (q)󵄩󵄩󵄩 ≤ 2ρ1 λm+1 −1

2 = λm+1 r1 ≤ 2ρ1 .

Now we prove that Tp is contractive. The differential expression (1.9.19) implies 𝜕Tp (q)η 𝜕q

1

= A− 2 η − A−1 Q[B(η, p + q) + B(p + q, η)].

Using (1.9.6) again, we get 󵄩󵄩 𝜕Tp (q)η 󵄩󵄩 − 1 − 43 − 21 󵄩󵄩 󵄩󵄩 ‖η‖ + 8ρ1 λ1 8 λm+1 ‖η‖ 󵄩󵄩 󵄩󵄩 ≤ λm+1 󵄩󵄩 𝜕q 󵄩󵄩 1 − 21 ≤ λm+1 r2 ‖η‖ ≤ ‖η‖. 2 Thus, Tp is contractive. Therefore, for any p ∈ B, there exists a unique fixed point Φs (p) : B → B⊥ of Tp which satisfies (1.9.17). According to [27], Φs is an analytic manifold of C. From (1.9.23), the selection of approximate inertial manifolds is implicit. In the following we construct an approximate function which approximates Φs . Theorem 1.9.4. Suppose that m is sufficiently large such that (1.9.18) holds. Define Φ0 (p) = 0,

∀p ∈ B,

Φi+1 (p) = Tp (Φi (p)),

∀p ∈ B, i = 1, 2, . . .

Then −3

4 ‖Φ1 ‖ ≤ K3 λm+1 ,

− 43

󵄩󵄩 s 󵄩 i 󵄩󵄩Φ (p) − Φ1 (p)󵄩󵄩󵄩 ≤ 2K3 α λm+1 , where −1

K3 = 4√2λ1 8 ρ21 ,

−1

2 α = r2 λm+1 .

(1.9.26) (1.9.27)

144 | 1 Discrete attractor and approximate calculation Proof. Let p ∈ B be fixed. It follows from (1.9.6) that − 43 󵄨 󵄩 󵄨 3 󵄨 󵄩󵄩 󵄨󵄨B(p, q)󵄨󵄨󵄨 󵄩󵄩Φ1 (p)󵄩󵄩󵄩 = 󵄨󵄨󵄨A 4 QB(p, q)󵄨󵄨󵄨 ≤ λm+1 󵄨 󵄨 −1

−3

4 ≤ 4√2λ1 8 ρ21 λm+1 ,

which implies (1.9.25). Since Tp is a contractive mapping with contractivity constant α, 󵄩 󵄩 󵄩 󵄩󵄩 k󵄩 k󵄩 󵄩󵄩Φk+1 (p) − Φk (p)󵄩󵄩󵄩 ≤ α 󵄩󵄩󵄩Φ1 (p) − Φ0 (p)󵄩󵄩󵄩 = α 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩, k = 0, 1, . . .

Due to ∞

Φs (p) − Φi (p) = ∑[Φk+1 (p) − Φk (p)], k=i

we derive ∞

󵄩󵄩 s 󵄩 󵄩 󵄩 k󵄩 i󵄩 󵄩󵄩Φ (p) − Φi (p)󵄩󵄩󵄩 ≤ ∑ α 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩 ≤ 2α 󵄩󵄩󵄩Φi (p)󵄩󵄩󵄩. k=i

Then (1.9.27) holds by substituting (1.9.26) into the above inequality. Theorem 1.9.5. Suppose that m is sufficiently large such that (1.9.18), (1.9.24) and −1

−7

1

1

−3

4 8 2 λm+1 + 4ρ1 λm+1 + √2ρ02 ρ12 λm+1 ≤

1 2

(1.9.28)

hold. Then for any ρ > 0 and every solution u(t) = p(t) + q(t) of (1.9.3), |u(0)| ≤ ρ, the following inequalities hold: 󵄨󵄨 󵄨 s −1 󵄨󵄨A(q(t)) − Φ (p(t))󵄨󵄨󵄨 ≤ 2K2 λm+1 , − 43 − 41 󵄩󵄩 󵄩 + 2K3 αi λm+1 , 󵄩󵄩q(t) − Φi (p(t))󵄩󵄩󵄩 ≤ 2K2 λm+1 where T ∗∗ and K2 are as in Theorem 1.9.2, K3 and α are as in Theorem 1.9.4. Proof. From (1.9.1), (1.9.2), we have 󵄨󵄨 󵄨 󵄨󵄨u(t)󵄨󵄨󵄨 ≤ ρ0 ,

󵄩󵄩 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩 ≤ ρ1 ,

∀t ≥ T ∗∗ .

So Φs (p(t)) is determined, and it satisfies ‖Φs (p(t))‖ ≤ 2ρ1 . Set Δ(t) = q(t) − Φs (p(t)), t ≥ T ∗∗ . From (1.9.5) and (1.9.23), 1

AΔ − A 2 Δ + Q[B(Δ, p + Φs (p(t))) + B(u, Δ)] + It follows from (1.9.6) that 1 1 󵄩 󵄩 󵄨 1 󵄨 |AΔ| ≤ 󵄨󵄨󵄨A 2 Δ󵄨󵄨󵄨 + √2|Δ| 2 ‖Δ‖ 2 󵄩󵄩󵄩p + Φs (p)󵄩󵄩󵄩

dq = 0. dt

(1.9.29) (1.9.30)

1.9 One-dimensional Kuramoto–Sivashinsky equation

| 145

󵄨󵄨 dq 󵄨󵄨 1 1 󵄨 󵄨 + √2|u| 2 ‖u‖ 2 ‖Δ‖ + 󵄨󵄨󵄨 󵄨󵄨󵄨. 󵄨󵄨 dt 󵄨󵄨 From above inequality and (1.9.20), one gets −1

−7

1

1

−3

−1 4 8 2 |AΔ| ≤ (λM+1 + 4ρ1 λm+1 + √2ρ02 ρ12 λm+1 )|AΔ| + k2 λM+1 .

Substituting (1.9.28) into the above inequality leads to (1.9.29). The estimate (1.9.30) follows from (1.9.29) and (1.9.27). Consider several approximate inertial manifolds and their numerical realizations: (1) Φ1 approximation Suppose that the equation is du + Au + B(u, u) = f . dt

(1.9.31)

dq + Aq + Q[B(p, p) + B(q, p) + B(p, q) + B(q, q)] = Qf . dt

(1.9.32)

It follows from (1.9.28) that

|, We can prove that u = p + q lies on the attractor 𝒜 for m sufficiently large. Since | dq dt |Q(B(q, p))|, |QB(q, p)| and |QB(q, q)| are relatively small, we can choose AIM as the graph of the function Φ1 (p) = −A−1 Q[f − B(p, p)],

(1.9.33)

−3

2 its error is λm+1 . Define Error(Φα ) = maxu∈𝒜 |Φ0 (p) − q|. (2) Euler–Galerkin AIM This time q = Ψτ (p) satisfies

q = −τ(τA + I)−1 QB(p + q). It is the fixed point of following contraction mapping: −1

ℱ : q → −τ(I + τAQ) QB(p + q).

(1.9.34)

The first iteration (with the initial value condition q0 ≡ 0, μ0 = Pm H) of mapping ℱ generates the explicit function Ψ1τ : Pm H → Qm H given by Ψ1τ = −τ(I + τAQ)−1 QF(p).

(1.9.35)

And then graph Ψ1τ = μτ is known as Euler AIM. (3) Quasi-steady state AIM From (1.9.2), we can estimate some terms of the function 1

Ψτ (p) = A− 2 Φ1 (p) − A−1 Q[B(p, p) + B(Φ1 , p) + B(p, Φ1 ) + B(Φ1 , Φ1 )].

(1.9.36)

146 | 1 Discrete attractor and approximate calculation Proposition 1.9.1. Suppose that m is sufficiently large and satisfies (1.9.18). Then for any u = p + q ∈ 𝒜, the following estimates hold: − 218 󵄨 󵄨 ; (i) 󵄨󵄨󵄨A−1 Qm B(Φ1 (p), Φ1 (p))󵄨󵄨󵄨 ≤ 2√2K23 λm+1

1 1 1 󵄨 󵄨 −2 (ii) 󵄨󵄨󵄨A−1 Qm B(Φ1 (p), p)󵄨󵄨󵄨 ≤ C3 ρ12 ρ04 ρ24 λm+1 ;

1 1 − 47 󵄨 󵄨 K3 , (iii) 󵄨󵄨󵄨A−1 Qm B(p, Φ1 (p))󵄨󵄨󵄨 ≤ √2ρ02 ρ12 λm+1

where ρ2 is as in Theorem 1.9.1, ρ0 , ρ1 are defined by (1.9.1) and (1.9.2), respectively. Proof. From Agmon inequality and (1.9.26), we get 󵄨󵄨 −1 󵄨 󵄩 󵄩 󵄩 −1 󵄩 󵄨󵄨A Qm B(Φ1 (p), Φ1 (p))󵄨󵄨󵄨 ≤ λm+1 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩∞ 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩 1 󵄩3 󵄨 󵄩 −1 √ 󵄨󵄨 ≤ λm+1 2󵄨󵄨Φ1 (p)󵄨󵄨󵄨 2 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩 2 1 󵄩2 −1 √ − 8 󵄩 ≤ λm+1 2λm+1 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩 −1

−3

−1 √ 8 2 2λm+1 K32 λm+1 . ≤ λm+1

For (ii), using (1.9.26) again, we derive 󵄨󵄨 −1 󵄨 󵄨 −1 󵄨 −2 󵄨󵄨A Qm B(Φ1 (p), p)󵄨󵄨󵄨 ≤ λm+1 󵄨󵄨󵄨Φ1 (p)󵄨󵄨󵄨‖px ‖∞ ≤ λm+1 ‖px ‖∞ . To remove ‖px ‖∞ , by Agmon inequality, we have 1 1 1 1 1 1 󵄨 1 󵄨1 ‖px ‖∞ ≤ C3 |pxx | 2 |px | 2 ≤ C3 ρ12 󵄨󵄨󵄨A 2 ρ󵄨󵄨󵄨 2 ≤ C3 ρ12 |p| 4 |Ap| 4 1

1

1

≤ C3 ρ12 ρ04 |Ap| 4 , thus (ii) follows from |Ap| ≤ ρ2 . Finally, by Agmon inequality, (1.9.26) and (1.9.1)–(1.9.2), we have 1 1 − 43 󵄨󵄨 −1 󵄨 󵄩 󵄩 −1 −1 󵄨󵄨A Qm B(p, Φ1 (p))󵄨󵄨󵄨 ≤ λm+1 ‖p‖∞ 󵄩󵄩󵄩Φ1 (p)󵄩󵄩󵄩 ≤ λm+1 √2|p| 2 ‖p‖ 2 K3 λm+1

−7

1

1

4 √ ≤ λm+1 2ρ02 ρ12 K3 ,

which completes the proof of (iii).

2 Some properties of global attractor In this chapter, we describe and discuss the following properties of the global attractor: (1) a finite number of modes, which determine the global properties of attractor as t → ∞; (2) oscillation of global attractor, measurement estimation of zeros of level set; (3) structure of a class of global attractors. For these results we can refer to the works of Foias, Kukavica, Gao, Guo, among others [26, 28, 71, 72, 70, 29, 23, 31].

2.1 Kuramoto–Sivashinsky equation We show that solutions of the 1D Kuramoto–Sivashinsky equation with periodic boundary conditions are asymptotically determined by their values at four points. That is, there exist x1 , x2 , x3 and x4 in the periodic domain Ω such that if 󵄨 󵄨 lim 󵄨󵄨u1 (xj , t) − u2 (xj , t)󵄨󵄨󵄨 = 0,

t→∞ 󵄨

j = 1, 2, 3, 4,

for two solutions u1 and u2 , then 󵄩 󵄩 lim 󵄩󵄩u1 (xj , t) − u2 (xj , t)󵄩󵄩󵄩L2 (Ω) = 0.

t→∞ 󵄩

Consider the following Kuramoto–Sivashinsky equation: 𝜕u 𝜕u 𝜕4 u 𝜕2 u + + +u =0 𝜕t 𝜕x4 𝜕x2 𝜕x

(2.1.1)

with periodic boundary condition u(x + L, t) = u(x, t), Denoting ux =

𝜕u , 𝜕x

uxx =

𝜕2 u , 𝜕x2

‖u‖L∞ (Ω󸀠 )

x ∈ ℝ, t ≥ 0, L > 0.

(2.1.2)

Ω = [0, L],

󵄨 󵄨 = sup 󵄨󵄨󵄨u(x)󵄨󵄨󵄨, 󸀠

2

1 2

‖u‖L2 (Ω󸀠 ) = (∫ u (x)dx) ,

x∈Ω

Ω󸀠

where Ω󸀠 ⊂ Ω, we introduce the set H = {u ∈ L2loc (ℝ) : ‖u‖L2 (Ω) < ∞, ∫ udx = 0, u is Ω -periodic}, Ω

which is a Hilbert space with the scalar product (u, v) = ∫Ω uvdx, u, v ∈ H. Introducing the operator Au = uxxxx on the domain D(A) = {u ∈ H : uxxxx ∈ H}, we may rewrite the KS equation in the form 1

u̇ + Au − A 2 u + B(u, u) = 0, https://doi.org/10.1515/9783110587265-002

(2.1.3)

148 | 2 Some properties of global attractor , B(u, v) = uvx . For any u0 ∈ H, there exists a unique solution u(t) = where u̇ = 𝜕u 𝜕t S(t)u0 , t ≥ 0, of (2.1.3) such that u(0) = u0 , and the solution operators S(t) are injective for every t ≥ 0. And there exist ρ0 = ρ0 (L) and ρ1 = ρ1 (L) such that 󵄩 󵄩 lim sup 󵄩󵄩󵄩S(t)u0 󵄩󵄩󵄩L2 (Ω) ≤ ρ0 ,

u0 ∈ H,

t→∞

󵄩󵄩 𝜕 󵄩󵄩 󵄩 󵄩 lim sup 󵄩󵄩󵄩 S(t)u0 󵄩󵄩󵄩 ≤ ρ1 , 󵄩 󵄩󵄩L∞ (Ω) 𝜕x t→∞ 󵄩

u0 ∈ H.

(2.1.4) (2.1.5)

As in [91], we deduce the existence of the global attractor 𝒜 = ⋂ S(t){u0 ∈ H : ‖u0 ‖L2 (Ω) ≤ 2ρ0 }. t≥0

The following properties of 𝒜 shall be needed: (1) 𝒜 is a compact subset of C[0, L] and of C 1 [0, L]. (2) limt→∞ dist(S(t)u0 , 𝒜) = 0, ∀u0 ∈ H, where the distance is taken in C[0, L] or in C 1 [0, L]. Also, S(t)𝒜 = 𝒜, ∀t ≥ 0. du (3) ‖u0 ‖L2 (Ω) ≤ ρ0 and ‖ dx0 ‖L∞ (Ω) ≤ ρ1 , u0 ∈ 𝒜. (4) For ∀u0 ∈ 𝒜, there exists a unique u(t) ∈ 𝒜 and u(0) = u0 . ∞ (5) Suppose that a sequence {un }∞ n=1 converges in C[0, 1] to u0 ∈ 𝒜. Then {S(t)un }n=1 converges in C[0, 1] to S(t)u0 , ∀t ∈ ℝ. For the KS equation, we recall a Gevrey class regularity result. For every r ≥ 0, the 1 4

operator erA is called the Gevrey operator with parameter r. It is well-known that there exists r = r(L) > 0 so that the global attractor 𝒜 consists of functions belonging to the Gevrey class. It is also well-known that functions belonging to some Gevrey class are real analytic. Theorem 2.1.1. There exist r = r(L) > 0 and M = M(L) such that 1

󵄩󵄩 rA 4 󵄩 󵄩󵄩e (u1 − u2 )󵄩󵄩󵄩L2 (Ω) ≤ M‖u1 − u2 ‖L2 (Ω) ,

u1 , u2 ∈ 𝒜.

The proof of this theorem can be found in [71]. Lemma 2.1.1. Let f ∈ H satisfy 1

󵄩󵄩 rA 4 󵄩󵄩 󵄩󵄩e f 󵄩󵄩L2 (Ω) ≤ M‖f ‖L2 (Ω) ,

r > 0, M > 0.

Then for every d > 0, there exists M 󸀠 = M 󸀠 (r, M, L, d) such that ‖f ‖L2 (Ω) ≤ M 󸀠 ‖f ‖L2 (Ω󸀠 ) , where Ω󸀠 ⊆ Ω, dim(Ω󸀠 ) ≥ d.

(2.1.6)

2.1 Kuramoto–Sivashinsky equation

| 149

Proof. First, (2.1.6) implies Mn! n! 󵄩 rA 41 󵄩 󵄩 n 󵄩 󵄩󵄩 (n) 󵄩󵄩 󵄩󵄩f 󵄩󵄩L2 (Ω) = 󵄩󵄩󵄩A 4 f 󵄩󵄩󵄩L2 (Ω) ≤ n 󵄩󵄩󵄩e f 󵄩󵄩󵄩L2 (Ω) ≤ n ‖f ‖L2 (Ω) , r r where f (n) =

dn f , dx n

n ∈ ℕ0 ,

(2.1.7)

ℕ0 = {0, 1, 2, . . . }. Hence, by Agmon’s inequality, 󵄩 (n) 󵄩 1 󵄩 1 (n) 󵄩 1 󵄩󵄩 (n) 󵄩󵄩 󵄩󵄩f 󵄩󵄩L∞ (Ω) ≤ 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 2 (Ω) 󵄩󵄩󵄩A 4 f 󵄩󵄩󵄩L2 2 (Ω) 󵄩 n 󵄩 1 󵄩 n+1 󵄩 1 = 󵄩󵄩󵄩A 4 f 󵄩󵄩󵄩L2 2 (Ω) 󵄩󵄩󵄩A 4 f 󵄩󵄩󵄩L2 2 (Ω) ≤

M(n + 1)! 1

r n+ 2

‖f ‖L2 (Ω) ,

∀n ∈ ℕ0 .

This implies that f can be extended to a holomorphic function f on the strip Πr = {z ∈ ℂ : |Im z| < r}. For every z ∈ Π r , we get 2

∞ |f (Re z)| M 󵄨󵄨 󵄨 |Im z|j ≤ 1 ‖f ‖L2 (Ω) ∑ (j + 1)z −j 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ ∑ j! r2 j=0 j=0 4M = M1 ‖f ‖L2 (Ω) , M1 = 1 . r2 ∞

(j)

Consider the function family 󵄨

󵄨

𝒩 = {g ∈ ℋ(Π r ) : ‖g‖L2 (Ω) = 1, 󵄨󵄨󵄨g(z)󵄨󵄨󵄨 ≤ M1 , z ∈ Π r }, 2

2

where ℋ(Π r ) denotes the set of holomorphic functions on Π r , and note that ‖f ‖ f2 2

2

L (Ω)

∈𝒩

(we may assume that f ≠ 0), and 𝒩 is a normal family of holomorphic functions on Π r . 2

The standard compactness argument implies that for every d > 0 there exists M 󸀠 > 0 so that ‖g‖L2 (Ω) ≥ M1 󸀠 , ∀Ω󸀠 ⊆ Ω, dim Ω󸀠 ≥ d. Letting g = ‖f ‖ f2 , we obtain our assertion. L (Ω)

Let n ∈ ℕ. A set {x1 , x2 , . . . , xn } ⊆ Ω is “determining” if for any pair of solutions u1 and u2 of the KS equation, 󵄨 󵄨 lim 󵄨󵄨u1 (xj , t) − u2 (xj , t)󵄨󵄨󵄨 = 0,

t→∞ 󵄨

j = 1, 2, . . . , n,

(2.1.8)

implies 󵄩 󵄩 lim 󵄩󵄩u1 (⋅, t) − u2 (⋅, t)󵄩󵄩󵄩L2 (Ω) = 0.

t→∞ 󵄩

(2.1.9)

Lemma 2.1.2. In order for a set {x1 , x2 , . . . , xn } ⊂ Ω to be “determining”, it is sufficient to check that for any two solutions u1 and u2 belonging to the global attractor, u1 (xj , t) = u2 (xj , t),

j = 1, 2, . . . , n, t ≤ 0,

(2.1.10)

implies u1 (⋅, 0) = u2 (⋅, 0).

(2.1.11)

150 | 2 Some properties of global attractor Proof. We assume on the contrary that the set {x1 , x2 , . . . , xn } is not “determining”, and we shall find solutions u1 and u2 belonging to the global attractor which satisfy (2.1.10) but not (2.1.11). Since {x1 , x2 , . . . , xn } is not determining, there are solutions v1 and v2 of the KS equation so that we have (2.1.8) but not (2.1.9), thus we can find a sequence 0 ≤ t1 < t2 < t3 < ⋅ ⋅ ⋅ so that {v1 (tn ) − v2 (tn )}∞ n=1 does not converge to 0. Passing to a subsequence and using (A1), (A2), we may assume that u01 = limn→∞ v1 (tn ) ∈ 𝒜 and u02 = limn→∞ v2 (tn ) ∈ 𝒜 exist and u01 ≠ u02 . By (A4), u1 (t) = S(t)u01 , u2 (t) = S(t)u02 ∈ 𝒜, ∀t ∈ ℝ, and, according to (A2) and (A5), the difference u1 (t) − u2 (t) vanishes on the set {x1 , x2 , . . . , xn } for all t ≤ 0, while u1 (0) ≠ u2 (0). The proof of Lemma 2.1.2 is completed. Remark 2.1.1. We will also need the following simple fact: if f ∈ C 1 [a, b], and f has a zero in [a, b], then 󵄩 󵄩 ‖f ‖L2 ([a,b]) ≤ (b − a)󵄩󵄩󵄩f 󸀠 󵄩󵄩󵄩L2 ([a,b]) ,

󵄩 󵄩 ‖f ‖L∞ ([a,b]) ≤ (b − a) 2 󵄩󵄩󵄩f 󸀠 󵄩󵄩󵄩L2 ([a,b]) . 1

Theorem 2.1.2. There exist ε1 = ε1 (L) > 0 and ε2 = ε2 (L) > 0 such that each set {x1 , x2 , x3 , x4 } ⊆ Ω is determining, where x1 < x2 < x3 < x4 , x4 − x1 = ε1 , x2 − x1 < ε2 , x4 − x3 < ε2 . Proof. Let {x1 , x2 , x3 , x4 } ⊆ Ω, ε1 ≥ ε2 > 0. In view of Lemma 2.1.2, let u1 and u2 be solutions belonging to the global attractor, for which u1 (xj , t) = u2 (xj , t),

j = 1, 2, 3, 4, t ≤ 0.

(2.1.12)

We want to prove that if ε1 > 0 is small enough, and if ε2 > 0 is small in comparison with ε1 , then we have u1 (⋅, 0) = u2 (⋅, 0). Note that v = u1 − u2 satisfies vt + vxxxx + vxx + vu1x + u2 vx = 0.

(2.1.13)

Multiplying (2.1.13) with v and integrating over Ω󸀠 = [x1 , x2 ], we get for t ≤ 0, 1 d 2 ‖v‖2 2 󸀠 − vx (x4 , t)vxx (x4 , t) + vx (x1 , t)vxx (x1 , t) + ∫ vxx dx − ∫ vx2 dx 2 dt L (Ω ) Ω󸀠

= − ∫ u1x v2 dx − ∫ u2 vvx dx = − ∫ u1x v2 dx + Ω󸀠

Ω󸀠

Ω󸀠

1 ∫ u2x v2 dx. 2 Ω󸀠

By virtue of (A3), we obtain 1 d ‖v‖2 2 󸀠 + ‖vxx ‖2L2 (Ω󸀠 ) − ‖vx ‖2L2 (Ω󸀠 ) − 2ρ1 ‖v‖2L2 (Ω󸀠 ) 2 dt L (Ω ) ≤ vx (x4 , t)vxx (x4 , t) − vx (x1 , t)vxx (x1 , t), t ≤ 0.

Ω󸀠

2.1 Kuramoto–Sivashinsky equation

| 151

Since v and vx have zeros in Ω󸀠 , ∀t ≤ 0, we have ‖v‖L2 (Ω󸀠 ) ≤ ε1 ‖vx ‖L2 (Ω󸀠 ) ,

‖vx ‖L2 (Ω󸀠 ) ≤ ε1 ‖vxx ‖L2 (Ω󸀠 ) ,

thus ‖vxx ‖2L2 (Ω󸀠 ) − ‖vx ‖2L2 (Ω󸀠 ) − 2ρ1 ‖v‖2L2 (Ω󸀠 ) ≥ (1 − ε12 − 2ρ1 ε14 )‖vxx ‖2L2 (Ω󸀠 ) ,

t ≤ 0.

We fix ε1 > 0 so small that the expression in parentheses is greater than 21 . Then, by Remark 2.1.1, we get ‖vxx ‖2L2 (Ω󸀠 ) − ‖vx ‖2L2 (Ω󸀠 ) − 2ρ1 ‖v‖2L2 (Ω󸀠 ) ≥

1 ‖v‖2 2 󸀠 , 2ε14 L (Ω )

and thus d 1 ‖v‖2 2 󸀠 − ‖v‖2 2 󸀠 ≤ 2(vx (x1 , t)vxx (x1 , t) − vx (x4 , t)vxx (x4 , t)), dt L (Ω ) ε14 L (Ω )

t ≤ 0.

(2.1.14)

It remains to estimate the right-hand side of (2.1.14). First, note that vx has at least one zero in each of the intervals [x1 , x2 ] and [x3 , x4 ], t ≤ 0. Hence, by Remark 2.1.1, 1 1 󵄨󵄨 󵄩 󵄩 󵄨 󵄩 󵄩 󵄨󵄨vx (x1 , t)󵄨󵄨󵄨 ≤ ε22 󵄩󵄩󵄩vxx (t)󵄩󵄩󵄩L2 ([x ,x ]) ≤ ε22 󵄩󵄩󵄩vxx (t)󵄩󵄩󵄩L2 (Ω) . 1

2

Similarly, we get 1 󵄩 󵄩 󵄨󵄨 󵄨 󵄨󵄨vx (x4 , t)󵄨󵄨󵄨 ≤ ε22 󵄩󵄩󵄩vxx (t)󵄩󵄩󵄩L2 (Ω) ,

∀t ≤ 0.

Clearly, also 1 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨vxx (xj , t)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩vxx (t)󵄩󵄩󵄩L∞ (Ω) ≤ L 2 ‖vxxx ‖L2 (Ω) ,

j = 1, 4, t ≤ 0.

Therefore, by (2.1.14), 1 1 d 1 ‖v‖2L2 (Ω󸀠 ) + 4 ‖v‖2L2 (Ω󸀠 ) ≤ 4ε22 L 2 ‖vxx ‖L2 (Ω) ‖vxxx ‖L2 (Ω) , dt ε1

t ≤ 0.

By virtue of Theorem 2.1.1 and Lemma 2.1.1, the right-hand side of the above inequality 1

1

≤ 4ε22 L 2 (

2M 6M ‖v‖L2 (Ω) )( 3 ‖v‖L2 (Ω) ). r r2

Applying Lemma 2.1.1 to f = v and d = ε1 , we obtain 1

1

48ε22 L 2 (MM 󸀠 )2 2 d 1 ‖v‖2L2 (Ω󸀠 ) + 4 ‖v‖2L2 (Ω󸀠 ) ≤ ‖v‖L2 (Ω󸀠 ) , dt r5 ε1

152 | 2 Some properties of global attractor where M 󸀠 and r depend on ε1 and L, but are independent of ε2 . Gronwall lemma and (A3) then imply that ∀t ≥ 0, 1

1

48ε22 L 2 (MM 󸀠 )2 1 󵄩2 󵄩 󵄩2 󵄩󵄩 t − 4 t) 󵄩󵄩v(0)󵄩󵄩󵄩L2 (Ω󸀠 ) ≤ 󵄩󵄩󵄩v(−t)󵄩󵄩󵄩L2 (Ω󸀠 ) exp( 5 r ε1 ≤

1 2

48ε2 4ρ20 exp(

1

L 2 (MM 󸀠 )2 1 t − 4 t). r5 ε1

Now, if ε2 is sufficiently small, we get ‖v(0)‖L2 (Ω󸀠 ) = 0, and thus v(⋅, 0) = 0.

2.2 Generalized Ginzburg–Landau equation We consider the following periodic initial problem for generalized Ginzburg–Landau equation: 𝜕t u + νux = χu + (γr + iγi )uxx − (βr + iβi )|u|2 u − (δr + iδi )|u|4 u − (λr + iλi )|u|2 ux − (μr + iμi )u2 ux ,

u(x, t) = u(x + 1, t), u(x, 0) = u0 (x),

x ∈ ℝ, t > 0,

x ∈ ℝ, t ≥ 0,

x ∈ ℝ.

(2.2.1) (2.2.2) (2.2.3)

We know that when γr , δr > 0,

4δr γr > (λi − μi )2 ,

(2.2.4)

1 there exist a unique global solution u ∈ Hper [0, 1] and finite-dimensional global attractor for the problem (2.2.1)–(2.2.3). So we are seeking for a finite set E ⊂ [0, 1] of points which completely determine the long time behavior of solutions, that is, if

󵄨 󵄨 lim 󵄨󵄨u1 (x, t) − u2 (x, t)󵄨󵄨󵄨 = 0,

x ∈ E,

󵄨 󵄨 lim 󵄨󵄨u1 (x, t) − u2 (x, t)󵄨󵄨󵄨 = 0,

x ∈ ℝ,

t→∞ 󵄨

one gets t→∞ 󵄨

where u1 and u2 are any two solutions to problem (2.2.1)–(2.2.3). Let H = L2per [0, 1] = {u ∈ L2 [0, 1], u(x + 1) = u(x)}, 1 V1 = Hper [0, 1] = {u : u ∈ H, ux ∈ H},

where (⋅, ⋅) and | ⋅ |0 denote the scalar product and norm in H, respectively. The norm in space V1 is |u|21 = |u|2V = |u|20 + |ux |20 .

2.2 Generalized Ginzburg–Landau equation

|

153

Theorem 2.2.1. Under the condition (2.2.4), u0 ∈ V1 , there exists a positive constant α1 which depends on the coefficients of (2.2.1) such that, if x1 < x2 , d1 = x2 − x1 < α1 , γr > 2d2 β1 (β1 will be given in the proof), and 󵄨 󵄨 lim 󵄨󵄨u1 (xi , t) − u2 (xi , t)󵄨󵄨󵄨 = 0,

t→∞ 󵄨

i = 1, 2,

(2.2.5)

where u1 , u2 are two solutions, then 󵄨 󵄨 lim 󵄨󵄨u1 (x, t) − u2 (x, t)󵄨󵄨󵄨 = 0,

t→∞ 󵄨

∀x ∈ ℝ.

In order to prove Theorem 2.2.1, we need the following lemmas: Lemma 2.2.1. Let Ω󸀠 = [x1 , x2 ], u ∈ C 1 (Ω󸀠 , ℂ), then for d = x2 − x1 , 󵄨 󵄨2 |u|20,Ω󸀠 ≤ 2󵄨󵄨󵄨u(x1 )󵄨󵄨󵄨 + 2d2 |ux |20,Ω󸀠 . Lemma 2.2.2. If a non-negative differentiable function f satisfies f 󸀠 (t) + αf (t) ≤ g(t), where α > 0, limt→∞ g(t) = 0, then limt→∞ f (t) = 0. The proof of the Lemma 2.2.1 and Lemma 2.2.2 are obvious. Proof of Theorem 2.2.1. Let ω = u2 − u1 . Then ω satisfies ωt + νωx = χω + (γr + iγi )ωxx

− (βr + iβi )(|u2 |2 u2 − |u1 |2 u1 )

− (δr + iδi )(|u2 |4 u2 − |u1 |4 u1 )

− (λr + iλi )(|u2 |2 u2x − |u1 |2 u1x ) − (μr + iμi )(u22 u2x − u21 u1x ).

Multiplying the latter equation by ω and integrating on Ω󸀠 = [x1 , x2 ], then taking the real part, we have 1 d x ∫ |ω|2 dx + γr ∫ |ωx |2 dx − Re[(γr + iγi )ωx ω]x2 1 2 dt Ω󸀠

Ω󸀠

ν 󵄨 󵄨2 󵄨 󵄨2 − χ ∫ |ω|2 dx + (󵄨󵄨󵄨ω(x2 )󵄨󵄨󵄨 − 󵄨󵄨󵄨ω(x1 )󵄨󵄨󵄨 ) 2 Ω󸀠

= Re(F(u2 ) − F(u1 ), ω), where F(u) = −(βr + iβi )|u|2 u − (δr + iδi )|u|4 u − (λr + iλi )|u|2 ux − (μr + iμi )u2 ux .

(2.2.6)

154 | 2 Some properties of global attractor Set ν 󵄨 x 󵄨2 󵄨2 󵄨 g1 (t) = Re[(γr + iγi )ωx ω]x2 − (󵄨󵄨󵄨ω(x2 )󵄨󵄨󵄨 − 󵄨󵄨󵄨ω(x1 )󵄨󵄨󵄨 ). 1 2 By (2.2.5), g1 (t) → 0 as t → ∞. By the results in [41], there exists T such that, for t ≥ T, |ui |∞ ≤ ρ1 ,

|uix |∞ ≤ ρ3 ,

i = 1, 2,

where ρ1 , ρ3 only depend on the coefficients of equation (2.2.1). Hence, for t ≥ T, (F(u2 ) − F(u1 ), ω) ≤ |βr | ⋅ ρ21 |ω|20,Ω󸀠 + 2√βr2 + βi2 ρ21 |ω|20,Ω󸀠 + 2ρ1 ρ3 |ω|20,Ω󸀠 + 4√δr2 + δi2 ρ41 |ω|20,Ω󸀠 + (√λr2 + λi2 + √μ2r + μ2i )ρ21 ∫ |ωx ||ω|dx, Ω󸀠

(√λr2 + λi2 + √μ2r + μ2i )ρ21 ∫ |ωx ||ω|dx ≤ Ω󸀠

γr 2 p2 |ω|0,Ω󸀠 + |ω|2 󸀠 , 2 2γr 0,Ω

where p = (√λr2 + λi2 + √μ2r + λi2 )ρ21 . Then (2.2.6) becomes d |ω|2 󸀠 + γr |ωx |20,Ω󸀠 ≤ 2(|χ| + |βr |ρ21 + 2√βr2 + βi2 ρ21 dt 0,Ω + 4√δr2 + δi2 ρ41 +

p2 2ρ ρ )|ω|20,Ω󸀠 + g1 (t). γr 1 3

Using Lemma 2.2.1, we get γr |ωx |20,Ω󸀠 ≥

γr γ |ω|2 󸀠 − r |ω(x1 )|2 . 2d2 0,Ω 2d2

Hence, by (2.2.7), γ γ 󵄨 d 󵄨2 |ω|2 󸀠 + ( r2 − β1 )|ω|20,Ω󸀠 ≤ g1 (t) + r2 󵄨󵄨󵄨ω(x1 )󵄨󵄨󵄨 , dt 0,Ω 2d 2d where β = 2(|χ| + |βr |ρ21 + 2√βr2 + βi2 ρ21 + 4√δr2 + δi2 ρ41 + By Lemma 2.2.2, lim |ω|20,Ω󸀠 = 0.

t→∞

p2 2ρ ρ ). γr 1 3

(2.2.7)

2.2 Generalized Ginzburg–Landau equation

| 155

In order to prove that solutions u1 and u2 are approximately equal by pointwise, we need to prove lim |ω|L∞ = 0.

t→∞

(2.2.8)

For this we choose an arbitrary sequence T < t1 < t2 < ⋅ ⋅ ⋅, i → ∞, ti → ∞. Due to the ̃1 = u ̃ 1 (x, t) ∈ 𝒜, u ̃2 = u ̃ 2 (x, t) ∈ 𝒜 compactness of the global attractor 𝒜, there exist u and a subsequence {tnk }∞ such that n=1 󵄨 ̃ 1 󵄨󵄨󵄨󵄨L∞ (Ω󸀠 ) = 0, lim 󵄨󵄨󵄨u1 (tnk ) − u 󵄨 ̃ 2 󵄨󵄨󵄨󵄨L∞ (Ω󸀠 ) = 0. lim 󵄨󵄨󵄨u2 (tnk ) − u k→∞ k→∞

̃=u ̃2 − u ̃1 Hence, for ω 󵄨 ̃1 󵄨󵄨󵄨󵄨L∞ (Ω󸀠 ) = 0. lim 󵄨󵄨󵄨ω(tnk ) − ω

k→∞

(2.2.9)

̃(x) = 0, x ∈ Ω󸀠 . Since ω ̃ = 0 is continuous, we get ω ̃(x) = 0, x ∈ Ω󸀠 . In By (2.2.8), ω ̃ = 0, so it is a real analytic function in x. Due to the fact according to Gevrey class, ω ̃(x) = 0, x ∈ ℝ. Hence (2.2.9) implies continuity of real analytic function, ω 󵄨 󵄨 lim 󵄨󵄨󵄨ω(tnk )󵄨󵄨󵄨L∞ = 0.

k→∞

Since {tnk } is arbitrary, 󵄨 󵄨 lim 󵄨󵄨󵄨ω(t)󵄨󵄨󵄨L∞ = 0.

k→∞

Next we consider the numbers of determining nodes for stationary solutions. Denote the stationary solution set of (2.2.1), (2.2.2) by S(Coeff). Theorem 2.2.2. Under condition (2.2.4), there exists a positive constant α2 which depends on the coefficients of (2.2.1) and has the following property: Let x1 < x2 , where d = x2 − x1 < α2 , let u1 , u2 ∈ S(Coeff), and denote ω = u1 − u2 , if each of the functions Re ω, Im ω has at least two zeros in Ω󸀠 = [x1 , x2 ], then ω = 0, i. e., u1 = u2 . For the proof we need the following lemmas: Lemma 2.2.3 (Poincaré inequality). Let Ω󸀠 = [x1 , x2 ] and d = x2 − x1 . (i) Let ω ∈ C 1 (Ω󸀠 , ℂ), y1 , y2 ∈ Ω󸀠 , then 󵄨 󵄨 |ω|∞,Ω󸀠 ≤ 󵄨󵄨󵄨Re ω(y1 ) + i Im ω(y2 )󵄨󵄨󵄨 + d|ωx |∞,Ω󸀠 . (ii) Let ω ∈ C 2 (Ω󸀠 , ℂ), and suppose that each of the functions Re ω, Im ω has at least two zeros on Ω󸀠 , then |ωx |L∞ (Ω󸀠 ) ≤ d|ωxx |L∞ (Ω󸀠 ) , |ω|L∞ (Ω󸀠 ) ≤ d2 |ωxx |L∞ (Ω󸀠 ) .

156 | 2 Some properties of global attractor Lemma 2.2.4. If condition (2.2.4) is satisfied, and u ∈ S(Coeff), then |u|L∞ ≤ ρ1 , 1

|ux |L∞ ≤ ρ3 ,

1

1

a2 ), α

where ρ1 2 = (1 + √aα− 2 )a, ρ3 = α− 2 aK12 (1 + following proof.

the constants α, a are defined in the

Proof. Take the scalar product of (2.2.1) with u. Then 1

1

1

0

0

0

ν Re ∫ ux udx + βr ∫ |u|4 dx + δr ∫ |u|6 dx 1

− χ|u|20 + γr |ux |20 − (λr + μr ) Re ∫ |u|2 ux udx 0

1

− (λi − μi ) Im ∫ |u|2 ux udx = 0, 0

since 1

1

Re ∫ |u|2 ux udx = 0,

Re ∫ ux udx = 0, 0

1

0

1

2

(λi − μi ) Im ∫ |u| ux udx ≤ |λi − μi | ∫ |u|3 |ux |dx 0

0 1

1 2

1

a2 b2 ≤ a1 b1 |ux |0 (∫ |u| dx) ≤ 1 ∫ |u|6 dx + 1 ‖ux ‖20 , 2 2 6

0

0

where a1 b1 = |λi − μi |. By condition (2.2.4), we can choose constants a1 , b1 such that α = 2γr − b21 > 0,

β = 2δr − a21 > 0,

and then α|ux |20

1

1

6

+ β ∫ |u| dx + 2βr ∫ |u|4 dx − 2χ|u|20 ≤ 0. 0

0

Let χ > 0, otherwise, discard this term, then α|ux |20

1

≤ −β ∫(|u|3 − 0

2

βr − 1 |u|) dx β

2.2 Generalized Ginzburg–Landau equation 1

− 2 ∫ |u|4 dx + 2χ|u|20 + 0

| 157

(βr − 1)2 2 |u|0 , β

namely, 2 2(|u|20 )

1

≤ 2 ∫ |u|4 dx ≤ (2χ + 0

(βr − 1)2 )|u|20 − α|ux |20 . β

Hence, |u|0 ≤ √a, 1

(β − 1)2 1 a = (2χ + r ), 2 β

∫ |u|4 dx ≤ a2 , 0

|ux |20 ≤

a2 , α

|u|L∞ ≤ ρ1 .

Take the scalar product of (2.2.1) with uxx , and choose the real part. Note that ut = 0, and then 2

|uxx |20 ≤ K1 (1 + |ux |20 ) . Therefore, |uxx |20 ≤ K1 (1 +

2

a2 ), α

|ux |L∞ ≤ ρ3 .

Proof of Theorem 2.2.2. Since |ωxx |∞,Ω󸀠 ≤

1 √γr2 + γi2

(|μ||ωx |∞,Ω󸀠 + χ|ω|∞,Ω󸀠 + 2√βr2 + βi2 ρ1 2 |ω|∞,Ω󸀠

+ 4√δr2 + δi2 ρ1 4 |ω|∞,Ω󸀠 (√λr2 + λi2 + √μ2r + μ2i ) × (ρ1 2 |ωx |∞,Ω󸀠 + 2ρ1 ρ3 |ω|∞,Ω󸀠 )), using Lemma 2.2.3, we derive |ωxx |∞,Ω󸀠 ≤

1 √γr2 + γi2

(|ν|d + χd2 + 2√βr2 + βi2 ρ1 2 d2

+ 4√δr2 + δi2 ρ1 4 d2 + (√λr2 + λi2 + √μ2r + μ2i )(ρ1 2 d + 2ρ1 ρ3 d2 )|ωxx |∞,Ω󸀠 ). It follows from the above bound that there exists a constant α2 which depends on the coefficients of equation (2.2.1) such that, as d < α2 , |ωxx |∞,Ω󸀠 = 0, then ωxx = 0, x ∈ Ω󸀠 . Since Re ωxx , Im ωxx are real analytic, ωxx (x) = 0, x ∈ ℝ. But the functions Re ωx , Im ωx , Re ω, Im ω have zeros, so we conclude that ω = 0. This completes the proof of Theorem 2.2.2.

158 | 2 Some properties of global attractor As a consequence of Theorem 2.2.2, any u ∈ S(Coeff) has the following property: Theorem 2.2.3. Let u ∈ S(Coeff), p ∈ ℂ, suppose that there exists an interval I = [x1 , x2 ] of length not greater than α2 such that each of the functions Re(u(x) − p), Im(u(x) − p) has at least three zeros on I. Then u is a constant, and consequently u = 0 in the case of βi ≠ 0, δi ≠ 0. Proof. The proof is the same as the proof of Theorem 2.3.4 in Section 2.3. Let {u} = {u(x) : x ∈ [0, 1]} ⊂ ℂ, then for any p ∉ {u} one can define the (integer valued) index of a closed differential path u : [0, 1] → ℂ around p by 1

u󸀠 (t) 1 dt. Indp (u) = ∫ 2πi u(t) − p 0

Corollary 2.2.1. Let u ∈ S(Coeff), p ∈ ℂ. (i) If p ∉ {u}, then |Indp (u)| ≤ 2[ α1 ] + 2. 2

(ii) If p ∈ {u} and u is not a constant function, then the function u−p has at most 2[ α1 ]+2 2 zeros on [0, 1].

Next we consider estimates of the fractal dimension of the set of stationary solutions S(Coeff). We have proved that every u ∈ S(Coeff) is completely determined by its values at any two sufficiently close points. This shows that S(Coeff) can be parameterized with four parameters, so one might suspect that the fractal dimension of S(Coeff) is not greater than 4. Lemma 2.2.5. Let (E, d) be a metric space, y ⊂ E, and suppose that for some k ∈ ℕ there exists a bounded mapping ϕ : Y → ℝk such that 󵄨󵄨 󵄨 󵄨󵄨ϕ(y1 ) − ϕ(y2 )󵄨󵄨󵄨 ≥ Cd(y1 , y2 ),

y1 , y2 ∈ Y,

(2.2.10)

for some positive constant C, then dF (Y) ≤ k, where dF (Y) is the fractal dimension of Y. Proof. The proof can be found in [91]. We are going to apply this lemma for E = C[0, 1], Y = S(Coeff), k = 4. Consider ϕ : S(Coeff) → ℂ2 , defined by ϕ(u) = (u(0), u(α2 )),

(2.2.11)

where α2 is given by Theorem 2.2.2. The mapping is obviously bounded, so condition (2.2.10) remains to be checked. For any ε > 0, take u1 , u2 ∈ S(Coeff), and assume that ω = u1 − u2 satisfies 󵄨󵄨 󵄨 󵄨󵄨ω(0)󵄨󵄨󵄨 < ε,

󵄨󵄨 α2 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨ω( )󵄨󵄨󵄨 < ε. 󵄨󵄨 2 󵄨󵄨

(2.2.12)

2.2 Generalized Ginzburg–Landau equation

|

159

We want to find a constant D = D(Coeff) such that (2.2.10) implies |ω|∞ ≤ Dε.

(2.2.13)

The proof of this fact will take two steps. Step 1. Estimate for ωx (0). Notice that if Ω󸀠 ⊂ ℝ is an arbitrary interval of length not greater than α = α2 , then by Lagrange theorem, there exist y1 , y2 ∈ Ω󸀠 such that 󵄨󵄨 󵄨 2ε 󵄨󵄨Re ωx (y1 )󵄨󵄨󵄨 < , d

󵄨󵄨 󵄨 2ε 󵄨󵄨Im ωx (y2 )󵄨󵄨󵄨 < . d

(2.2.14)

By Theorem 2.2.2, Lemma 2.2.3 and (2.2.14), we have |ωxx |∞,Ω󸀠 ≤

1 4ε + 2 |ω|∞,Ω󸀠 . d 4d

(2.2.15)

Let Ω󸀠 = [0, d], by Lemma 2.2.3 (i) and (2.2.15), 󵄨 󵄨 |ω|∞,Ω󸀠 ≤ 󵄨󵄨󵄨ω(0)󵄨󵄨󵄨 + d|ωx |∞,Ω󸀠 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨ω(0)󵄨󵄨󵄨 + d(󵄨󵄨󵄨Re ωx (y1 )󵄨󵄨󵄨 + 󵄨󵄨󵄨Im ωx (y2 )󵄨󵄨󵄨) + d2 |ωxx |∞,Ω󸀠 1 ≤ ε + 4ε + 4dε + |ω|∞,Ω󸀠 . 4 Hence, 4ε 4ε + 2, d d 4ε 4ε 8ε ≤ + 4ε + = 4ε + . d d d

|ω|∞,Ω󸀠 ≤ 16ε, |ω|∞,Ω󸀠

|ωxx |∞,Ω󸀠 ≤

Therefore, 󵄨󵄨 󵄨 󵄨󵄨ω(0)󵄨󵄨󵄨 < ε,

󵄨󵄨 α2 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨ω( )󵄨󵄨󵄨 < ε, 2 󵄨󵄨 󵄨󵄨

8ε 󵄨󵄨 󵄨 󵄨󵄨ωx (0)󵄨󵄨󵄨 < 4ε + . d

Step 2: Assume that |ω(x0 )| < ε1 , |ωx (x0 )| < ε1 + Lemma 2.2.3, 󵄨󵄨 󵄨 󵄨󵄨ω(x)󵄨󵄨󵄨 < 8ε1 ,

x0 ∈ ℝ1 , ε1 > 0. Then from

ε 󵄨󵄨 󵄨 󵄨󵄨ωx (x)󵄨󵄨󵄨 < 3(ε1 + 1 ) d

x ∈ [x0 , x0 + d] = Ω󸀠 . Step 1 and Step 2 show that (2.2.12) implies

|ω|∞ ≤ Dε, which completes the proof of (2.2.13).

ε1 , d

160 | 2 Some properties of global attractor Theorem 2.2.4. dF (S(Coeff)) ≤ 4. Remark 2.2.1. dF (S(Coeff)) ≤ 4 cannot be improved. Let SM be the set of complex-valued solutions of the following problem: uxx + 4π 2 u = 0,

{

u(x + 1) = u(x).

x ∈ ℝ, |u|L∞ ≤ M,

(2.2.16)

Obviously, u = a cos πx + b sin πx, a, b ∈ ℂ, is a solution of (2.2.16), dF (SM ) = 4 which means that the upper bound 4 cannot be improved.

2.3 Upper bound estimate for the winding number Consider the following periodic initial value problem for Ginzburg–Landau equation: 𝜕u 𝜕2 u − (1 + iν) 2 + (1 + iμ)|u|2 u − au = 0, 𝜕t 𝜕x u(x + 1, t) = u(x, t), x ∈ ℝ, t ≥ 0,

(2.3.2)

u(x, 0) = u0 (x),

(2.3.3)

(2.3.1)

x ∈ ℝ,

where a > 0, ν, μ are given real numbers, u0 (x) ∈ H = {u(x) ∈ L2loc (ℝ), u(x + 1) = u(x)}. Definition 2.3.1. Let f : ℝ → ℂ be a differentiable periodic function with period 1, and denote {f } = {f (x) : x ∈ [0, 1]} ⊆ ℂ. Then the winding number of f around a point p ∉ {f } is defined as 1

1 f 󸀠 (x) indp (f ) = dx. ∫ 2πi f (x) − p 0

In the following we will give an upper bound of the winding number indp (f ) for all elements (different from stationary solution) of the global attractor. Meanwhile, we will also give an estimate for the analyticity radius of solutions with respect to the space variable and a generalization of the strong squeezing property for higher order derivatives. Let H be a Hilbert space, on which we introduce the scalar product (u, v) = Re ∫ uv∗ dx, Ω 1

where v∗ indicates the complex conjugate of v. The norm is |u| = (u, u) 2 . From Fourier expansion, ∞

u = ∑ uj e2πijx , j=−∞

uj ∈ ℂ,

2.3 Upper bound estimate for the winding number

| 161

which satisfies ∞

|u|2 = ∑ |uj |2 < ∞. j=−∞

Let A be a closed unbounded operator defined (together with its positive powers) by ∞

As u = ∑ |2πj|2s uj e2πijx , j=−∞

the domains of these operators are ∞

∞

j=−∞

j=−∞

D(As ) = {u = ∑ uj e2πijx : ∑ |2πj|2s |uj |2 < ∞}. Notice that 󵄨 m 󵄨 󵄨󵄨 m/2 󵄨󵄨 󵄨󵄨 (m) 󵄨󵄨 󵄨󵄨󵄨 𝜕 u 󵄨󵄨󵄨 󵄨󵄨A u󵄨󵄨 = 󵄨󵄨u 󵄨󵄨 = 󵄨󵄨 m 󵄨󵄨, 󵄨󵄨 𝜕x 󵄨󵄨 If we denote B(u, v, w) = uvw∗ and ut =

𝜕u , 𝜕t

u ∈ D(Am/2 ), m ∈ ℕ.

equation (2.3.1) gets the form

ut + (1 + iν)Au + (1 + iμ)B(u, u, u) − au = 0.

(2.3.4)

In order to establish estimates for the nonlinear term B, we use the following version of the Agmon’s inequality: 󵄨 󵄨 2 |u|2∞ = (sup 󵄨󵄨󵄨u(x)󵄨󵄨󵄨) ≤ |u||ux | + |u|2 x∈Ω

(2.3.5)

for functions u ∈ C 1 (Ω, ℂ) which are 1-periodic. When ∫Ω udx = 0, one has |u|2∞ ≤ |u||ux |.

(2.3.6)

󵄨󵄨 󵄨 2 󵄨󵄨(B(u1 , u2 , v), v)󵄨󵄨󵄨 ≤ |u1 ||u2 ||v|∞ ≤ |u1 ||u2 |(|v||vx | + |v|2 ).

(2.3.7)

󵄨󵄨 󵄨 2 󵄨󵄨(B(v, u1 , u2 ), v)󵄨󵄨󵄨 ≤ |u1 ||u2 ||v|∞ ≤ |u1 ||u2 |(|v||vx | + |v|2 ).

(2.3.8)

Therefore,

Similarly,

For the solution u(t) = S(t)u0 , t ≥ 0 to the periodic initial value problem (2.3.1)– (2.3.3), the properties of S which will be used in the sequel are:

162 | 2 Some properties of global attractor (i) S(t)H ⊆ H, ∀t ≥ 0, limt→0+ S(t)u0 = S(0)u0 = u0 , u0 ∈ H; (ii) S(t)S(s)u0 = S(t + s)u0 , s, t ≥ 0, u0 ∈ H; (iii) There exist constants ρ0 = ρ0 (a, ν, μ), t0 = t0 (a, ν, μ) such that 󵄨 󵄨󵄨 󵄨󵄨S(t)H 󵄨󵄨󵄨 ⊆ B = {u ∈ H : |u| ≤ ρ0 },

S(t)B ⊆ B,

t ≥ t0 ,

∀t ≥ 0, it is possible to take ρ0 = 2√a.

Definition 2.3.2. The definition of the global attractor is 𝒜 = ⋂ S(t)B. t>0

The properties of the attractor needed below are: (1) 𝒜 ≠ 0; (2) S(t)𝒜 = 𝒜, ∀t ≥ 0; (3) |u0 | ≤ ρ0 , u0 ∈ 𝒜; (4) limt→∞ supt0 ∈H dist(S(t)u0 , 𝒜) = 0. For τ > 0 we consider the operator ∞

1 2

eτA u = ∑ uj eτ|2πj| e2πijx , j=−∞

∞

∀u = ∑ uj e2πijx j=−∞

(2.3.9)

such that ∞

1

󵄨 2 󵄨2 |u|2τ = 󵄨󵄨󵄨eτA u󵄨󵄨󵄨 = ∑ e2τ|2πj| |uj |2 < ∞.

(2.3.10)

j=−∞

1 2

1 2

This set D(eτA ), which is the domain of definition for eτA , is called the Gevrey class of functions with parameter τ. It is a Hilbert space over the field ℝ with the scalar product 1 2

1 2

(u, v)τ = (eτA u, eτA v),

1 2

u, v ∈ D(eτA ),

1

and |u|τ = (u, u)τ2 . The solutions of (2.3.1)–(2.3.3) belong to the Gevrey regularity class. 1 2

Let uk ∈ D(eτA ), k = 1, 2, 3, 4, τ > 0, ∞

uk = ∑ uk e2πijx , j=−∞ ∞

∞

j=−∞

j=−∞

̃ k = ∑ |ukj |eτ|2πj| e2πijx = ∑ u ̃ kj e2πijx , u

τ > 0.

Lemma 2.3.1. One has 󵄨󵄨 󵄨 ̃1 , u ̃2 , u ̃ 3 ), u ̃ 4 ). 󵄨󵄨(B(u1 , u2 , u3 ), u4 )τ 󵄨󵄨󵄨 ≤ (B(u

2.3 Upper bound estimate for the winding number

| 163

Proof. 󵄨󵄨 󵄨󵄨 ∞ 󵄨 󵄨󵄨 󵄨󵄨 ∗ ∗ 2τ|2πm| 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨(B(u1 , u2 , u3 ), u4 )τ 󵄨󵄨󵄨 = 󵄨󵄨󵄨Re ∑ u1,j u2,k u3,j+k−m u4,m e 󵄨󵄨 󵄨󵄨 󵄨 j,m,k=−∞ 󵄨 ≤ = ≤

∞

∑

j,m,k=−∞ ∞

∑

j,m,k=−∞ ∞

∑

j,m,k=−∞

󵄨 󵄨󵄨 󵄨 |u1,j ||u2,k |󵄨󵄨󵄨u∗3,j+k−m 󵄨󵄨󵄨󵄨󵄨󵄨u∗4,m 󵄨󵄨󵄨e2τ|2πm| ̃ 2,k u ̃ 3,j+k−m u ̃ 4,m exp(2πτ(|m| − |j + k − m| − |j| − |k|)) ̃ 1,j u u ̃ 1,j u ̃ 2,k u ̃ 3,j+k−m u ̃ 4,m u

̃1 , u ̃2 , u ̃ 3 ), u ̃ 4 ). = (B(u From (2.3.10) we see that ̃ ix | = 󵄨󵄨󵄨󵄨A 2 ui 󵄨󵄨󵄨󵄨τ , |u 1

̃ i | = |u ̃ i |τ , |u

τ ≥ 0, i = 1, 2, 3, 4,

and this, together with Lemma 2.3.1 and (2.3.7)–(2.3.8), implies 󵄨󵄨 󵄨 󵄨 1 󵄨 2 󵄨󵄨(B(u1 , u2 , v), v)τ 󵄨󵄨󵄨 ≤ |u1 |τ |u2 |τ (|v|τ 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨τ + |v|τ ), 1 󵄨󵄨󵄨(B(v, u1 , u2 ), v) 󵄨󵄨󵄨 ≤ |u1 |τ |u2 |τ (|v|τ 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨 + |v|2 ). τ 󵄨τ 󵄨 󵄨 τ󵄨

(2.3.11) (2.3.12)

Theorem 2.3.1. There exists a constant ρ1 = ρ1 (a, ν, μ) such that the following is true: If |u0 | ≤ ρ0 , u(t) = S(t)u0 , then 1

󵄨󵄨 ( 83 ρ1 )tA 2 󵄨 u(t)󵄨󵄨󵄨 ≤ 2ρ0 , 󵄨󵄨e 1

󵄨󵄨 ρ1 A 2 󵄨 u(t)󵄨󵄨󵄨 ≤ 2ρ0 , 󵄨󵄨e

t≥

0≤t≤

8ρ21 = t1 , 3

8ρ21 = t1 . 3

(2.3.13) (2.3.14)

It will be seen from the proof that an appropriate choice is ρ1 =

3 −1 (2a + 8ρ20 √1 + μ2 + 16ρ40 (1 + μ2 )) 2 . 8

Proof. Let ϕ(t) = (u, u)αt . For a fixed α > 0, we have 1 1 1 1 1 1 󸀠 2 2 2 2 ϕ (t) = (eαtA ut , eαtA u) + α(A 2 eαtA u, eαtA u) 2 = −(Au, u)αt − ((1 + iμ)B(u, u, u), u)αt 󵄨 1 󵄨2 + α|u|2αt + α󵄨󵄨󵄨A 4 u󵄨󵄨󵄨αt ,

(2.3.15)

164 | 2 Some properties of global attractor since (iνAu, u)αt = 0, using (2.3.11) and the interpolation inequality, we get 1 󸀠 󵄨 1 󵄨2 󵄨 1 󵄨 󵄨 1 󵄨 ϕ (t) + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨αt ≤ √1 + μ2 (|u|4αt + |u|3αt 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨αt ) + α|u|2αt + α|u|αt 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨αt 2 1 1 󵄨 1 󵄨2 ≤ √1 + μ2 |u|4αt + (1 + μ2 )|u|6αt + 󵄨󵄨󵄨A 2 u󵄨󵄨󵄨αt 2 2 1 󵄨󵄨 21 󵄨󵄨2 1 2 2 2 + α|u|αt + α |u|αt + 󵄨󵄨A u󵄨󵄨αt . 2 2 Hence, ϕ󸀠 (t) ≤ (2a + α2 )ϕ(t) + 2√1 + μ2 ϕ(t)2 + (1 + μ2 )ϕ(t)3 . Notice that by assumption, ϕ(0) ≤ ρ20 . As long as ϕ(t) ≤ 4ρ20 , one has ϕ󸀠 (t) ≤ 4ρ20 (2a + α2 + 8ρ20 √1 + μ2 + 16ρ40 (1 + μ2 )) = 8α2 ρ20 ,

where 1

α = (2a + 8ρ20 √1 + μ2 + 16ρ40 (1 + μ2 )) 2 . This shows that ϕ(t) ≤ ϕ(0) + 8α2 ρ20 t,

t≤

3 = t1 , 8α2

since ρ1 = 8α3 , (2.3.13) follows. Inequality (2.3.14) is a consequence of (2.3.13) and the 1 properties of S(t). Corollary 2.3.1 (Space analyticity). For every u0 ∈ H such that |u0 | ≤ ρ0 , and every fixed t ≥ t1 , the functions Re Sq(t)u0 and Im S(t)u0 are real analytic in the space variable, with the analyticity radius at least ρ1 . Also, we have the estimates: ∀t ≥ t1 , 󵄨󵄨 dn 󵄨󵄨 󵄨󵄨 󵄨 −n 󵄨󵄨 n Re S(t)u0 󵄨󵄨󵄨 ≤ n!2ρ0 ρ1 , 󵄨󵄨 dx 󵄨󵄨 󵄨󵄨 dn 󵄨󵄨 󵄨󵄨 󵄨 −n 󵄨󵄨 n Im S(t)u0 󵄨󵄨󵄨 ≤ n!2ρ0 ρ1 , 󵄨󵄨 dx 󵄨󵄨

n = 0, 1, 2, . . . , n = 0, 1, 2, . . .

Proof. The two estimates follow from (2.3.14) and (2.3.3). To obtain real analyticity, fix an arbitrary ε ∈ (0, 1), take n ≥ 1, and use (2.3.6): 1

1

󵄨󵄨 dn 󵄨󵄨 󵄨󵄨 dn 󵄨󵄨 2 󵄨󵄨 dn+1 󵄨󵄨 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 n Re S(t)u0 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 n Re S(t)u0 󵄨󵄨󵄨 󵄨󵄨󵄨 n+1 Re S(t)u0 󵄨󵄨󵄨 󵄨󵄨 dx 󵄨󵄨∞ 󵄨󵄨 dx 󵄨󵄨 󵄨󵄨 dx 󵄨󵄨 −1

≤ (n + 1)! ⋅ 2ρ0 ⋅ ρ1 2 ρ−n 1

2.3 Upper bound estimate for the winding number

−1

≤ n! ⋅ 2ϵ−1 ρ0 ρ1 2 ⋅ (

| 165

n

1+ε ) . ρ1

For n = 0, we use (2.3.5) to get 1 󵄨 󵄨󵄨 2 󵄨󵄨Re S(t)u0 󵄨󵄨󵄨∞ ≤ (2ρ0 ) (1 + ), ρ1 then it is easy to find that Re S(t) is real analytic. The same argument obviously works also for Im S(t)u0 . The known results of strong squeezing property for GLE equations are as follows: Theorem 2.3.2 ([72]). There exist constants ρ1 = ρ2 (a, ν, μ) and ρ3 = ρ3 (a, ν, μ) with the following property: If u1 , u2 are solutions of the GLE such that |u1 (0)| ≤ ρ0 , |u2 (0)| ≤ ρ0 , v = u1 − u2 , then at least one of the following assertions is true: (1) |v(t)| ≤ |v(0)|e−ρ2 t , t ≥ 0; (2) there exists t 󸀠 > 0 such that 󵄨2 󵄨 󵄨2 󵄨󵄨 21 󵄨󵄨A v(t)󵄨󵄨󵄨 ≤ ρ3 󵄨󵄨󵄨v(t)󵄨󵄨󵄨 ,

t ≥ t󸀠.

(2.3.16)

Moreover, if (2.3.16) holds for some t = t 󸀠󸀠 , it holds for any t ≥ t 󸀠󸀠 . 1

Corollary 2.3.2. If u1 , u2 ∈ 𝒜, v = u1 − u2 , then |A 2 v|2 ≤ ρ2 |v|2 . Now we will give the strong squeezing property for higher order derivatives. Theorem 2.3.3. There exist constants ρ4 = ρ4 (a, ν, μ) and t2 = t2 (a, ν, μ) such that the following is true: If u1 , u2 are solutions of the GLE which satisfy 1

󵄨 󵄨󵄨 ρ1 A 2 ui (t)󵄨󵄨󵄨 ≤ 2ρ0 , 󵄨󵄨e

t ≥ 0, i = 1, 2

(2.3.17)

and if 󵄨2 󵄨󵄨 21 󵄨 󵄨2 󵄨󵄨A v(t)󵄨󵄨󵄨 ≤ ρ3 󵄨󵄨󵄨v(t)󵄨󵄨󵄨 ,

t ≥ 0,

(2.3.18)

then 1

󵄨 󵄨 󵄨 󵄨󵄨 ρ4 A 2 v(t)󵄨󵄨󵄨 ≤ 2󵄨󵄨󵄨v(t)󵄨󵄨󵄨. 󵄨󵄨e

(2.3.19)

Proof. It is easy to see that vt + (1 + iν)Av + (1 + iμ)(B(u1 , u2 , v) + B(v, u1 + u2 , u2 )) − av = 0. Denote ϕ(t) = (v, v)αt for a fixed α > 0, then for αt ≤ ρ1 we get 1 1 1 1 1 1 󸀠 2 2 2 2 ϕ (t) = (eαtA vt , eαtA v) + α(A 2 eαtA v, eαtA v) 2

(2.3.20)

166 | 2 Some properties of global attractor = −(Av, v)αt − ((1 + iμ)B(u1 , u2 , v), v)αt

󵄨 1 󵄨2 − ((1 + iμ)B(v, u1 + u2 , u2 ), v)αt + a|v|2αt + α󵄨󵄨󵄨A 4 v󵄨󵄨󵄨αt .

Using the estimates (2.3.11), (2.3.12) and (2.3.17), we get 1 󸀠 󵄨 1 󵄨2 󵄨 1 󵄨 ϕ (t) + 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt ≤ √1 + μ2 |u1 |2αt (|v|αt 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt + |v|2αt ) 2 󵄨 1 󵄨 + √1 + μ2 |u1 + u2 |αt |u2 |αt (|v|αt 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt + |v|2αt ) 󵄨 1 󵄨 + a|v|2αt + α|v|αt 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt 󵄨 1 󵄨 ≤ |v|αt 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt (12ρ20 √1 + μ2 + α) + |v|2αt (12ρ20 √1 + μ2 + a)

1 󵄨 1 󵄨2 1 󵄨 1 󵄨2 1 ≤ 72ρ40 (1 + μ2 )|v|2αt + 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt + α2 |v|2αt + 󵄨󵄨󵄨A 2 v󵄨󵄨󵄨αt 2 2 2 + |v|2αt (12ρ20 √1 + μ2 + a),

αt ≤ ρ1 .

Hence, ϕ󸀠 (t) ≤ ϕ(t)(14ρ40 (1 + μ2 ) + α2 + 24ρ20 √1 + μ2 + 2a) = ϕ(t)(λ1 + α2 ),

αt ≤ ρ1

(2.3.21)

for the obvious choice of λ1 . On the other hand, if we take the scalar product of (2.3.20) with v, (2.3.18) implies 1 d 2 󵄨 1 󵄨2 |v| = −󵄨󵄨󵄨A 2 v󵄨󵄨󵄨 − ((1 + iμ)B(u1 , u2 , v), v) 2 dt

− ((1 + iμ)B(u1 , u1 + u2 , v), v) + a|v|2

󵄨 1 󵄨 ≥ −ρ3 |v|2 − 12ρ20 √1 + μ2 (|v|󵄨󵄨󵄨A 2 v󵄨󵄨󵄨 + |v|2 ) 1

≥ −|v|2 (ρ3 + 12ρ20 ρ32 √1 + μ2 + 12ρ20 √1 + μ2 ) =−

λ2 2 |v| , 2

t≥0

(2.3.22)

for the obvious choice of λ2 . We obtain |v(t)|2 ≥ e−λ2 t |v(0)|2 , and hence, together with (2.3.21), 1

2 󵄨󵄨 αtA 2 󵄨2 󵄨 󵄨2 v(t)󵄨󵄨󵄨 ≤ e(λ1 +λ2 +α )t 󵄨󵄨󵄨v(t)󵄨󵄨󵄨 , 󵄨󵄨e

ρ

αt ≤ ρ1 . ρ

Choose α = max{√λ1 , √λ2 }, t1 = t2 = min{ 4λ1 , 4λ1 , √λ1 , √λ1 }, and ρ4 = αt2 . Then 1

1

2

1

2

1 1 1 󵄨 󵄨2 󵄨 󵄨2 󵄨󵄨 ρ4 A 2 󵄨2 v(t2 )󵄨󵄨󵄨 ≤ e 4 + 4 + 4 󵄨󵄨󵄨v(t2 )󵄨󵄨󵄨 ≤ 4󵄨󵄨󵄨v(t2 )󵄨󵄨󵄨 , 󵄨󵄨e

2.3 Upper bound estimate for the winding number

| 167

and (2.3.19) is established for t = t2 . Using a translation in time, we obtain (2.3.19) for every t ≥ t2 . Remark 2.3.1. If we ignore the dependence on μ and ν, then 1

ρ0 = O(a 2 ),

ρ3 = O(a4 ),

−2 ρ−1 4 = O(a ),

ρ−1 1 = O(a),

−2 ρ−1 2 = O(a ),

λ1 = O(a2 ),

for a → ∞.

λ2 = O(a4 ),

Corollary 2.3.3. Let u1 , u2 ∈ 𝒜, v = u1 − u2 . Then (i) |v|ρ4 ≤ 2|v|; (ii) |v(n) | ≤ 2n!ρ−n 4 |v|, n = 0, 1, 2, . . . ; (iii) |v(n) |∞ ≤

M n!( ρ2 )n |v|∞ , 2 4

−1

n = 0, 1, 2, . . . , where M = max{4ρ4 2 , 2(1 +

2 21 ) }. ρ4

̃ u1 , u ̃ u2 ∈ 𝒜 such Proof. By property (2) of attractor from Definition 2.3.2, there exist u ̃ that S(t1 + t2 )uui = ui , i = 1, 2. Using property (3) and Theorem 2.3.1, we have 1

󵄨󵄨 ρ1 A 2 ̃ i 󵄨󵄨󵄨󵄨 ≤ 2ρ0 , S(t)u 󵄨󵄨e

t ≥ t1 , i = 1, 2,

and by Corollary 2.3.2, 󵄨󵄨 21 ̃ u1 − S(t)u ̃ u2 )󵄨󵄨󵄨󵄨 ≤ ρ3 󵄨󵄨󵄨󵄨S(t)u ̃ u1 − S(t)u ̃ u2 󵄨󵄨󵄨󵄨, 󵄨󵄨A (S(t)u

t ≥ t1 .

By an application of Theorem 2.3.3, (i) follows. Claim (ii) follows easily from (i). To show (iii), for n ≥ 1, it follows from (2.3.6) and (ii) that − 1 −n 󵄨󵄨 (n) 󵄨󵄨 󵄨 (n) 󵄨 1 󵄨 (n+1) 󵄨󵄨 21 󵄨󵄨 ≤ 2(n + 1)!ρ4 2 ρ4 |v| 󵄨󵄨v 󵄨󵄨∞ ≤ 󵄨󵄨󵄨v 󵄨󵄨󵄨 2 󵄨󵄨󵄨v n 2 −1 ≤ 2n!ρ4 2 ( ) |v|. ρ1

In the case n = 0 we use (2.3.5) and (ii), then 1

1

|v|∞ ≤ |v| 2 (|v| + |vx |) 2 ≤ |v|(1 +

1

2 2 ) , ρ4

|v| ≤ |v|∞ ,

so (iii) follows. With the help of Corollary 2.3.3, we can extend functions to holomorphic functions on a strip containing the real axis. More precisely, Corollary 2.3.4. Let u1 , u2 ∈ 𝒜, v = u1 − u2 . The functions Re v and Im v can be extended to functions vr and vi , respectively, which are holomorphic on πδ = {z ∈ ℂ, |Im z| < δ},

δ=

ρ4 4

168 | 2 Some properties of global attractor such that 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨vr (z)󵄨󵄨󵄨 ≤ M|v|∞ = M sup 󵄨󵄨󵄨v(x)󵄨󵄨󵄨, x∈Ω 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨vi (z)󵄨󵄨󵄨 ≤ M|v|∞ = M sup 󵄨󵄨󵄨v(x)󵄨󵄨󵄨, x∈Ω

z ∈ πδ , z ∈ πδ .

Proof. The functions Re v and Im v are real analytic by Corollary 2.3.1. Hence, they can be extended to holomorphic functions on π2δ by Corollary 2.3.3. To obtain the remaining assertions, take z ∈ πδ , and use the Taylor expansion for vr : 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨󵄨 Re v(n) (Re z)(z − Re z)n 󵄨󵄨󵄨 󵄨󵄨vr (z)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ 󵄨󵄨 󵄨󵄨󵄨 n=0 n! 󵄨 ∞ 1 1 1 |v| )δn = M|v|∞ . ≤ ∑ ( Mn! n! 2 (2δ)n ∞ n=0 The same argument can be used also for vi . Now we will give an upper bound for the winding number. Proposition 2.3.1. Let v be a function which satisfies the assertions of Corollary 2.3.4, and suppose that each of the functions Re v and Im v has at least m zeros in Ω (counting multiplicities), for some nonnegative integer m. If m ≥ max{0,

log(M √2) π + 1} = m0 , − log(tanh 4δ )

then v = 0. In the proof we will use the following version of the Schwarz lemma. Lemma 2.3.2. Let f be a holomorphic function on the strip πδ , δ > 0, and suppose that 󵄨󵄨 󵄨 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ M,

z ∈ πδ .

If z1 , . . . , zm ∈ πδ are zeros of f (counting multiplicities), then m 󵄨󵄨 π(z − zj ) 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 )󵄨󵄨󵄨, 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ M ∏ 󵄨󵄨󵄨 tanh( 󵄨󵄨 󵄨󵄨 4δ j=1

z ∈ πδ .

Proof of Proposition 2.3.1. Let vr and vi be holomorphic extensions of Re v and Im v as in Corollary 2.3.4. Let v1 be one of the functions vr or vi , and let x1 , . . . , xm be the zeros of v1 , m ≥ m0 , then by Schwarz lemma, m 󵄨󵄨 π(z − zj ) 󵄨󵄨󵄨 󵄨 󵄨 󵄨 |v1 |∞ = sup 󵄨󵄨󵄨v1 (x)󵄨󵄨󵄨 ≤ M|v|∞ ∏ sup 󵄨󵄨󵄨 tanh( )󵄨󵄨󵄨 󵄨 󵄨󵄨 4δ x∈Ω j=1 x∈Ω 󵄨

2.3 Upper bound estimate for the winding number

≤ M(tanh(

| 169

m

π )) |v|∞ . 4δ

Therefore, 󵄨2 1 󵄨2 󵄨 󵄨 |v|∞ = sup(󵄨󵄨󵄨Re v(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨Im v(x)󵄨󵄨󵄨 ) 2 x∈Ω

≤ M √2(tanh(

m

π )) |v|∞ . 4δ

π m If m ≥ m0 , then M √2(tanh( 4δ )) < 1. Thus, v = 0 on Ω, and Proposition 2.3.1 follows.

Finally, we are in a position to state and prove our main result: Theorem 2.3.4. Let u ∈ 𝒜, p ∈ ℂ. Then (i) If each of the functions Re(u − p) and Im(u − p) has at least m0 + 1 different zeros in Ω, then u(x) = p, x ∈ Ω. (ii) If p ∉ {u}, then |indp (u)| ≤ m0 + 1. Proof. Obviously, (i) implies (ii). Now we give proof of (i). Let u ∈ 𝒜 and suppose that each of the functions Re(u − p) and Im(u − p) has at least m0 + 1 zeros. Since Re(u − p) and Im(u − p) are real analytic functions, we can choose 0 ≤ x1 < x2 < ⋅ ⋅ ⋅ < xm0 +1 ≤ 1 and 0 ≤ y1 < y2 < ⋅ ⋅ ⋅ < ym0 +1 ≤ 1 such that x1 , . . . , xm0 +1 are all the zeros of Re(u − p) in [x1 , xm0 +1 ] and y1 , . . . , ym0 +1 are all the zeros of Im(u − p) in [y1 , ym0 +1 ]. Choose a positive number ϵ < min{ min (xj+1 − xj ), min (yj+1 − yj )}, 1≤j≤m0

1≤j≤m0

(2.3.23)

and let u1 (x) = u(x),

u2 (x) = u(x − ϵ),

x ∈ ℝ.

Since the GLE is invariant under translations in the space variable, we have u1 , u2 ∈ 𝒜. Take v = u1 − u2 , let i ∈ {1, . . . , m0 }, and compute Re v(xi + ϵ) = Re u(xi + ϵ) − Re u(xi ) = Re u(xi + ϵ) − Re p,

Re v(xi+1 ) = Re u(xi+1 ) − Re u(xi+1 − ϵ) = Re p − Re u(xi+1 ).

Since xi +ϵ, xi+1 ∈ [xi , xi+1 ], Re v vanishes at least once in (xi , xi+1 ). Consequently, at least m0 times in Ω. Similarly, Im v vanishes at least m0 times in Ω. By Proposition 2.3.1, it follows that v = 0, i. e., u(x) = u(x − ϵ), ∀ϵ > 0 satisfying (2.3.23). Therefore, u is a constant, and the proof is finished.

170 | 2 Some properties of global attractor

2.4 Oscillations of solutions of the Kuramoto–Sivashinsky equation Consider the following periodic initial value problem for Kuramoto–Sivashinsky equation: 𝜕u 𝜕4 u 𝜕2 u 𝜕u + 4 + 2 +u = 0, 𝜕t 𝜕x 𝜕x 𝜕x u(x + L, t) = u(x, t), x ∈ ℝ, t ≥ 0,

(2.4.2)

u(x, 0) = u0 (0),

x ∈ ℝ,

(2.4.3)

∫ u(x, t)dx = 0,

t ≥ 0,

(2.4.4)

(2.4.1)

L

0

where L > 0. For the solutions of (2.4.1)–(2.4.4), we prove that every solution either converges to the stationary solution, or after some time the cardinalities of the zero sets of all its derivatives are bounded by certain constants, which are independent of solutions. Denote ∞

∞

j=−∞

j=−∞

H = {u = ∑ uj eiqjx , |u|2 = L ∑ |uj |2 < ∞, u0 = 0, uj = u∗−j , j ∈ ℤ}, iqjx where q = 2π/L, and ∗ denotes the complex conjugation. If u = ∑∞ ∈ H, j=−∞ uj e iqjx v = ∑∞ ∈ H, then j=−∞ vj e

∞

(u, v) = L ∑ uj vj . j=−∞

In order to rewrite the KS equation in the usual functional form, we introduce a closed operator A, and at the same time its powers, by ∞

As u = q4s ∑ |j|4s uj eiqjx ,

s ≥ 0,

j=−∞

where ∞

∞

j=−∞

j=−∞

u ∈ D(As ) = {v = ∑ vj eiqjx ∈ H, ∑ |j|8s |vj |2 < ∞}. 1

2

1

Note that A 2 u = − 𝜕𝜕xu2 , u ∈ D(A 2 ), Au = can be written as

𝜕4 u , 𝜕x4 1

u ∈ D(A), u󸀠 =

u󸀠 + Au − A 2 u + B(u, u) = 0.

𝜕u , 𝜕t

𝜕v B(u, v) = u 𝜕x . The KSE

(2.4.1󸀠 )

2.4 Oscillations of solutions of the Kuramoto–Sivashinsky equation

| 171

As we all know, the global existence and uniqueness of solutions of (2.4.1󸀠 ) was established, hence, for every u0 ∈ H, there exists a unique solution u(t) = S(t)u0 , u(0) = u0 , and for t ≥ 0, the mapping S(t) : H → H is continuous, and the dissipativity of the KSE was proved, namely there exists a constant ρ1 = ρ1 (L) > 0 such that for every u0 ∈ H we have S(t)u0 ∈ Bρ1 = {v0 ∈ H : |v0 | ≤ ρ1 },

t ≥ t1 , 8

where t1 = t1 (L, |u0 |) ≥ 0, S(t)Br ⊆ B2r , t ≥ 0, r ≥ ρ1 , ρ1 = C1 L 5 . We also obtain some known results on the Gevrey class regularity. For every τ ≥ 0, we introduce the operator ∞

1 4

eτA u = ∑ uj eτq|j| eiqjx , j=−∞

∞

1

∞

1

4 󵄨 4 󵄨2 u ∈ D(eτA ) = {v = ∑ vj eiqjx ∈ H : 󵄨󵄨󵄨eτA v󵄨󵄨󵄨 = L ∑ e2τq|j| |vj |2 < ∞}.

j=−∞

j=−∞

One can prove the following theorem. Theorem 2.4.1 ([15]). There exist positive constants ρ2 = ρ2 (L) and t2 = t2 (L) > 0 such that 1

󵄨󵄨 τρ2 A 4 󵄨 S(t)u0 󵄨󵄨󵄨 ≤ 2|u0 |, 󵄨󵄨e

t ∈ [0, t2 ], u0 ∈ B2ρ1 .

Theorem 2.4.2. There exists constants ρ3 = ρ3 (L) > 0, t2 = t2 (L) > 0, Mj = Mj (L) > 0, j = 1, 2, . . . , with the following property: For every u0 ∈ Bρ1 , there is t3 = t3 (L, u0 ) ∈ [0, ∞] such that (1) |S(t)u0 | ≤ |u0 |e−ρ3 t , 0 ≤ t ≤ t3 ; (2) for every fixed t ≥ t2 + t3 , j ∈ ℕ0 = {0, 1, 2, . . . }, the number of zeros of [0, L] is less than or equal to Mj .

dj (S(t)u0 ) dxj

in

Before we will prove the theorem, we will first prove the following corollary. Corollary 2.4.1. If u0 ∈ 𝒜\{0}, where the global attractor is 𝒜 = ⋂t≥0 S(t)Bρ1 , then the number of zeros of

dj u0 dxj

in [0, L] is less than or equal to Mj , j ∈ ℕ0 .

Proof. Suppose the assertion is not true for some u0 ∈ 𝒜. We will show that necessarily u0 = 0. For any T > 0, let uT ∈ Bρ1 be such that S(T)uT = u0 . By Theorem 2.4.2, we have T < t2 (L) + t3 (L, uT ). Consequently, 󵄨󵄨 󵄨 −ρ t 󵄨󵄨S(t)uT 󵄨󵄨󵄨 ≤ |uT |e 3 ,

t ≤ T − t2 .

Letting t = T − t2 , we obtain |ut2 | ≤ ρ1 e−ρ3 (T−t2 ) .

172 | 2 Some properties of global attractor Note that t2 depends only on L. So, as we let T → ∞, we conclude that ut2 = 0. Therefore, u0 = S(t2 )ut2 = 0. Before the proof of Theorem 2.4.2, we will prove several auxiliary results. Lemma 2.4.1. There exist positive constants ρ3 = ρ3 (L) and ρ4 = ρ3 (L) such that for every u0 ∈ Bρ1 , there is t3 = t3 (L, u0 ) ∈ [0, ∞] so that 1

(1) |A 2 S(t)u0 | > ρ4 |S(t)u0 | and |S(t)u0 | ≤ |u0 |e−ρ3 t , 0 < t < t3 ; 1 (2) |A 2 S(t)u0 | ≤ ρ4 |S(t)u0 |, t ≥ t3 . We will strengthen part (2) in Lemma 2.4.1.

Lemma 2.4.2. There exists a positive constant ρ5 = ρ5 (L) such that if S(t)u0 ∈ B2ρ1 , 1

|A 2 S(t)u0 | ≤ ρ4 |S(t)u0 |, t ≥ 0, then 1

2 󵄨󵄨 ρ5 A 4 󵄨 󵄨 󵄨 S(t)u0 󵄨󵄨󵄨 ≤ 2eρ4 t2 󵄨󵄨󵄨S(t)u0 󵄨󵄨󵄨, 󵄨󵄨e

t ≥ t2 .

Note that the assertion of the lemma implies 2 󵄨 2ρ t2 󵄨 󵄨 󵄨󵄨 4n 󵄨󵄨A S(t)u0 󵄨󵄨󵄨 ≤ 4n 󵄨󵄨󵄨S(t)u0 󵄨󵄨󵄨, ρ5

t ≥ t2 , n ∈ ℕ0 ,

(2.4.5)

since n 1 󵄨󵄨 αA 4 󵄨󵄨 a 󵄨󵄨 4n 󵄨󵄨 v0 󵄨󵄨 ≥ 󵄨󵄨e 󵄨A u0 󵄨󵄨, n! 󵄨

n

∀a ≥ 0, n ∈ ℕ0 , v0 ∈ D(A 4 ).

Proof. We take the scalar product of (2.4.1) with u and obtain 1 d 2 󵄨 1 󵄨2 󵄨 1 󵄨2 󵄨 1 󵄨2 |u| = −󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 + 󵄨󵄨󵄨A 4 u󵄨󵄨󵄨 > −󵄨󵄨󵄨A 2 u󵄨󵄨󵄨 ≥ −ρ24 |u|2 , 2 dt where we also used (B(u, u), u) = 0. Hence, 󵄨󵄨 󵄨 󵄨 󵄨 ρ2 τ 󵄨󵄨S(t)u0 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨S(t + τ)u0 󵄨󵄨󵄨e 4 ,

t, τ ≥ 0.

On the other hand, Theorem 2.4.1 implies 1

󵄨 󵄨 󵄨 󵄨 󵄨󵄨 τρ2 A 4 󵄨 2 S(t + τ)u0 󵄨󵄨󵄨 ≤ 2󵄨󵄨󵄨S(t)u0 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨S(t + τ)u0 󵄨󵄨󵄨eρ4 τ , 󵄨󵄨e t ≥ 0, τ ∈ [0, t2 ]. Denoting ρ5 = t2 ρ2 , we get 1

2 󵄨 󵄨 󵄨 󵄨󵄨 ρ5 A 4 S(t)u0 󵄨󵄨󵄨 ≤ 2eρ4 t2 󵄨󵄨󵄨S(t)u0 󵄨󵄨󵄨, 󵄨󵄨e

and our Lemma 2.4.2 is established.

t ≥ t2 ,

2.4 Oscillations of solutions of the Kuramoto–Sivashinsky equation

| 173

A consequence of Lemma 2.4.2 is the following: 1

Lemma 2.4.3. Suppose that S(t)u0 ∈ B2ρ1 and |A 2 S(t)u0 | ≤ ρ4 |S(t)u0 |, t ≥ 0. Then for each fixed t ≥ t2 , the function u = S(t)u0 can be (as a function of the x variable) extended ̃ which is holomorphic on the strip to a function u πδ = {z ∈ ℂ : |Im z| < δ},

δ=

ρ5 , 2

moreover, it satisfies 󵄨󵄨 󵄨󵄨 dn 󵄨󵄨 ̃ (z)󵄨󵄨󵄨󵄨 ≤ ρ(n) 󵄨󵄨 n u 6 |u|, 󵄨󵄨 󵄨󵄨 dz n+ 21

2

where ρ(n) = 2(n+3) eρ4 t (n + 1)!ρ5 6 rem 2.4.1 and Lemma 2.4.2.

z ∈ πδ , n ∈ ℕ0 ,

. The constants t2 and ρ5 were introduced in Theo-

Proof. Assumptions of Lemma 2.4.2 are satisfied. Therefore, by Agmon’s inequality, ρ2 t 󵄨󵄨 dn u 󵄨󵄨 󵄨󵄨 󵄨 󵄨 n 󵄨 1 󵄨 n+1 󵄨 1 2e 4 2 (n + 1)! |u|, 󵄨󵄨 n 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨A 4 u󵄨󵄨󵄨 2 󵄨󵄨󵄨A 4 u󵄨󵄨󵄨 2 ≤ 󵄨󵄨 dx 󵄨󵄨∞ n+ 1 ρ5 2

and thus we obtain the first conclusion. Next, using the power series expansions, 󵄨󵄨 dn 󵄨󵄨 ∞ |z − Re z|j 󵄨󵄨 dn+j u 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 ̃ (z)󵄨󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨 n u 󵄨󵄨 n+j 󵄨󵄨󵄨 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dx 󵄨󵄨∞ j! j=0 ≤ =

2eρ24 t2 n+ 1 ρ5 2

∞

(n + j + 1)! −j 2 j! j=0

|u| ∑ 2

2n+3 eρ4 t2 (n + 1)! n+ 21

ρ5

|u|,

∀z ∈ πδ , n ∈ ℕ0 .

Proof of Theorem 2.4.1. We can assume u0 ≠ 0 since otherwise t3 = ∞ obviously works. Part (1) follows from Lemma 2.4.1. We fix an arbitrary t ≥ t2 + t3 , t2 is as in Lemma 2.4.2, and t3 is as in Lemma 2.4.1, and we denote u = S(t)u0 . Fix j ∈ ℕ0 , and let j j ̃ is x1 , x2 , . . . , xNj be all the zeros of ddxuj in [0, L]. From Schwarz lemma applied to ddxũj (u as in Lemma 2.4.3), Lemma 2.4.1, and Lemma 2.4.3, we get N 󵄨󵄨 dj u 󵄨 πL j 󵄨󵄨 ̃ 󵄨󵄨󵄨 (j) ) |u|, 󵄨󵄨 j (x)󵄨󵄨 ≤ ρ6 (tanh 󵄨󵄨 dx 󵄨󵄨 2ρ5

Therefore, N

πL j 21 󵄨󵄨 4j 󵄨󵄨 (j) ) L |u| 󵄨󵄨A u󵄨󵄨 ≤ ρ6 (tanh 2ρ 5

x ∈ [0, L].

174 | 2 Some properties of global attractor

(j)

≤ ρ6 (tanh (j) 󵄨 j 󵄨 = ρ7 󵄨󵄨󵄨A 4 u󵄨󵄨󵄨.

N

j

πL j 21 L 󵄨󵄨 4j 󵄨󵄨 ) L ( ) 󵄨󵄨A u󵄨󵄨 2ρ5 2π l

Since u ≠ 0 and u cannot be a polynomial (u is periodic with period L, and ∫0 udx = 0), (j)

we have ρ7 ≥ 1. This shows that 1

(j)

Nj ≤ lg[ρ5 L 2 (

j

−1

L πL ) ](lg coth ) 2π 2ρ5

= Mj ,

∀j ∈ ℕ0 .

Remarks. (1) If t ∈ [t3 , t2 + t3 ], j ∈ ℕ0 , then the number of zeros of less than or equal to 1

(j)

lg[ρ6 L 2 (

j+ 21

j

t L )( 2 ) 2π t − t3

](lg coth

dj (S(t)u0 ) dxj

in [0, L] is

πLt2 ). 2ρ5 (t − t3 )

(2) The analysis of dependence of Mj on j shows that for a fixed L, Mj = O(j lg j),

j → ∞.

2.5 Hausdorff length of level sets Consider the 2D complex Ginzburg–Landau equation ut − (1 + iν)Δu + (1 + iμ)|u|2 u − au = 0,

(2.5.1)

where the bifurcation parameters are ν ∈ ℝ, μ ∈ ℝ, a ≥ 0, and u : ℝ2 → ℂ is unknown. We consider periodic boundary condition on Ω = [0, 1]2 given by u(x + 1, y, t) = u(x, y + 1, t) = u(x, y, t),

x, y ∈ ℝ, t ∈ [0, ∞)

(2.5.2)

and u(x, y, 0) = u0 (x, y),

x, y ∈ ℝ,

(2.5.3)

where u0 is a given Ω-periodic, locally square integrable function. We establish an upper bound for an appropriate Hausdorff measure of the zero set of (2.5.1). Let Ω = [0, L]n , for some L > 0, n ∈ ℕ, and let L2per (Ω) = {f : ℝn → ℝ : ∫ f 2 < ∞, f is Ω-periodic}. Ω

We identify each f ∈

L2per (Ω)

with its Fourier expansion f (x) = ∑ fj eiqj⋅x , j∈ℤn

x ∈ ℝn ,

(2.5.4)

2.5 Hausdorff length of level sets | 175

where fj ∈ ℂ are the Fourier coefficients and q = 2π/L. (In particular, fj = f−j∗ , ∀j ∈ ℤn , where ∗ denotes the complex conjugation.) Note that ‖f ‖2L2 = ∫ f 2 dx = Ln ∑ |fj |2 . j∈ℤn

Ω

Denoting |j| = |j1 | + |j2 | + ⋅ ⋅ ⋅ + |jn |,

1

|j|2 = (|j1 |2 + |j2 |2 + ⋅ ⋅ ⋅ + |jn |2 ) 2 , j = (j1 , j2 , . . . , jn ) ∈ ℤn ,

define 2 ‖f ‖2m = Ln ∑ |qj|2m 2 |fj | , j∈ℤn

m ≥ 0.

(2.5.5)

Define the Sobolev space m Hper (Ω) = {f (x) = ∑ fj eiqj⋅x , ‖f ‖2H m = ‖f ‖2m + q2m ‖f ‖2L2 < ∞}. j∈ℤn

We introduce the Gevrey class of real-valued functions for which the Fourier coefficients converge with an exponential rate, let Γ(r) = {f = ∑ fj eiqj⋅x : ‖f ‖2Γ(r) = Ln ∑ e2q|j|2r |fj |2 < ∞}, j∈ℤn

j∈ℤn

r ≥ 0.

(2.5.6)

For any function f on a set S, denote by W(f , S) = {x ∈ S : f (x) = 0} its zero set. We need to prove the following theorem: Theorem 2.5.1. Let r > 0, and suppose that some f ∈ Γ(r), which is not the zero function, satisfies ‖f ‖Γ(r) ≤ M‖f ‖L2 ,

(2.5.7)

where M > 0. Then there exists α1 = α1 (L, M, r, n) such that ℋ

n−1

(N(f , Ω)) ≤ α1 ,

where α1 = C1 Ln−1 (1 + lg M)eC2 L/r . Here and below, C1 , C2 , . . . denote various constants which may depend only on dimension n of the space.

176 | 2 Some properties of global attractor In order to prove the theorem, we recall the following lemma: Lemma 2.5.1. Let f be a holomorphic function on Πδ = {z ∈ ℂ : |Im z| < δ},

δ > 0,

which is not the constant function 0, and suppose it satisfies 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ M max 󵄨󵄨󵄨f (x)󵄨󵄨󵄨, x∈[0,L]

z∈Πδ

M ≥ 1, l ≥ 0.

(2.5.8)

Then card(N(f , [0, L])) ≤ α2 ,

α2 = α2 (δ, M, L).

Proof. Suppose x1 , . . . , xN ∈ [0, L] are the distinct zeros of f in [0, L]. Schwarz lemma and (2.5.8) imply N 󵄨󵄨 π(x − xj ) 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨f (x)󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ∏ 󵄨󵄨󵄨 tanh 󵄨 4δ 󵄨󵄨󵄨 z∈Πδ j=1 󵄨

≤ M(tanh

N

πL 󵄨 󵄨 ) max 󵄨󵄨󵄨f (y)󵄨󵄨󵄨, y∈[0,L] 4δ

∀x ∈ [0, L].

Therefore, M(tanh

N

πL ) ≥ 1. 4δ

We thus get N≤

log M ≤ (log M)eπL/2δ = α2 , log coth(πL/4δ)

since M ≥ 1, 1/ lg coth x ≤ e2x , ∀x ∈ ℝ holds. Before the proof of Theorem 2.5.1, we also recall Agmon’s inequality 1 1 n 󵄨 󵄨 ‖f ‖∞ = sup 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ≤ C3 (‖f ‖L2 2 ‖f ‖n2 + q 2 ‖f ‖L2 ),

x∈Ω

n f ∈ Hper (Ω).

Note that (2.5.4)–(2.5.5) imply 󵄩󵄩 𝜕|α| f 󵄩󵄩2 󵄩󵄩 󵄩 n 2m 󵄨 α 󵄨2 2 󵄩󵄩 α 󵄩󵄩󵄩 = L ∑ |qj|2 󵄨󵄨󵄨(iqj) 󵄨󵄨󵄨2 |fj | n 󵄩󵄩 𝜕x 󵄩󵄩m j∈ℤ ≤ Ln ∑ |qj|2(m+|α|) |fj |2 ≤ 2 j∈ℤn

Ln (2(m + |α|))! ∑ |fj |2 e2q|j|2r (2r)2(m+|α|) j∈ℤn

(2.5.9)

2.5 Hausdorff length of level sets | 177

≤

Ln (m + |α|)!2 ∑ |fj |2 e2q|j|2r (r)2(m+|α|) j∈ℤn

n+|α| ∀α ∈ ℕN0 , m ∈ ℕ0 , f ∈ Hper (Ω). Hence, by (2.5.6)

󵄩󵄩 𝜕|α| f 󵄩󵄩 (m + |α|)! 󵄩 󵄩󵄩 ‖f ‖Γ(r) , 󵄩󵄩 α 󵄩󵄩󵄩 ≤ 󵄩󵄩 𝜕x 󵄩󵄩m r m+|α| α ∈ ℕN0 , m ∈ ℕ0 , f ∈ Γ(r).

(2.5.10)

Proof of Theorem 2.5.1. With the help of (2.5.9) and (2.5.10), we get 1

1

󵄩󵄩 𝜕|α| f 󵄩󵄩 󵄩󵄩 𝜕|α| f 󵄩󵄩 2 󵄩󵄩 𝜕|α| f 󵄩󵄩 2 n󵄩 󵄩󵄩 𝜕|α| f 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 α 󵄩󵄩󵄩 ≤ C3 (󵄩󵄩󵄩 α 󵄩󵄩󵄩 󵄩󵄩󵄩 α 󵄩󵄩󵄩 + q 2 󵄩󵄩󵄩 α 󵄩󵄩󵄩 ) 󵄩󵄩 𝜕x 󵄩󵄩∞ 󵄩󵄩 𝜕x 󵄩󵄩L2 󵄩󵄩 𝜕x 󵄩󵄩L2 󵄩󵄩 𝜕x 󵄩󵄩n 1

1

≤

n C3 |α|! 2 (n + |α|)! 2 ( + q 2 |α|!)‖f ‖Γ(r) |α| |α| r r

≤

n C3 |α|!2(n+|α|)/2 n! 2 ( n + q 2 )‖f ‖Γ(r) |α| r r2

1

≤ α3

2|α| |α|! ‖f ‖Γ(r) , r |α|

(2.5.11)

where n

α3 = C3 2 2 (

1

n! 2 r

n 2

n

+ q 2 ).

Like in the estimate (2.5.12) below, it follows that given any x0 ∈ ℝn the series α ∑α∈ℕ2 (𝜕|α|f /𝜕x )(x − x0 )α converges for all x ∈ ℝn such that |x − x0 | < r/2n. This shows 0 that f , modified on a set of measure 0, is real analytic at every point, analyticity radius at each point being at least r/2n. It can thus be extended to a holomorphic function ̃f on Πnδ = {z = (z1 , z2 , . . . , zn ) : |Im zj | ≤ δ, ∀j ∈ {1, . . . , n}}, where δ = r/4n. Now, by (2.5.7) and (2.5.11), we have 󵄨󵄨 𝜕|α| f 󵄨󵄨 δα 󵄨󵄨̃ 󵄨󵄨 󵄨 󵄨 󵄨󵄨f (z)󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨 α (Re z)󵄨󵄨󵄨 󵄨󵄨 α! n 󵄨󵄨 𝜕x α∈ℕ0 ≤ Mα3 ‖f ‖L2 ∑

α∈ℕn0

2|α| |α|!δα r |α| α!

n

= ML 2 α3 ‖f ‖L∞ ∑

α∈ℕn0

n

|α|! (2n)|α| α!

∞

1 |α|! ∑ k α! (2n) k=0 |α|=k

= ML 2 α3 ‖f ‖L∞ ∑ n

∞

= ML 2 α3 ‖f ‖L∞ ∑ 2−k = α4 ‖f ‖L∞ , k=0

178 | 2 Some properties of global attractor n

where α4 = 2ML 2 α3 , ℕ0 = {0, 1, 2, . . . }. Note that if n = 1, Lemma 2.5.1 gives our assertion. If n > 1, we first find x0 ∈ Ω for which 1 󵄨 󵄨 󵄨 󵄨󵄨̃ sup 󵄨󵄨̃f (z)󵄨󵄨󵄨. 󵄨󵄨f (x0 )󵄨󵄨󵄨 ≥ α4 z∈Πnδ 󵄨

(2.5.12)

Using a simple translation, we can assume that x0 = 0. By [23], there exists a singular set S ⊆ Ω with H n−1 (S) = 0 such that N(f , Ω)\S is a countable union of (n − 1)-dimensional analytic manifolds with ℋ

n−1

(N(f , Ω) = ℋn−1 N(f , Ω)\S) < ∞.

Now let x1 = (−L, 0, 0, 0, . . . , 0), x2 = (−L, L, 0, 0, . . . , 0), x3 = (−L, 0, L, 0, . . . , 0), . . . , xn = (−L, 0, 0, 0, . . . , L). For j = 1, 2, . . . , n and ω ∈ Sn−1 , denote by lj (ω) the number of intersections of the line through xj having direction ω with the manifold N(f , Ω)\S. A simple geometrical consideration shows that ℋ

n−1

n

(N(f , Ω)) ≤ C4 Ln−1 ∑ ∫ lj (ω)dω, j=1

(2.5.13)

Sn−1

where C4 depends only on n. For the rest of the proof, we fix j ∈ {1, 2, . . . , n} and ω ∈ Sn−1 , and we intend to estimate lj (ω). We only need to consider ω ∈ Sn−1 for which lj (ω) < ∞. Indeed, considering the function gj : Ω → Sn−1 defined by gj (x) =

x − xj

|x − xj |

,

x ∈ Ω,

we obtain by [23] ∫ lj (ν)dν ≤ (Lip gj )n−1 ℋn−1 (N(f , Ω)) < ∞, Sn−1

where Lip gj is the Lipschitz constant of gj ; this shows that lj (ν) < ∞ for almost every ν ∈ Sn−1 . Thus, we may assume that ϕ(t) = ̃f (xj + tω),

t ∈ Πδ

is not the zero function. By (2.5.12), 1 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨̃f (z)󵄨󵄨󵄨 ≥ sup 󵄨󵄨ϕ(t)󵄨󵄨󵄨. 󵄨󵄨ϕ(0)󵄨󵄨󵄨 = 󵄨󵄨󵄨f (xj )󵄨󵄨󵄨 = 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 ≥ α4 z∈Πnδ 󵄨 α4 t∈Πnδ 󵄨 Lemma 2.5.1 implies lj (ω) ≤ α2 (δ, α4 , L√n + 3), since maxx∈Ω dist(xj , x) = L√n + 3. Combining this with (2.5.13), we finally obtain ℋ

n−1

(N(f , Ω)) ≤ C4 Ln−1 α2 (δ, α4 , L√n + 3),

and the proof is concluded.

2.5 Hausdorff length of level sets | 179

Now we describe some known results of GL equation and verify the conditions which satisfy Theorem 2.5.1. Let Ω = [0, 1]2 , and let H be the space of complex-valued, Ω-periodic functions which are square integrable on Ω. We identify each f ∈ H with its Fourier expansion f (x) = ∑ fj e2πij⋅x , j∈ℤ2

x ∈ ℝ2 ,

(2.5.14)

where fj ∈ ℂ, j ∈ ℤ2 . The set H is a Hilbert space over the field ℝ if equipped with the scalar product (f , g) = Re ∫ fg ∗ ,

f , g ∈ H,

Ω

and the norm 1

‖f ‖L2 = (f , f ) 2 ,

f ∈ H.

If f is as in (2.5.14), then ‖f ‖2L2 = ∑j∈ℤ2 |fj |2 . Let s

As f = ∑ (|2πj|22 + 1) fj e2πij⋅x ,

s ≥ 0,

j∈ℤ2

where f is as in (2.5.14), and where, recall, |j|22 = (j(1) )2 + (j(2) )2 , ∀j = (j(1) , j(2) ) ∈ ℤ2 . For every s ≥ 0, As is a positive closed operator with the domain of definition 2s 󵄨 󵄨2 D(As ) = {g = ∑ gj e2πij⋅x ∈ H : 󵄨󵄨󵄨As g 󵄨󵄨󵄨 = ∑ |gj |2 (|2πj|22 + 1) < ∞}. j∈ℤ2

j∈ℤ2

Note that Af = −Δf + f for f ∈ D(A). Also, we have immediately 󵄩󵄩 21 󵄩󵄩 󵄩󵄩A f 󵄩󵄩L2 ≥ ‖f ‖L2 ,

1

f ∈ D(A 2 ).

Denoting u󸀠 = ut and B(u1 , u2 , u3 ) = u1 u2 u∗3 , we may rewrite the GL equation in the form u󸀠 + (1 + iν)Au + (1 + iμ)B(u, u, u) − (a + 1 + iν)u = 0.

(2.5.15)

We recall some inequalities involving the nonlinear term B. They are 󵄨󵄨 󵄨 󵄨󵄨(B(u1 , u2 , u3 ), u4 )󵄨󵄨󵄨 ≤ ‖u1 ‖L2 ‖u2 ‖L∞ ‖u3 ‖L∞ ‖u4 ‖L2 , 󵄨󵄨 󵄨󵄨 󵄨󵄨(B(u1 , u2 , u3 ), u4 )󵄨󵄨 ≤ ‖u1 ‖L∞ ‖u2 ‖L∞ ‖u3 ‖L2 ‖u4 ‖L2 ,

(2.5.16)

1 󵄩 1 󵄩1 󵄨󵄨 󵄨 󵄨󵄨(B(u1 , u2 , u3 ), u4 )󵄨󵄨󵄨 ≤ C6 ∏ ‖uj ‖L2 2 󵄩󵄩󵄩A 2 uj 󵄩󵄩󵄩L2 2 .

(2.5.17)

4

j=1

For every u0 ∈ H there exists a solution u(t) = S(t)u0 (t ≥ 0), u(0) = u0 . The properties of the solution operator S are:

180 | 2 Some properties of global attractor (i) S(t)H ⊂ H, ∀t ≥ 0, S(t) : H → H is injective and continuous. (ii) S(t + s)u0 = S(t)S(s)u0 , ∀s, t ≥ 0, u0 ∈ H. (iii) There exist constants ρ1 , ρ2 , ρ3 and t, depending only on ν, μ and a such that 󵄩 󵄩󵄩 󵄩󵄩S(t)u0 󵄩󵄩󵄩L2 ≤ ρ1 , u0 ∈ H, t ≥ t1 , 󵄩󵄩 21 󵄩 󵄩󵄩A S(t)u0 󵄩󵄩󵄩L2 ≤ ρ2 , u0 ∈ H, t ≥ t1 , 󵄩 󵄩󵄩 󵄩󵄩S(t)u0 󵄩󵄩󵄩L∞ ≤ ρ3 , u0 ∈ H, t ≥ t1 . Denoting Bρ1 = {u0 ∈ H : ‖u0 ‖L2 ≤ ρ1 }, we recall the definition of the global attractor: 𝒜 = ⋂ S(t)Bρ1 . t≥0

The global attractor has the following properties: 1 (1) 𝒜 is a nonempty, compact subset of D(A 2 ); (2) S(t)𝒜 = 𝒜, ∀t ≥ 0; (3) ‖u0 ‖L2 ≤ ρ1 , u0 ∈ 𝒜; (4) limt→∞ sup‖u0 ‖ 2 ≤M infv0 ∈𝒜 ‖S(t)u0 − v0 ‖L2 = 0, ∀M > 0. L

Now we consider Gevrey class regularity. We introduce the operator 1 2

2

1 2r

erA f = ∑ fj e(|2πj|2 +1) e2πij⋅x , j∈ℤ2

r ≥ 0,

1 2

f = ∑ fj e2πij⋅x ∈ D(ArA ) j∈ℤ2

1

1

2 2r 󵄩 2 󵄩2 = {g = ∑ gj e2πij⋅x : ‖g‖2G(r) = 󵄩󵄩󵄩erA g 󵄩󵄩󵄩L2 = ∑ |gj |2 e2(|2πj|2 +1) < ∞},

j∈ℤ2

(f , g)G(r) = (e

j∈ℤ2

1 rA 2

f,e

1 rA 2

g),

f , g, ∈ D(e

1 rA 2

).

Proposition 2.5.1 ([29]). There exist constants t2 = t2 (ν, μ, a) and α5 = α5 (ν, μ, a) such 1 that if ‖A 2 S(t)u0 ‖L2 ≤ αρ2 , ∀t ≥ 0, then 󵄩󵄩 21 󵄩 󵄩󵄩A u(t)󵄩󵄩󵄩G(α5 t/t2 ) ≤ 2ρ2 , 0 ≤ t ≤ t2 , 1 󵄩󵄩󵄩A 2 u(t)󵄩󵄩󵄩 󵄩 󵄩G(α5 ) ≤ 2ρ2 , t ≥ t2 . Proposition 2.5.2. There exist constant α6 = α6 (ν, μ, a) and α7 = α7 (ν, μ, a) > 0 such that for any two solutions u1 and u2 , for which 󵄩 󵄩󵄩 21 󵄩󵄩A uj (t)󵄩󵄩󵄩L2 ≤ ρ2 ,

t ≥ 0, j = 1, 2,

there exists a unique t3 = t3 (ν, μ, a, u1 , u2 ) ∈ [0, ∞] such that 󵄩 󵄩 󵄩 󵄩󵄩 21 󵄩󵄩A (u1 (t) − u2 (t))󵄩󵄩󵄩L2 > α6 󵄩󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩L2 ,

t ∈ [0, t3 ),

2.5 Hausdorff length of level sets | 181

󵄩 −α t 󵄩 󵄩 󵄩󵄩 󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩L2 ≤ 󵄩󵄩󵄩u1 (0) − u2 (0)󵄩󵄩󵄩L2 e 7 , t ∈ [0, t3 ), 󵄩󵄩 21 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩A (u1 (t) − u2 (t))󵄩󵄩L2 ≤ α6 󵄩󵄩u1 (t) − u2 (t)󵄩󵄩L2 , t ∈ [t3 , ∞). Next we study the Gevrey class properties of differences of two solutions. For this purpose, we shall obtain some inequalities involving the trilinear form B. Let uk ∈ 1 2

D(erA ), k = 1, 2, 3, 4, and r > 0 be given by uk = ∑ uk e2πij⋅x , j∈ℤ2

2

1 2

̃ k = ∑ ukj e2πij⋅x = ∑ |ukj |e(|2πj |2 +1) r e2πij⋅x , u j=−∞

j∈ℤ2

and note that 󵄩󵄩 s 󵄩󵄩 󵄩 s ̃ 󵄩󵄩 󵄩󵄩A u󵄩󵄩G(r) = 󵄩󵄩󵄩A u k󵄩 󵄩L2 ,

s ≥ 0, k = {1, 2, 3, 4}.

(2.5.18)

Lemma 2.5.2. With the notation in the above paragraph, we have 󵄨󵄨 󵄨 ̃1 , u ̃2 , u ̃ 3 ), u ̃ 4 ). 󵄨󵄨(B(u1 , u2 , u3 ), u4 )G(r) 󵄨󵄨󵄨 ≤ (B(u 1

Proof. Let ϕ(j) = (|2πj|22 + 1) 2 r, j ∈ ℤ2 . We have 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∗ ∗ 2φ(j+k−m) 󵄨󵄨󵄨 󵄨󵄨(B(u1 , u2 , u3 ), u4 )G(r) 󵄨󵄨󵄨 = 󵄨󵄨󵄨Re ∑ u1,j u2,j u3,j u4,j+k−m e 󵄨󵄨 󵄨󵄨 󵄨󵄨 j,k,m∈ℤ2 ≤ = ≤

∑

󵄨 󵄨󵄨 󵄨 |u1,j ||u2,j |󵄨󵄨󵄨u∗3,j 󵄨󵄨󵄨󵄨󵄨󵄨u∗4,j+k−m 󵄨󵄨󵄨e2φ(j+k−m)

∑

̃ 4,j+k−m eφ(j+k−m)−φ(j)−φ(k)−φ(m) ̃ 3,j u ̃ 2,j u ̃ 1,j u u

∑

̃ 1,j u ̃ 2,j u ̃ 3,j u ̃ 4,j+k−m u

j,k,m∈ℤ2 j,k,m∈ℤ2 j,k,m∈ℤ2

̃1 , u ̃2 , u ̃ 3 ), u ̃ 4 ). = (B(u The last inequality holds since φ is even and since φ(j + k) ≤ φ(j) + φ(k),

j, k ∈ ℤ2 .

Lemma 2.5.2 and (2.5.17) imply 4

1 󵄩󵄩 21 󵄩󵄩 21 󵄨󵄨 󵄨 2 󵄩󵄩A uj 󵄩󵄩G(r) , 󵄨󵄨(B(u1 , u2 , u3 ), u4 )G(r) 󵄨󵄨󵄨 ≤ C6 ∏ ‖uj ‖G(r)

j=1

1

1 2

provided u1 , u2 , u3 , u4 ∈ D(A 2 erA ).

(2.5.19)

182 | 2 Some properties of global attractor Lemma 2.5.3. Let u1 and u2 be two solutions of (2.5.15) which satisfy 󵄩󵄩 21 󵄩 󵄩󵄩A uj (t)󵄩󵄩󵄩G(α5 ) ≤ 2ρ2 , t ≥ 0, j = 1, 2, 󵄩󵄩 󵄩󵄩 󵄩󵄩uj (t)󵄩󵄩L∞ ≤ ρ3 , t ≥ 0, j = 1, 2.

(2.5.20) (2.5.21)

There exist constants α8 = α8 (ν, μ, a) and t4 = t4 (ν, μ, a) such that if 󵄩󵄩 21 󵄩 󵄩 󵄩 󵄩󵄩A (u1 (t) − u2 (t))󵄩󵄩󵄩L2 ≤ α6 󵄩󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩L2 ,

t ≥ 0,

(2.5.22)

then 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩G(α5 ) ≤ α8 󵄩󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩L2 ,

t ≥ t4 ,

where α5 , α6 , ρ2 , ρ3 have been introduced in the previous section. Proof. Let v = u1 − u2 , and note that v󸀠 + (1 + iν)Av + (1 + iμ)(B(u1 , u1 , v) + B(v, u1 + u2 , u2 )) − (a + 1 + iν)v = 0.

(2.5.23)

Fix any α > 0, and denote ϕ(t) = (v, v)G(αt) . Then, if a < αt < α5 , we have 1 1 󸀠 φ (t) = (v󸀠 , v)G(αt) + α(A 2 v, v)G(αt) 2 = −(Av, v)G(αt) − ((1 + iμ)B(u1 , u1 , v), v)G(αt)

󵄩 1 󵄩2 − ((1 + iμ)B(v, u1 + u2 , u2 ), v)G(αt) + (a + 1)‖v‖2G(αt) + α󵄩󵄩󵄩A 4 v󵄩󵄩󵄩G(αt) ,

since (iνv, v)G(αt) = (iνAv, v)G(αt) = 0. Hence, using (2.5.19), 1 1 󸀠 󵄩 1 󵄩2 󵄩 1 󵄩 󵄩 1 󵄩 φ (t) + 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) ≤ C6 (1 + μ2 ) 2 ‖u1 ‖G(αt) 󵄩󵄩󵄩A 2 u1 󵄩󵄩󵄩G(αt) ‖v‖G(αt) 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) 2 1 1 1 1 2 󵄩󵄩󵄩A 2 (u1 + u2 )󵄩󵄩󵄩 2 + C6 (1 + μ2 ) 2 ‖u1 + u2 ‖G(αt) 󵄩 󵄩G(αt) 1 󵄩󵄩 21 󵄩󵄩 21 󵄩 1 󵄩 2 × ‖u2 ‖G(αt) 󵄩󵄩A u2 󵄩󵄩G(αt) ‖v‖G(αt) 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) 1 󵄩 1 󵄩 2 + α‖v‖G(αt) 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) + (a + 1)‖v‖G(αt) 󵄩 1 󵄩2 ≤ C62 (1 + μ2 )‖u1 ‖2G(αt) 󵄩󵄩󵄩A 2 u1 󵄩󵄩󵄩G(αt) ‖v‖2G(αt) 󵄩 1 󵄩 + C62 (1 + μ2 )‖u1 + u2 ‖G(αt) 󵄩󵄩󵄩A 2 (u1 + u2 )󵄩󵄩󵄩G(αt) 󵄩 1 󵄩 × ‖u2 ‖G(αt) 󵄩󵄩󵄩A 2 u2 󵄩󵄩󵄩G(αt) ‖v‖2G(αt) 1 󵄩 1 󵄩2 1 󵄩 1 󵄩2 + 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) + 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) 4 4 α2 1 󵄩 1 󵄩2 + (a + 1)‖v‖2G(αt) + ‖v‖2G(αt) + 󵄩󵄩󵄩A 2 v󵄩󵄩󵄩G(αt) 2 2 0 ≤ αt ≤ α5 .

2.5 Hausdorff length of level sets | 183

Therefore, (2.3.20) implies φ󸀠 (t) ≤ (α9 + α2 )φ(t),

t ∈ [0,

α5 ], α

where α9 = C7 (1 + μ2 )ρ42 + 2a + 2. Hence, 󵄩2 󵄩󵄩 (α +α2 )t 󵄩 󵄩󵄩v(0)󵄩󵄩󵄩2 2 , 󵄩󵄩v(t)󵄩󵄩󵄩G(αt) ≤ e 9 󵄩L 󵄩

t ∈ [0,

α5 ]. α

1

1

Choosing α = α92 and t = α5 /α92 , we obtain 1

󵄩󵄩 󵄩 󵄩 α α2 󵄩 󵄩󵄩v(t4 )󵄩󵄩󵄩G(α5 ) ≤ e 5 9 󵄩󵄩󵄩v(0)󵄩󵄩󵄩L2 , 1

where t4 = α5 /α92 . Translating in time, we are led to 1

󵄩󵄩 󵄩 󵄩 α α2 󵄩 󵄩󵄩v(t + t4 )󵄩󵄩󵄩G(α ) ≤ e 5 9 󵄩󵄩󵄩v(t)󵄩󵄩󵄩L2 , 5

t ≥ 0.

(2.5.24)

On the other hand, we may take the scalar product of the both sides of (2.5.23) with v and use (2.5.16) in order to get 1 d 󵄩 1 󵄩2 ‖v‖2L2 = −󵄩󵄩󵄩A 2 v󵄩󵄩󵄩L2 − ((1 + iμ)B(u1 , u1 , v), v) 2 dt − ((1 + iμ)B(v, u1 + u2 ), u2 ) − (a + 1)‖v‖2L2 1

≥ −α62 ‖v‖2L2 − (1 + μ2 ) 2 ‖u1 ‖2L∞ ‖v‖2L2 1

− (1 + μ2 ) 2 ‖u1 ‖L∞ ‖u1 + u2 ‖L∞ ‖v‖L2 + (a + 1)‖v‖2L2 1

≥ −(α62 + 3ρ23 (1 + μ2 ) 2 − (a + 1))‖v‖2L2 = −α10 ‖v‖2L2 with the obvious choice for α10 . This and (2.5.24) imply 1

󵄩󵄩 󵄩 󵄩 α α2 󵄩 󵄩󵄩v(t + t4 )󵄩󵄩󵄩G(α5 ) ≤ e 5 9 󵄩󵄩󵄩v(t)󵄩󵄩󵄩L2 1

2 󵄩 󵄩 ≤ eα5 α9 +α10 t4 󵄩󵄩󵄩v(t + t4 )󵄩󵄩󵄩L2 ,

t ≥ 0.

1 2

Letting α8 = eα5 α9 +α10 t4 , we complete the proof. Theorem 2.5.2. Let u01 , u02 ∈ 𝒜. Then 󵄩󵄩 0 󵄩 0 0󵄩 0󵄩 󵄩󵄩u1 − u2 󵄩󵄩󵄩G(α5 ) ≤ α8 󵄩󵄩󵄩u1 − u2 󵄩󵄩󵄩L2 . The constants α5 and α8 have been introduced in Proposition 2.5.1 and Lemma 2.5.2, respectively.

184 | 2 Some properties of global attractor Proof. Let u1 and u2 be the solutions of the GLE for which u1 (0) = u01 and u2 (0) = u02 . Since u01 , u02 ∈ 𝒜, we may assume that u1 and u2 are also defined for t < 0. By the properties of global attractor, we obtain 󵄩 1 󵄩 󵄩 󵄩󵄩 󵄩󵄩uj (t)󵄩󵄩󵄩L2 ≤ ρ1 , 󵄩󵄩󵄩A 2 uj (t)󵄩󵄩󵄩 ≤ ρ2 , 󵄩󵄩 󵄩󵄩 󵄩󵄩uj (t)󵄩󵄩L∞ ≤ ρ3 , t ∈ ℝ, j = 1, 2. Proposition 2.5.1 implies 󵄩󵄩 21 󵄩 󵄩󵄩A uj (t)󵄩󵄩󵄩G(α5 ) ≤ 2ρ2 ,

t ∈ ℝ, j = 1, 2,

while Proposition 2.5.2 shows that 󵄩 󵄩 󵄩󵄩 21 󵄩 󵄩󵄩A (u1 (t) − u2 (t))󵄩󵄩󵄩L2 ≤ α6 󵄩󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩L2 ,

t ∈ ℝ.

Hence, by Lemma 2.5.3, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩G(α5 ) ≤ α8 󵄩󵄩󵄩u1 (t) − u2 (t)󵄩󵄩󵄩L2 ,

t ∈ ℝ,

which for t = 0 gives our assertion. In order to estimate the Hausdorff length of level sets, we assume n = 2 and L = 1. Lemma 2.5.4. If ‖f ‖G(r) ≤ M‖f ‖L2 for some f ∈ H, then at least one of the following assertions is true: (i) ‖Re f ‖Γ(r) ≤ M‖Re f ‖L2 ; (ii) ‖Im f ‖Γ(r) ≤ M‖Im f ‖L2 ; if Re f = 0, then (ii) holds, and if Im f = 0, then (i) holds. Note that if f = ∑j∈ℤ2 fj e2πij⋅x , fj ∈ ℂ, then Re f = ∑

1 (f + f ∗ )e2πij⋅x , 2 j −j

Im f = ∑

1 i(−fj + f−j∗ )e2πij⋅x . 2

j∈ℤ2

j∈ℤ2

Proof. We have ‖Re f ‖2Γ(r) + ‖Im f ‖2Γ(r) =

1 ∑ e4π|j|2 r (|fj + f−j∗ |2 + |fj − f−j∗ |2 ) 4 j∈ℤ2

= ∑ |fj |2 e4π|j|2 r j∈ℤ2

2

≤ ∑ |fj |2 e2(|2πj|2 +1) j∈ℤ2

1 2

r

= ‖f ‖2G(r)

≤ M 2 ‖f ‖2L2 = M 2 (‖Re f ‖2L2 + ‖Im f ‖2L2 ), and the lemma follows easily.

2.5 Hausdorff length of level sets |

185

For any differentiable function f , denote by fω󸀠 the directional derivative of f in direction ω ∈ S1 . Lemma 2.5.5. Let v1 = Re u0 and v2 = Im u0 for some u0 ∈ 𝒜. There exists α11 = α11 (ν, μ, a) such that for every ω ∈ S1 if (u0 )󸀠ω ≠ 0, then 1

󸀠

ℋ (N((vj )ω , Ω)) ≤ α11 holds for at least one j ∈ {1, 2}.

(2.5.25)

Proof. Let ω ∈ S1 be arbitrary. The GLE is invariant with respect to translations in the space variable. Thus, Theorem 2.5.2 gives α 󵄩 1 󵄩󵄩 󵄩 󵄩 󵄩󵄩u0 (⋅ + hω) − u0 (⋅)󵄩󵄩󵄩G(α5 ) ≤ 8 󵄩󵄩󵄩u0 (⋅ + hω) − u0 (⋅)󵄩󵄩󵄩L2 h h h ∈ ℝ\{0}. As h → 0, we conclude 󵄩󵄩 󵄩 󸀠 󵄩 󸀠 󵄩 󵄩󵄩(u0 )ω 󵄩󵄩󵄩G(α5 ) ≤ α8 󵄩󵄩󵄩(u0 )ω 󵄩󵄩󵄩L2 .

(2.5.26)

If (v1 )󸀠ω ≠ 0 and (v2 )󸀠ω ≠ 0, then by Lemma 2.5.1, 󵄩 󸀠󵄩 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩(vj )ω 󵄩󵄩Γ(α5 ) ≤ α8 󵄩󵄩󵄩(vj )ω 󵄩󵄩󵄩L2 holds for at least one j ∈ {1, 2}.

(2.5.27)

For this j, Theorem 2.5.1 gives (2.5.25) with α11 = α1 (1, α8 , α5 , 2). If, on the other hand, (vk )󸀠ω = 0 for some k ∈ {1, 2}, then (2.5.26) implies (2.5.27) with j = k. Again, Theorem 2.5.1 gives (2.5.25). Theorem 2.5.3. Suppose u0 ∈ 𝒜 is a non-constant function. Then there exists α12 = α12 (ν, μ, a) such that min{ℋ1 (N(Re(u0 − λ), Ω)), ℋ1 (N(Im(u0 − λ), Ω))} ≤ α12 ,

λ ∈ ℂ.

(2.5.28)

Corollary 2.5.1. Suppose u0 ∈ 𝒜 is a non-constant function. Then either 1

λ1 ∈ ℝ

(2.5.29)

1

λ2 ∈ ℝ,

(2.5.30)

ℋ (N(Re u0 − λ1 , Ω)) ≤ α12 ,

or ℋ (N(Im u0 − λ2 , Ω)) ≤ α12 ,

or both. If Re u0 is a constant function, then we have (2.5.30), while if Im u0 is a constant function, we have (2.5.29). We prove the corollary first. Suppose the inequality in (2.5.29) fails for some λ1 ∈ ℝ. For an arbitrary λ2 ∈ ℝ, let λ = λ1 + iλ2 . Then (2.5.28) implies (2.5.30). Similarly, we can prove that if (2.5.30) is not true, then (2.5.29) holds.

186 | 2 Some properties of global attractor Now, suppose Re u0 is a constant function. Let λ = λ1 + iλ2 where λ1 = Re u0 and λ2 is arbitrary. Then (2.5.28) leads to (2.5.30). Similarly, we can prove that Im u0 being constant implies (2.5.29). Before the proof of Theorem 2.5.3, we introduce another piece of notation. Let f ∈ ̄ the number H, ω = (ω(1) , ω(2) ) ∈ S1 , ω⊥ = (−ω(2) , ω(1) ) ∈ S1 . For any t ∈ ℝ, denote by l(t) ⊥ of zeros (no multiplicity) of f |Ω on the line passing through tω with direction ω. (Note ̄ = 0 if t > √2.) Denote that l(t) ̄ l(ω, f ) = ∫ l(t)dt. This integral exists by [23]. Proof of Theorem 2.5.3. Since u0 is a non-constant function, there exist directions ω1 , ω2 , ω3 ∈ S1 such that the angle between any two of them is 2π/3 and (u0 )󸀠ωj is not the zero function for any j ∈ {1, 2, 3}. Denote u1 = Re u0 and u2 = Im u0 , and λ = λ1 + iλ2 ∈ ℂ, λ1 , λ2 ∈ ℝ. We wish to find an upper bound for m = min ℋ1 (N(uj − λj , Ω)). j=1,2

If both u1 − λ1 and u2 − λ2 are not constant zero functions, the sets N(u1 − λ1 , Ω) and N(u2 −λ2 , Ω) consist of countably many analytic curves and, possibly, of a finite number of singular points [23]. These analytic curves at each point form an angle of at least π6 with at least two of the lines with directions ω1 , ω2 and ω3 . Hence, m ≤ 2(l(ωk , u1 − λ1 ) + l(ωl , u1 − λ1 )),

(2.5.31)

m ≤ 2(l(ωk , u2 − λ2 ) + l(ωl , u2 − λ2 )),

k, l ∈ {1, 2, 3}, k ≠ l.

(2.5.32)

The last statement holds also if one of the functions u1 −λ1 or u2 −λ2 is the zero function. By (2.5.31), there exist k1 , k2 ∈ {1, 2, 3} with k1 ≠ k2 such that l(ωkj , u1 − λ1 ) ≥

m , 4

j = 1, 2,

by (2.5.32), there are k3 , k4 ∈ {1, 2, 3}, k3 ≠ k4 for which l(ωkj , u2 − λ2 ) ≥

m , 4

j = 3, 4.

m , 4

j = 1, 2.

Then, taking k ∈ {k1 , k2 } ∩ {k3 , k4 }, we get l(ωk , uj − λj ) ≥

By Rolle theorem from elementary calculus, we arrive at l(ωk , (uj )󸀠ωk ) ≥

m √ − 2, 4

j = 1, 2,

2.5 Hausdorff length of level sets | 187

hence, by [23], 1

󸀠

ℋ ((uj )ωk , Ω) ≥

m √ − 2, 4

j = 1, 2.

Lemma 2.5.5 then finally gives m ≤ 4(α11 + √2) = α12 . Regarding the nodal sets of solutions which do not belong to the global attractor, we have the following statement: Theorem 2.5.4. Let u1 and u2 be solutions of the GLE, and let v = u1 − u2 . There exists α13 = α13 (ν, μ, a) such that at least one of the following possibilities occurs: (i) limt→∞ ‖v(t)‖L2 = 0; (ii) there exists t5 = t5 (ν, μ, a, u1 , u2 ) such that min{ℋ1 (N(Re v(t), Ω)), ℋ1 (N(Im v(t), Ω))} ≤ α13 ,

t ≥ t5 .

(2.5.33)

Proof. Proposition 2.5.2 implies the inequalities in (2.5.20)–(2.5.21), for t ≥ t1 + t2 . Assume that (i) does not hold. Then Proposition 2.5.1 implies t3 < ∞ and the inequality in (2.5.22), for t ≥ t1 + t2 + t3 . Theorem 2.5.1, Lemma 2.5.1 and Lemma 2.5.3, then give (2.5.33) for a suitable α13 . Remark 1. If we discuss the purely real case ν = μ = 0 of the GLE, the initial datum u0 is also real valued. In that case, we have the following consequence of Theorem 2.5.3. If v0 ∈ 𝒜 is a non-constant function, then ℋ1 (N(v0 − λ, Ω)) ≤ α12 , λ ∈ ℝ. Remark 2. We proceed differently when extending the conclusion of Theorem 2.5.4 to arbitrary solutions. For every n ∈ ℕ, we construct a solution un of the GLE such that 1

ℋ (N(un (⋅, ⋅, t), Ω)) = n + 1,

t ∈ [0, ∞).

(2.5.34)

Consider 󵄨 󵄨2 ϕ󸀠 (t) + (2πn)2 (1 + iν)ϕ(t) + (1 + iμ)󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 ϕ(t) − aϕ(t) = 0 with the initial condition ϕ(0) = ϕ0 , ϕ0 ∈ ℂ, φ0 ≠ 0. Since 1 d 󵄨󵄨 󵄨2 󵄨2 󵄨 󵄨4 󵄨 󵄨2 2󵄨 󵄨󵄨ϕ(t)󵄨󵄨󵄨 + (2πn) 󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 + 󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 − a󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 = 0, 2 dt there exists a solution ϕ : [0, ∞) → ℂ\{0} of the above initial value problem. It is easy to check that u(x, y, t) = ϕ(t)e2πinx , is a solution of the GLE which satisfies (2.5.34).

x, y ∈ ℝ, t ≥ 0

188 | 2 Some properties of global attractor

2.6 The structure of the global attractor and lower bound estimate of its dimension Let E be a Banach space with norm ‖ ⋅ ‖ and consider a continuous semigroup operator S(t) : E → E, S(t + s) = S(t)S(s),

∀s, t ≥ 0,

(2.6.1)

S(0) = I.

(2.6.2)

The mapping (t, u0 ) → S(t)u0 is continuous from ℝ+ × E to E, z is the fixed point of S(t) satisfying s(t)z = z,

(2.6.3)

∀t ∈ ℝ+ .

Let the mapping u → S(t)u be Fréchet differentiable for all t ∈ ℝ+ on the neighborhood O of z, and it satisfies Hölder condition: 󵄩󵄩 󸀠 󵄩 󸀠 󵄩󵄩S (t)u1 − S (t)u2 󵄩󵄩󵄩 ≤ C3 (T)‖u1 − u2 ‖σ , 0 < α ≤ 1, ∀u1 , u2 ∈ O, ∀t ∈ [0, T],

(2.6.4)

where constant C3 depends on T, and does not depend on u1 and u2 . Definition 2.6.1. Let z be a fixed point of S(z), S(z) = z. Then z is called the hyperbolic fixed point of S(z) if it satisfies the following two conditions: (i) The spectrum σ(S󸀠 (z)) of S󸀠 (z) and circle {λ ∈ ℂ, |λ| = 1} is disjoint; (ii) E+ is finite-dimensional, where E+ = E+ (z), E− = E− (z) are the linear subspaces of E corresponding to subsets of S󸀠 (z) included in {λ ∈ ℂ, |λ| > 1} and {λ ∈ ℂ, |λ| < 1}, respectively. Definition 2.6.2. We say that z is the hyperbolic fixed point of S(t), t ∈ ℝ+ , if it satisfies the following two conditions: (i) By Definition 2.6.1, z is the hyperbolic fixed point of S(t), ∀t > 0; (ii) The linear invariant subspaces E+ and E− corresponding to operator S󸀠 (t)(z) do not depend on t. The stable and unstable manifolds at z are defined as follows: μ− (z) = {u0 ∈ E, ∀t ≤ 0, ∃u(t) ∈ S(−t)−1 u0 S(t)u0 → z, t → +∞},

μ+ (z) = {u0 ∈ E, ∀t ≤ 0, ∃u(t) ∈ S(−t)−1 u0 u(t) → z, t → −∞}.

Obviously, S(t)μ+ (z) = μ+ (z),

S(t)μ− (z) = μ− (z),

∀t ≥ 0.

(2.6.5)

2.6 The structure of the global attractor and lower bound estimate of its dimension

| 189

Set S1 = S(1). Then W+ (z) denotes the unstable manifold of hyperbolic fixed point of mapping S1 . Similar to S1 , we can define W+R (z), W−R (z) as follows: W+R (z) = {u0 ∈ OR (z), ∀n ∈ ℕ, ∃un ∈ OR (z), { { { S1n un = s(n)un = u0 , un → z, n → ∞}, { { { n R { W− (z) = {u0 ∈ OR (z), S1 u0 → z, n → ∞}.

(2.6.6)

For the structure of W+R (z) near z, we have the following lemma: Lemma 2.6.1. Let R be sufficiently small. Then there exist mappings g+ , g− satisfying g+ : OR (z) ∩ E+ (z) → E− (z), g− : OR (z) ∩ E− (z) → E+ (z),

g+ (z) = 0, g− (z) = 0,

such that the sets W+R (z), W−R (z) can be represented as R

W+ (z) = {u ∈ E, u = u+ + g+ (u+ ), { { { { { u+ ∈ OR (z) ∩ E+ (z)}, { { R { { W− (z) = {u ∈ E, u = u− + g− (u− ), { { { u− ∈ OR (z) ∩ E− (z)}. {

(2.6.7)

The mappings g+ and g− are Fréchet differentiable, their differentials satisfy Hölder conditions with same index α, and g−󸀠 (z) = g+󸀠 (z) = 0. Now assume that semigroup S(t) has attractor 𝒜. Then we have the following theorem: Theorem 2.6.1. Let E be a Banach space, and assume S(t) is a semigroup operator for t ∈ ℝ+ satisfying conditions (2.6.1)–(2.6.4). Suppose that S(t) has a global attractor 𝒜, and z is the hyperbolic fixed point of S(t). Then R

𝒜 ⊃ μ+ (z) ⊃ W+ (z),

(2.6.8)

where R > 0 is sufficiently small, W+R (z) can be represented as (2.6.6). In particular W+R (z) is ε1,α manifold with the same dimension as E+ (z). Definition 2.6.3. The Lyapunov functional of semigroup S(t) on a set ℱ ⊂ E is a continuous function F: ℱ → ℝ, which satisfies (i) For any u0 ∈ ℱ , function z → F(S(t)u0 ) is decreasing; (ii) If for a τ > 0, F(s(τ)u1 ) = F(u1 ), then u1 is a fixed point of semigroup S(t). Theorem 2.6.2. Assume that the given semigroup S(t) has properties (2.6.1)–(2.6.2). Suppose S(t) has Lyapunov functional F, and F is continuous in ℱ ⊂ E. The global attractor is 𝒜 ⊂ ℱ . Let ε denote the fixed point set of the semigroup. Then 𝒜 = μ+ (ε).

(2.6.9)

190 | 2 Some properties of global attractor Furthermore, if ε is discrete, then 𝒜 is the union of heteroclinic orbits from a point in ε to another one, namely 𝒜 = ⋃ μ+ (z).

(2.6.10)

z∈ε

Proof. Since ε ∈ 𝒜, it is easy to verify μ+ (ε) ⊂ μ+ (𝒜), this implies μ+ (𝒜) = 𝒜. So μ+ (ε) ⊂ 𝒜. Now we prove that 𝒜 ⊂ μ+ (ε) holds. Let u0 ∈ 𝒜, then u0 belongs to the complete orbit {u(t), t ∈ ℝ, u0 = u(0)} contained in 𝒜. Since 𝒜 is compact, then by continuity of S(t), we can deduce that the set γ = ⋂ {u(t), t < s} s s}. s>0

γ 󸀠 is ω-limit set ω(u0 ) of u0 . It implies that ω(u0 ) ⊂ ε is an accumulation point. Example. Consider 𝜕u − dΔu + g(u) = 0, 𝜕t u = 0, 𝜕Ω × ℝ+ .

Ω × ℝ+ ,

(2.6.13) (2.6.14)

2.6 The structure of the global attractor and lower bound estimate of its dimension

| 191

Let G be the original function of g given by 2p−1

G(s) = ∑ j=1

bj

sj+1 .

j+1

Let F(u) =

d 2 ‖u‖ + G(u), 2

then d F(u(t)) = d(u(t), u󸀠 (t)) + ∫ g(u(t))u󸀠 (t)dx dt 󵄨󵄨 du 󵄨󵄨2 󵄨 󵄨 = −󵄨󵄨󵄨 (t)󵄨󵄨󵄨 ≤ 0. 󵄨󵄨 dt 󵄨󵄨

Ω

So the Lyapunov functional of problem (2.6.13)–(2.6.14) exists, and is continuous in H01 (Ω) ∩ L2p (Ω). We choose ℱ as a bounded set of H 2 (Ω) ∩ L∞ (Ω) included in 𝒜, F is continuous on ℱ . By Theorem 2.6.1, we may estimate a lower bound of the attractor. In fact, under the assumption of Theorem 2.6.1, z is the hyperbolic fixed point of semigroup. By (2.6.8), R

𝒜 ⊃ W+ (z),

where W+R (z) is the manifold of ε1,α with dimension n = dim E+ (z). Hence, dim 𝒜 ≥ n, where dim 𝒜 is Hausdorff or fractal dimension of 𝒜. For example, we consider the problem (2.6.13)–(2.6.14), then z = 0 is a steady state solution. For g 󸀠 (0) = b1 , the linearized equation of (2.6.13)–(2.6.14) is 𝜕v − dΔv + b1 v = 0, 𝜕t v(0) = ξ ,

(2.6.15) (2.6.16)

namely, L(t, 0) ⋅ ξ = v(t), L(t, 0) = S󸀠 (t)(0). Let λh , h ∈ ℕ represent the eigenvalues of the following Dirichlet problem: −Δωh = λh ωh ,

{

ωh ∈ H01 (Ω),

0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅, λh → ∞, h → ∞.

(2.6.17)

192 | 2 Some properties of global attractor Similarly, μh , h ∈ ℕ represent the eigenvalue sequence of the linearized operator − dΔϕn + b1 ϕh = μh ϕh ,

ϕh ∈ H01 (Ω).

(2.6.18)

∀h.

(2.6.19)

Comparing ϕh = ωh , we have μh = b1 + dλh ,

The spectrum of S󸀠 (t) consists of its eigenvalues and 0, the eigenvalue of S󸀠 (t) is the number of exp{−μh t}, the unstable one corresponds to μh < 0, the stable one corresponds to μh > 0. If μh ≠ 0, ∀h, then the system is hyperbolic. Assume that −

b1 ∉ ⋃λ d j∈ℕ j

(2.6.20)

holds, the number n of eigenvalues of S󸀠 (t) is equal to k in {|λ| < 1}, where λh < −

b1 , d

(2.6.21)

which implies dim 𝒜 ≥ n. Certainly, if b1 < 0 and b is sufficiently large, then n is sufficiently large.

(2.6.22)

3 Structures of small dissipative dynamical systems For near-integral systems it is a very interesting problem how to apply the theory of finite-dimensional systems, such as Melnikov method, center manifold and normal hyperbolicity theory, to infinite-dimensional systems. Applying these theories to infinite-dimensional systems, we can obtain more detailed geometric structures. Mclaughlin, Li, Kovancic, Wiggins and others [74, 69] have made an in-depth study on a perturbed nonlinear Schrödinger equation and a damped Sine–Gordon equation. In the small perturbation case, the problems of, for example, how the structure of a steady state solution will evolve with time, and what the structure of perturbed dynamical systems is, are worth studying. Recently, Doelman, Keefe, and Kapitula et al. [22, 68, 92] have studied the Ginzburg–Landau equation. In [45, 44], and [63], we have considered dynamical systems of the cubic–quintic Ginzburg–Landau equation and the derivative Ginzburg–Landau equation. It is worth mentioning that structures of the augmented unstable bundle are relative to the first Chern number, see [1]. Meanwhile, we introduced the Evans function into the stability problem of the quintic Ginzburg–Landau equation, see [63].

3.1 Quintic Ginzburg–Landau equation Consider the following Ginzburg–Landau equation [45, 44]: Wt = c0 W + (c0 + iεc1 )Wxx − (

c0 c + iεc2 )|W|2 W − ( 0 + iεc3 )|W|4 W, 2 2

(3.1.1)

where W(x, t) is a complex function of the space and time (x, t) ∈ ℝ × ℝ, c0 , c1 , c2 and c3 are real constants, and ε is a small parameter. When c0 = 0, equation (3.1.1) is a general Hamiltonian nonlinear Schrödinger equation: iWt + εc1 Wxx − εc2 |W|2 W − εc3 |W|4 W = 0.

(3.1.2)

We will search for the following type of periodic solutions of (3.1.1): W(x, t) = Rei(kx−ωt) .

(3.1.3)

Substituting (3.1.3) into (3.1.1) and rewriting the result into the real and imaginary parts, we obtain R4 + R2 = 2(1 − k 2 ),

{

ω = ε(c1 k 2 + c2 R2 + c3 R4 ).

(3.1.4)

Then for ε = 0, this type of the solution is stationary, and for ε ≠ 0, we know that ω = O(ε). The solutions of (3.1.1) are assumed to be of the following form: W(x, t) = ρ(x)ei[θ(x)−εωt] . https://doi.org/10.1515/9783110587265-003

(3.1.5)

194 | 3 Structures of small dissipative dynamical systems Substituting (3.1.5) into (3.1.1) and rewriting the result into the real and imaginary parts gives 2 3 5 {−εωρ = c0 (2ρx θx + ρθxx ) + εc1 (ρxx − ρθx ) − εc2 ρ − εc3 ρ , c c { c ρ + c0 (ρxx − ρθx2 ) − εc1 (2ρx θx + ρθxx ) − 0 ρ3 − 0 ρ5 = 0. { 0 2 2

(3.1.6)

By (3.1.6), we obtain εc ρ c c { { 2ρx θx + ρθxx = 2 0 2 2 [(c1 − ω) + (c2 − 1 )ρ2 + ρ4 (c3 − 1 )], { { 2 2 c0 + ε c1 { (3.1.7) { { c02 c02 ρ { 2 2 2 2 2 4 { {ρxx − ρθx2 = − 2 [(c + ε c ω) − ( + ε c c )ρ − ( + ε c c )ρ ]. 1 1 2 1 3 0 2 2 c0 + ε2 c12 { Letting ε = 0 in (3.1.7) yields an integrable system 2ρx θx + ρθxx = 0,

1 1 ρxx − ρ2 θx2 = −ρ + ρ3 + ρ5 , 2 2

(3.1.8) (3.1.9)

with integrals ρ2 θx = Ω, ρ2x + ρ2 +

(3.1.10) 4

6

ρ Ω2 ρ − − = K, 4 6 ρ2

(3.1.11)

where Ω and K are integral constants. In the following we shall consider Ω = Ω(x) as a slow variable of the perturbed system. Setting v = ρx , from (3.1.8) and (3.1.9), we get ρx = v, vx =

(3.1.12) 2

Ω 1 1 − ρ + ρ3 + ρ5 2 2 ρ3 +

ε2 c1 ρ c c [(c1 − ω) + (c2 − 1 )ρ2 + (c3 − 1 )ρ4 ], 2 2 + ε2 c12

c02

Ωx = ρ2

εc0 c c [(c1 − ω) + (c2 − 1 )ρ2 + (c3 − 1 )ρ4 ]. 2 2 c02 + ε2 c12

(3.1.13) (3.1.14)

First we discuss the equilibria of system (3.1.12)–(3.1.14) and, moreover, study the properties of the equilibria. Let ε = 0 in (3.1.12)–(3.1.14). Then ρx = v, { { { { { Ω2 1 1 v = − ρ + ρ3 + ρ5 , x { 3 { 2 2 ρ { { { Ω = 0. { x

(3.1.15)

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| 195

Also let g(p) =

1 4 1 3 p + p − p2 + Ω20 , 4 2

(3.1.16)

where Ω0 is a real constant and p = ρ2 . It is clear that the roots of (3.1.16) are the equilibria of (3.1.15) except for p = 0 and Ω0 = 0. The graph of the function g(p) is shown in Figure 3.1 for different constants Ω0 .

Figure 3.1: The relation between p and g(p). √ √73+3 )(p − 73−3 ). Since p ≥ 0, g(p) has the minimum g(p0 ) = 8 8 √73−3 ̃ 0 , and Ω ̃ 0 = √ 827−73√73 , p1 , p2 are the , for 0 < |Ω0 | < Ω 8 1024

Obviously, g(p) = 2p(p + 73√73−827 1024

+ Ω20 at p0 = two positive roots of g(p) in the plane Ω = Ω0 . For Ω0 = 0, p1 = 0, p2 = 1 are the two ̃ 0 , p = p0 is the only root of g(p). Then we have nonnegative roots of g(p). For Ω = Ω Lemma 3.1.1. The unperturbed system (3.1.15) has (i) three equilibria (±1, 0, 0) and (0, 0, 0) for Ω0 = 0; ̃ 0 ) for Ω0 = Ω ̃0; (ii) one pair of equilibria (±√p0 , 0, ±Ω ̃ 0,j ) for 0 < |Ω0 | < Ω ̃ 0,j , j = 1, 2; (iii) two pairs of equilibria (±√pj , 0, ±Ω ̃ (iv) no equilibrium for Ω0 > Ω0 . Moreover, 0 < p1 < p0 < p2 < 1, and p1 (Ω0 ) is an increasing function with Ω0 and p2 (Ω0 ) is a decreasing function with Ω0 , and they satisfy lim p1 (Ω0 ) =

̃0 |Ω0 |→Ω

lim p2 (Ω0 ) = p0 ;

̃0 |Ω0 |→Ω

lim p1 (Ω0 ) = 0; lim p2 (Ω0 ) = 1.

|Ω0 |→0

|Ω0 |→0

196 | 3 Structures of small dissipative dynamical systems The eigenvalue equation of (3.1.15) at the equilibria (±p∗j , 0, ±Ω∗j ) is given by λ2 −

1 5 4 3 3 ( p + p − p2j − 3Ω20 ) = 0. pj 2 j 2 j

(3.1.17)

Note that q(pj ) =

√73 + 3 5 4 3 3 pj + pj − p2j − 3Ω20 = 4p2j (pj − p0 )(pj + ). 2 2 8

When pj < p0 , we have q(pj ) < 0, which implies that λ1,2 (pj ) are pure imaginary roots. Therefore pj is a stable center fixed point. On the other hand, if pj > p0 , q(pj ) > 0, then pj is an unstable saddle fixed point, and when pj = p0 , q(pj ) > 0. By the analysis for (3.1.17), we obtain the following result: Lemma 3.1.2. The properties of the critical points of the unperturbed system (3.1.15) are determined as follows: ̃ 0 , the critical points (±√p1 , 0, ±√|Ω0 |) are stable centers and the (i) when 0 < |Ω0 | < Ω critical points (±√p2 , 0, ±√|Ω0 |) are unstable saddles; (ii) when Ω0 = 0, the critical point (1, 0, 0) is an unstable saddle and (0, 0, 0) is a center; ̃ 0 , the critical points (±√P0 , 0, ±Ω ̃ 0 ) are non-hyperbolic. (iii) when Ω0 = Ω We define K(ρ, v) as follows: K(ρ, v) = v2 +

Ω2 1 1 + ρ2 − ρ4 − ρ6 , 4 6 ρ2

the relation of K(ρ, 0) and the phase plane of (3.1.15) is shown in Figure 3.2.

Figure 3.2: Phase portrait of equation (3.1.15).

(3.1.18)

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̃ 0 by numerical simulation; (b) Phase portrait of Figure 3.3: (a) Phase portrait for 0 < |Ω0 | < Ω equation (3.1.15).

By the integral values of (3.1.18) and Lemma 3.1.2, we get the phase planes of (3.1.15) as follows: Lemma 3.1.3. (i) There are two families of periodic orbits and two homoclinic orbits ̃ 0 , see Figures 3.2(a) and 3.3(a). which are symmetric about v-axis for 0 < |Ω0 | < Ω (ii) There are a one-parameter family of periodic orbits and two heteroclinic orbits for Ω0 = 0, see Figures 3.2(b) and 3.3(b). (iii) There are two families of periodic orbits which are symmetric about v-axis for Ω0 = ̃ 0 , see Figure 3.2(c). Numerical simulation of (3.1.15) exhibits the phase portraits Ω ̃ 0 and for Ω0 = 0 which are shown in Figures 3.2(a) and 3.3(b), for 0 < |Ω0 | < Ω respectively. Set Ki (Ω0 ) = K(ρi (Ω0 ), 0) to be the values at the critical points ρi (Ω0 ), i = 1, 2. The periodic solutions in the Ω0 plane correspond to the values of K between K1 (Ω0 ) and K2 (Ω0 ) (K1 < K2 for all Ω0 ); these periodic solutions correspond to the solutions W(x, t), quasiperiodic in x, slowly periodic in t. Solutions with the values of K ⊈ [K1 , K2 ] are unbounded. The integral values (K, Ω) of the periodic solutions of (3.1.1) form a bounded region E in (K, Ω) space, symmetric with respect to K axis. The region E, describing the relation between K and Ω, is shown in Figure 3.4. Set 𝜕E be the boundary of E which consists of points corresponding to the critical points ρi (Ω), i = 1, 2: 1 1 𝜕Ei = {(K, Ω) : Ω2 = y2 − y3 − y4 , 2 2 3 2 2 3 K = 2y − y − y , y = ρ2i , i = 1, 2}. 4 3 Region E in (K, Ω) space corresponds to a volume S in (ρ, v, Ω) space, whose boundary 𝜕S is given by the family of homoclinic loops. Inside S all solutions are periodic.

198 | 3 Structures of small dissipative dynamical systems

Figure 3.4: Region E showing relation between K and Ω.

We consider the perturbed system (3.1.12)–(3.1.14), and then have Lemma 3.1.4. (i) The flow induced by (3.1.12)–(3.1.14) is invariant under symmetry transformations: x → −x, v → −v, Ω → −Ω and ρ → −ρ. (ii) If p and Ω are the critical points of (3.1.12)–(3.1.14), then P satisfies ap2 + bp + c = 0, 1 1 p2 − p3 − p4 = Ω2 , 2 2

(3.1.19) (3.1.20)

where a = c3 − c21 , b = c2 − c21 and c = c1 − ω, p = ρ2 ; p and Ω are also the critical points of unperturbed system, if p∗ is a critical point, then 0 < p∗ < 1; the position of a critical point is independent of c0 and ε. Lemma 3.1.5. A non-negative real number p∗ is a solution of (3.1.20) if and only if 1 0 ≤ Ω2 = − p2∗ (p∗ − 1)(p∗ + 2) 2

⇐⇒

0 ≤ p∗ ≤ 1.

For a ≠ 0, we rewrite (3.1.19)–(3.1.20) as follows: p2 + rp + s = 0, 1 1 p2 − p3 − p4 = Ω2 , 2 2

(3.1.21) (3.1.22)

where r = ba and s = ac . Solving for P and Ω from (3.1.21)–(3.1.22), we get the following relation between roots and coefficients: 2

Lemma 3.1.6. (i) In case s = r4 and 0 > r ≥ −2, there exists one multiple positive root p0 = |r| for (3.1.21) (see segment AO in Figure 3.5). 2 2

(ii) In case 0 < s < r4 , r + s > −1 and r < 0, there exist two positive roots 0 < p1 < p2 < 1 for (3.1.21) (see region (I) in Figure 3.5). (iii) In case s = 0 and −1 < r < 0, there exist one zero root p1 = 0 and one positive root p2 = |r| for (3.1.21) (see segment OC in Figure 3.5).

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Figure 3.5: The relation between r and s.

(iv) In case s < 0 and r + s > −1, there exists one positive root 0 < p1 < 1 for (3.1.21) (see region (II) in Figure 3.5). (v) In case r + s = −1 and −2 < r, there exists one positive root for (3.1.21) (see segment AB in Figure 3.5). (vi) In case r = 0 and s = 0, there exists only one zero root for (3.1.21). (vii) In case a = 0 and 0 < − bc < 1, there exists only one positive root p = − bc < 1 for (3.1.21). By Lemma 3.1.6, we have Theorem 3.1.1. Suppose that f (p) = p2 − 21 p3 − 21 p4 . Then (i)

2

In case s = r4 and 0 > r ≥ −2, if either ω = c1 − Δ < c2 + c3 and 2c2 < c1 < 2c3 or ω = c1 + Δ > c2 + c3 and 2c3 < c1 < 2c2 , then there exists one multiple positive c1 −2c2 root 0 < p0 = 2(2c < 1 for (3.1.21) and (3.1.22), and there exist four equilibria: −c ) 3

1

(ρ0 , 0, ±Ω0 ) and (−ρ0 , 0, ±Ω0 ) for (3.1.12)–(3.1.14) where Δ =

c1 2 ) 2 c 4(c3 − 21 )

(c2 −

, ±ρ0 = ±√p0

and ±Ω0 = ±√f (p0 ). 2 (ii) In case 0 < s < r4 , r + s > −1 and r < 0, if either c1 − Δ < ω < min{c1 , c2 + c3 } and 2c2 < c1 < 2c3 or max{c1 , c2 + c3 } < ω < c1 + Δ and 2c3 < c1 < 2c2 , then there exist two positive roots 0 < p1 < p2 < 1 for (3.1.21) and there exist eight equilibria: (ρ1 , 0, ±Ω1 ), (−ρ1 , 0, ±Ω1 ), (ρ2 , 0, ±Ω2 ), (−ρ2 , 0, ±Ω2 ) for (3.1.12)–(3.1.14), where ±ρi = ±√pi , i = 1, 2, Ω2 = Ω2i = (−Ω2i ), i = 1, 2. (iii) In case s = 0 and −1 < r < 0, if either 2c2 < ω = c1 < min{c2 + c3 , 2c3 } or max{c2 + c3 , 2c3 } < ω = c1 < 2c2 , then there exist one zero root p = 0 and one positive root p1 = (3.1.19)–(3.1.20) and there exist five equilibria: (0, 0, 0), (ρ1 , 0, ±Ω1 ), (−ρ1 , 0, ±Ω1 ) for (3.1.12)–(3.1.14), where ±ρ1 = ±√p1 , ±Ω1 = ±√f (p1 ).

c1 −2c2 2c3 −c1

< 1 for

200 | 3 Structures of small dissipative dynamical systems (iv) In case s < 0 and r + s > −1, if either ω ≤ c2 + c3 and c1 < min{2c3 , ω} or ω ≥ c2 + c3 ,

c1 > max{2c3 , ω},

then there exists one positive root 0 < p < 1 for (3.1.21) and there exist four equilibria: (ρ1 , 0, ±Ω1 ), (−ρ1 , 0, ±Ω1 ) for (3.1.12)–(3.1.14), where ±ρ1 = ±√p1 , ±Ω1 = ±√f (p1 ). (v) In case r + s = −1 and −2 < r, if either c1 − Δ < ω = c2 + c3 < c1 < 2c3 or 2c3 < c1 < ω = c2 + c3 < c1 + Δ, then there exists one positive root p = 1 for (3.1.12)–(3.1.14), thus there exist two equilibria: (1, 0, 0) and (−1, 0, 0) for (3.1.12)–(3.1.14). (vi) In case r = 0 and s = 0, for c1 ≠ 2c3 and ω = c1 = 2c2 , there exists only one equilibrium (0, 0, 0) for (3.1.12)–(3.1.14). (vii) In case a = 0 and 0 < − bc < 1, there exists only one positive root p = − bc < 1 for (3.1.19)–(3.1.20), thus there exist two equilibria (±√− bc , 0, ±Ω) for (3.1.12)–(3.1.14).

When the coefficients a = c3 − c21 ≠ 0, the numbers of the equilibria of system (3.1.12)–(3.1.14) depend upon r and s. Figure 3.5 shows the relation between r and s. In the regions (I) and (II) there exist two positive roots and one positive root of (3.1.19), respectively, expect for one curve. On that curve, there exists one positive root, and at the origin, point O(0, 0), there is only zero (0, 0, 0) root. Figure 3.6 shows the eight equilibria, they are symmetric about Ω and ρ axes in the plane v = 0. We discuss the properties of equilibria for the perturbed system (3.1.12)–(3.1.14), the eigenvalue equation is given by λ3 − Aj λ − Bj Cj = 0,

Figure 3.6: The portrait of eight equilibria.

(3.1.23)

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| 201

where Aj = −3 Bj =

Ω2j ρ4j

2Ωj ρ3j

,

ε2 c1 3 5 − 1 + ρ2j + ρ4j + 2 [c + 3bρ2j + 5aρ4j ], 2 2 c0 + ε2 c12 Cj =

εc0 ρj

c02

+ ε2 c12

[2c + 4bρ2j + 6aρ4j ],

j = 1, 2.

Because ap2j + bpj + c = 0,

1 1 p2j − p3j − p4j = Ω2j , 2 2

we have Aj = 4(pj − p0 )(pj + Bj Cj =

4Ωj εc0

c02 + ε2 c12

2ε2 c1 pj √73 + 3 (b + 2apj ), )+ 2 8 c0 + ε2 c12

(b + 2apj ).

For convenience, we assume that c1 > 0 and εc0 > 0. Case I. Two roots of equations (3.1.19) and (3.1.20). Assume that (3.1.19) and (3.1.20) have two positive roots, then it is clear that p1 < b − 2a < p2 , i. e., 2ap1 + b < 0 and 2ap2 + b > 0. If Ωj > 0, then B1 C1 < 0,

B2 C2 > 0,

A1 < 0,

A2 > 0.

(3.1.24)

First we discuss the stability for the smaller positive root p1 . We assume that Ωj > 0. Set λ11 , λ21 and λ31 to be the three roots of (3.1.23). Then by the relation between the roots and the coefficients, we get λ11 + λ21 + λ31 = 0,

λ11 λ21 + λ11 λ31 + λ21 λ31 = λ11 λ21 λ31 = B1 C1 < 0.

(3.1.25) −A1 > 0,

(3.1.26) (3.1.27)

Equation (3.1.27) implies that there exists one negative root, say λ31 < 0. Let λ11 = α + iβ λ1

and λ21 = α − iβ. Then substituting λj1 into equation (3.1.25), we have α = − 23 > 0. Therefore Re λ11 = Re λ21 > 0,

λ31 < 0.

Next we analyze the stability for the bigger positive root p2 . Assume that f2 (λ) = λ − A2 λ − B2 C2 . Then by f2󸀠 (λ) = 3λ2 − A2 , f2 (λ) has two stationary points ±√ A32 . 3

202 | 3 Structures of small dissipative dynamical systems Since B2 C2 = ϑ(ε), A A 2 { { f2 (−√ 2 ) = A2 √ 2 − B2 C2 > 0, { { { 3 3 3 { { { { {f (√ A2 ) = −( 2 A √ A2 + B C ) < 0. 2 2 2 3 3 2 3 { Because f2 (0) = −B2 C2 < 0, λ12 < 0, λ22 < 0 and λ32 > 0 (λ12 < −√ A32 < λ22 , λ32 > √ A32 ). By the above analysis we get the following theorem. Theorem 3.1.2. Assume that (3.1.21), (3.1.22) or (3.1.12)–(3.1.14) have two different types of the equilibrium. If εc0 > 0, Ωj > 0 and c1 > 0, then p1 is a saddle-focus and p2 is a saddle point. Moreover, (i) for the equilibria p11 (ρ1 , 0, Ω1 ) and p21 (−ρ1 , 0, Ω1 ), we have s dim Wloc ((±ρ1 , 0, Ω1 )) = 1,

u dim Wloc ((±ρ1 , 0, Ω1 )) = 2 (see Figure 3.7(a));

(ii) for the equilibria p31 (ρ1 , 0, −Ω1 ) and p41 (−ρ1 , 0, −Ω1 ), we have s dim Wloc ((±ρ1 , 0, −Ω1 )) = 2,

u dim Wloc ((±ρ1 , 0, −Ω1 )) = 1 (see Figure 3.7(b));

Figure 3.7: (a) p21 (−ρ1 , 0, 0) or p11 (ρ1 , 0, Ω1 ) (W s = 1, W u = 2); (b) p41 (−ρ1 , 0, −Ω1 ) or p31 (ρ1 , 0, −Ω1 ) (W s = 2, W u = 1); (c) p22 (−ρ2 , 0, −Ω2 ) or p22 (ρ2 , 0, Ω2 ) (W s = 2, W u = 1); (d) p42 (−ρ2 , 0, −Ω2 ) or p32 (ρ2 , 0, −Ω2 ) (W s = 2, W u = 1).

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| 203

(iii) for the equilibria p12 (ρ2 , 0, Ω2 ) and p22 (−ρ2 , 0, Ω2 ), we have s dim Wloc ((±ρ2 , 0, Ω2 )) = 2,

u dim Wloc ((±ρ2 , 0, Ω2 )) = 1

(see Figure 3.7(c));

(iv) for the equilibria p32 (ρ2 , 0, −Ω2 ) and p42 (−ρ2 , 0, −Ω2 ), we have s dim Wloc ((±ρ2 , 0, −Ω2 )) = 1,

u dim Wloc ((±ρ2 , 0, −Ω2 )) = 2 (see Figure 3.7(d)), s u where dim Wloc ((±ρj , 0, −Ωj )) and dim Wloc ((±ρ2 , 0, −Ω2 )) are the stable and unstable manifolds of the equilibria (±ρj , 0, Ωj ), j = 1, 2, respectively.

Case II. One root of equations (3.1.19) and (3.1.20). 1 is a positive root of (3.1.19) and (3.1.20). By equation If a = 0, then p∗ = cω−c c − 1 2

(3.1.23), we have

2

f (λ) = λ3 − Aλ − BC = 0, where A = 4(p∗ − p0 )(p∗ + BC =

4Ω∗ εc0 b , c02 + ε2 c12

√73 + 3 2ε2 c b ) + 2 12 2 p∗ , 8 c0 + ε c1

p0 =

√73 − 3 . 8

(i) Assume that Ω∗ εc0 b > 0. If c0 , b are given, then the sign of Ω∗ εc0 b is determined by the sign of Ω∗ . Moreover, if 0 < p∗ < p0 , then A < 0 + O(ε2 ), BC > 0 + O(ε), and Re λ2 = Re λ3 = −λ1 < 0.

(3.1.28)

If p0 < p∗ ≤ 1, then A > 0 + O(ε2 ), BC > 0 + O(ε), and λ1 < 0,

λ2 < 0,

λ3 > 0.

(3.1.29)

(ii) Assume that Ω∗ εc0 b > 0. If 0 < p∗ < p0 , then A < 0 + O(ε2 ), BC < 0 + O(ε) and Re λ2 = Re λ3 = −λ1 > 0.

(3.1.30)

If p0 < p∗ ≤ 1, then A > 0 + O(ε2 ), BC < 0 + O(ε), and λ1 < 0,

λ2 > 0,

λ3 > 0.

(3.1.31)

204 | 3 Structures of small dissipative dynamical systems We have the following results: Theorem 3.1.3. If a = 0 (c1 = 2c3 ) and εc0 b > 0, then there exist four equilibria: (±ρ∗ , 0, Ω∗ ), (±ρ∗ , 0, −Ω∗ ) for (3.1.12)–(3.1.14). Moreover, (i) if (±ρ∗ )2 = p∗ < p0 , then the equilibria are saddle-focus and s dim Wloc ((±ρ∗ , 0, Ω∗ )) = 2,

u dim Wloc Wloc ((±ρ∗ , 0, Ω∗ )) = 1 s dim Wloc ((±ρ∗ , 0, −Ω∗ )) u dim Wloc ((±ρ∗ , 0, −Ω∗ ))

(similar to Figure 3.7(b));

= 1, =2

(similar to Figure 3.7(a));

(ii) if P0 < (±ρ∗ )2 = P∗ ≤ 1, then the equilibria are saddles and s dim Wloc ((±ρ∗ , 0, Ω∗ )) = 2, u dim Wloc ((±ρ∗ , 0, Ω∗ )) = 1

s dim Wloc ((±ρ∗ , 0, −Ω∗ )) u dim Wloc ((±ρ∗ , 0, −Ω∗ ))

(similar to Figure 3.7(c));

= 1, = 2 (similar to Figure 3.7(d)).

When a ≠ 0, we have the following theorem: Theorem 3.1.4. Assume that (3.1.21), (3.1.22) have one root. If c1 ≠ 2c2 (a ≠ 0), εc0 > 0 and c1 > 0, then there are four equilibria for (3.1.21), (3.1.22) or (3.1.12)–(3.1.14). Moreover, (i) if (±ρ∗ )2 = p∗ < p0 , then the equilibria are saddle-focus and s dim Wloc ((±ρ∗ , 0, Ω∗ )) = 1,

u dim Wloc ((±ρ∗ , 0, Ω∗ )) = 2

s dim Wloc ((±ρ∗ , 0, −Ω∗ )) u dim Wloc ((±ρ∗ , 0, −Ω∗ ))

(similar to Figure 3.7(a));

= 2, = 1 (similar to Figure 3.7(b));

(ii) if p0 < (±ρ∗ )2 = P∗ < 1, then the equilibria are saddles and s dim Wloc ((±ρ∗ , 0, Ω∗ )) = 2, u dim Wloc ((±ρ∗ , 0, Ω∗ )) = 1

s dim Wloc ((±ρ∗ , 0, −Ω∗ )) u dim Wloc ((±ρ∗ , 0, −Ω∗ ))

(similar to Figure 3.7(c));

= 1, = 2 (similar to Figure 3.7(d)),

s u where Wloc and Wloc are the stable and unstable manifolds of the equilibria, respectively.

It is clear that (3.1.12)–(3.1.14) have a fixed point ρ = R, { { { Ω = R2 k, { { { {υ = 0,

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| 205

where R and k satisfy (3.1.4). This fixed point corresponds to the periodic solution of (3.1.1). We discuss the existence and nonexistence of periodic orbits for system (3.1.12)–(3.1.14). By the first equation of (3.1.10) and the last of (3.1.14), we have Kx =

2Ωεc0 c c [(c1 − ω) + ρ2 (c2 − 1 ) + ρ4 (c3 − 1 )], 2 2 2 2 + ε c1

c02

Ωx = ρ2

εc0 c c [(c1 − ω) + ρ2 (c2 − 1 ) + ρ4 (c3 − 1 )]. 2 2 c02 + ε2 c12

(3.1.32) (3.1.33)

The integrals K and Ω of the unperturbed system are slow variables of the perturbed system (see (3.1.10) and (3.1.14)): Kx , Ωx = O(ε). A Poincaré map P on an O(ε) neighborhood of E is denoted by Eε , and we can use the map to detail the dynamic behavior of a bounded solution for the unperturbed system (3.1.12)–(3.1.14). Assume that (K0 , Ω0 ) ∈ Eε and consider the solution of (3.1.12)–(3.1.14) with initial values υ(0) = 0, Ω(0) = Ω0 and ρ(0) such that K(0) = K0 . Let (ρ, 0, Ω) be the next intersection point of Γε (x) with the υ = 0 plane having dυ/dx < 0 (if such a point exists). Then P(K0 , Ω0 ) = (K(ρ, 0, Ω), Ω) = (ρ2 +

2

Ω

2

ρ

−

1 4 1 6 ρ − ρ , Ω). 4 6

Define ΔK(K0 , Ω0 ) and ΔΩ(K0 , Ω0 ) by P(K0 , Ω0 ) = (K, Ω) = (K0 + ΔK(K0 , Ω0 ), Ω0 + ΔΩ(K0 , Ω0 )). The representations of ΔK and ΔΩ can be calculated to O(ε) as Xε (K0 ,Ω0 )

ΔK(K0 , Ω0 ) =

Kx (Γε (x))dx,

∫ 0

Xε (K0 ,Ω0 )

ΔΩ(K0 , Ω0 ) =

Ωx (Γε (x))dx,

∫ 0

with Xε (K0 , Ω0 ) the return time of Γε (“time = x”). In fact, X(K0 ,Ω0 )

ΔK =

∫ 0

c c 2Ωε 2 (ρ − ρ2∗ )[(c2 − 1 ) + (ρ2 + ρ2∗ )(c3 − 1 )]dx + O(ε2 ). c0 2 2

The orbit (ρ0 (x), υ0 (x)) in the Ω0 plane is described by the K integral ((3.1.10) and (3.1.11)), which intersects the ρ axis (υ = ρx ) at two points 0 < ρ1 (K0 , Ω0 ) < ρ2 (K0 , Ω0 ) (see Figure 3.3(a)). Hence, set ρ2 = R, ρ21 = R1 , and ρ22 = R2 , we get ΔK = −

Ω0 ε c0

R2 (K0 ,Ω0 )

∫ R1 (K0 ,Ω0 )

(ρ2∗ − R)[(c2 − √K0 R −

R2

c1 ) 2

−

+ (ρ2∗ + R)(c3 −

Ω20

+

1 3 R 4

+

1 4 R 6

c1 )] 2

dR + O(ε2 ).

(3.1.34)

206 | 3 Structures of small dissipative dynamical systems When ρ2 (x) < ρ2x , ΔK < 0, and similarly we get ε ΔΩ = − c0

R2 (K0 ,Ω0 )

∫ R1 (K0 ,Ω0 )

(ρ2∗ − R)[(c2 − √K0 R −

R2

c1 ) 2

−

+ (ρ2∗ + R)(c3 −

Ω20

+

1 3 R 4

+

1 4 R 6

c1 )] 2

dR + O(ε2 ),

(3.1.35)

and when ρ2 (x) < ρ2∗ , ΔΩ < 0. Therefore 2Ωεc0 c d 2 (ρ2 − ρ2∗ ) (ρ2 − ρ2 ∗2 )(c3 − 1 ). (Ω2 − ρ2∗ K) = 2 dx 2 c0 + ε2 c12 For the sake of convenience, we assume that a = c3 −

c1 2

> 0, and then

2ΩΔΩ − ρ2∗ ΔK > 0 (Ωεc0 < 0).

(3.1.36)

Equation (3.1.36) implies that for Ω0 ≠ 0, ΔΩ = ΔK = 0 is impossible. Hence Poincaré map P has no fixed points for Ω ≠ 0. Theorem 3.1.5. For Ω ≠ 0 and ε ≠ 0, all periodic solutions of the perturbed system (3.1.12)–(3.1.14) are collapsed and all quasiperiodic solutions of (3.1.1) blow-up due to the perturbation. For Ω = 0, there exists a K0 such that ΔK(K0 , 0) = ΔΩ(K0 , 0) = 0. Hence Theorem 3.1.6. There exist solutions of the Ginzburg–Landau equation with small complex coefficients, which are slowly periodic in time and space, and not of the type Rei(kx−ωt) . We will discuss the heteroclinic orbits. System (3.1.12)–(3.1.14) exhibits heteroclinic orbits for some special values. We start from the two-dimensional unstable manifold Γu+ at the equilibrium points (±√pj , 0, Ωj ). Define Γu+ = {(Ki+ , Ω+i )}i∈I ,

I = {1, 2, 3, . . .},

with (K1+ , Ω+1 ) ∈ E∗ the first intersection point of Γu+ with υ = 0 plane (and υτ < 0), + (Ki+1 , Ω+i+1 ) = p(Ki+ , Ω+i ). Similarly we define γs+ to be the set of points in E∗ symmetric (in the K axis) to γu+ , while γs− corresponds to the one-dimensional stable manifold Γs− of (∓√pj , 0, ±Ωj ) or (±√pj , 0, ∓Ωj ). We search for coefficients c0 , c1 , c2 , c3 , ε and ω, such that Γu+ = Γs− . By the symmetry of Lemma 3.1.4, similar to the proof in [22], by the Poincaré mapping, we have Theorem 3.1.7. For any sufficiently small ε there exist (at least) the coefficients c0 , c1 , c2 and c3 dependent on j, j = 1, 2, such that system (3.1.12)–(3.1.14) has a heteroclinic orbit connecting (√pj , 0, Ωj ) and (±√pj , 0, −Ωj ), and another one connecting (−√pj , 0, Ωj ) and (−√pj , 0, −Ωj ), see Figure 3.8.

3.1 Quintic Ginzburg–Landau equation

| 207

Figure 3.8: Geometry of the homoclinic orbits.

Figure 3.9: A symmetric pair of homoclinic orbits.

Remark 1. It is probable that the there exist some heteroclinic orbits or homoclinic orbits such as shown in Figures 3.8 and 3.9. Remark 2. There exist the double pulse homoclinic orbits as shown in Figure 3.10. Remark 3. Also there are the double heteroclinic orbits found by numerical simulation and shown in Figure 3.14. In Figure 3.15, we drew the heteroclinic orbit for c0 = 1, c1 = 4.5, c2 = 2, c3 = 3 2(c +c2 −c1 )(c2 −c1 /2)(c3 −c2 /2−c1 /4) and ε = 0.002, Ω = c1 − 3 (c . In this case, we took the ini−c /2)(c −2c +c /2) 3

1

3

2

1

208 | 3 Structures of small dissipative dynamical systems

Figure 3.10: Double pulse homoclinic orbits.

Figure 3.11: Heteroclinic orbit.

√−b+√b2 −4ac

tial data as ρ = + ε, where c = c1 − ω, b = c2 − 0.5c1 , a = c3 − 0.5c1 , 2a v = 0, and Ω = ((−0.5ρ − 0.5)ρ + 1)ρ2 . From Figure 3.15 we can see that the heteroclinic orbit starts from one equilibrium point (0.73107084, 0, 0.58700412) and proceeds to the other equilibrium point (0.73107084, 0, −0.58700412). First, the orbit rotates clockwise. When ρ → 0, v changes very sharply. And then the orbit rotates anticlockwise, finally connecting to the other equilibrium point. In Figure 3.11, we show the projection of the heteroclinic orbit in Figure 3.15 on the plane Ω = 0. Moreover, in Figure 3.12, < 0 for the heteroclinic orbit in Figwe give the Poincaré truncation at v = 0 and dv dt ure 3.15. For the cubic–quintic Ginzburg–Landau equation (3.1.1), we simulate the solution numerically by the pseudo-spectral method. And FFT (fast Fourier transformation) is used to solve (3.1.1). Numerical results show that the solution of (3.1.1) is not periodic (see Figure 3.13) when ε ≠ 0. In Figure 3.13, we draw the flow on the phase space (|W(x0 , t)|, |W(x0 , t)|t ), where x0 = π/q and q = 1.0931. We take the initial data as W(x, 0) = 1 + 0.02 exp(i π4 ) cos(qx) and ε = 0.0025. In Figure 3.14, we give the picture of the flow with ε = 0.

3.1 Quintic Ginzburg–Landau equation

Figure 3.12: Projection of the heteroclinic orbit on Ω = 0.

Figure 3.13: Poincaré truncation of the heteroclinic orbit at v = 0 and

dv dt

< 0.

Figure 3.14: Flow of W (x, t) on the phase plane (|W (x0 , t)|, |W (x0 , t)|t ) with ε = 0.025.

| 209

210 | 3 Structures of small dissipative dynamical systems

Figure 3.15: Flow of W (x, t) on the phase plane (|W (x0 , t)|, |W (x0 , t)|t ) with ε = 0.

3.2 The derivative Ginzburg–Landau equations Consider the generalized Ginzburg–Landau equation in one spatial dimension (see [63]): ̃ ̃ 2 ̃ + (1 + iβ)u ̃ x = (μ̃ + iσ)U uτ + νu xx + (η̃ + iδ)|u| u

̃ 2 ux − (m̃ + in)u ̃ 2 ū x , ̃ 4 u − (α̃ + iq)|u| + (ξ ̃ + iγ)|u|

(3.2.1)

where (x, τ) ∈ ℝ × ℝ+ and u(x, t) is an unknown complex function; ν,̃ μ,̃ σ,̃ β,̃ η,̃ δ,̃ ξ ̃ , γ,̃ α,̃ q,̃ m̃ and ñ are constants, i = √−1. Now we consider the structure of traveling wave solutions of equation (3.2.1). Let z = x − εcτ, where 0 < ε ≤ 1. Let ν̃ = εν, μ̃ = 1, η̃ = 21 , β̃ = εβ, σ̃ = εσ, δ̃ = εδ, ξ ̃ = − 21 , γ̃ = εγ, α̃ = εα, q̃ = εq, m̃ = εm and ñ = εn. For (z, τ) variables equation (3.2.1) becomes 1 uτ = ε(c − ν)uz + (1 + iεβ)uzz + (1 + iεσ)u + (− + iεδ)|u|2 u 2 1 + (− + iεγ)|u|4 u − ε(α + iq)|u|2 uz − ε(m + in)u2 ū z . 2

(3.2.2)

It will be convenient to rewrite (3.2.2) in polar coordinates. Setting u(z, τ) = r(z, τ)e−iθ(z,τ) ,

(3.2.3)

and t = ε2 τ,

Z = εz,

Q = εθ,

(3.2.4)

into equation (3.2.2), we have ε2 rt = ε2 rZZ + ε2 βrQZZ + ε2 (c + 2βQZ − ν)rZ

1 1 + r(1 − r 2 − r 4 − Q2Z ) − ε2 (α + m)r 2 rZ − r 3 ε(q − n)QZ , 2 2

(3.2.5)

3.2 The derivative Ginzburg–Landau equations |

r Q rZZ + 2 Z Z + (c + βQZ − ν)QZ − (σ + δr 2 + γr 4 ) r r − (α − m)r 2 QZ + ε(q + n)rrZ .

211

Qt = QZZ − ε2 β

(3.2.6)

Suppose that rZ , rZZ , rt , QZ and QZZ are O(1) terms for all (z, τ) ∈ ℝ × ℝ+ and all ε > 0. By letting ε → 0 one then gets 1 2 1 4 2 { {1 − 2 r − 2 r − QZ = 0, { rZ QZ { 2 4 2 {Qt = QZZ + 2 r + (c + βQZ − ν)QZ − (σ + δr + γr ) − (α − m)r QZ .

(3.2.7)

Solve for r 2 from the first equation of (3.2.7) in terms of QZ , and then substitute the result into the second equation of (3.2.7). We also denote Z be z, Q be θ. For the sake of convenience, it will be assumed that β = 0 in (3.2.5), and (3.2.6) then yields the equation for θ, ε2 rt = ε2 rzz + ε2 (c − ν − (α + m)r 2 )rz { { { { { { 1 1 { + r(1 − r 2 − r 4 − θz2 ) − r 3 ε(q − n)θz , 2 2 { { { { { { { θ = θ + (c − ν − (α − m)r 2 + 2 γz )θ − (σ + δr 2 + γr 4 ) + ε(q + n)rr . z z t zz r {

(3.2.8)

We consider the steady-state solutions to (3.2.8) when 0 < ε ≪ 1. The steady-state equation of (3.2.8) can be written as two systems, one is a slow system given by 󸀠

εr { { { { { { 󸀠 { { εs { { { { { θ󸀠 { { { { { { { ϕ󸀠 { where 󸀠 = given by

= s, 1 1 = −ε(c − ν − (α + m)r 2 )s − r(1 − r 2 − r 4 − ϕ2 ) + εr 3 (q − n)ϕ, 2 2 = ϕ,

= −(c − ν − (α + m)r 2 + 2

s )ϕ + (σ + δr 2 + γr 4 ) − (q + n)rs, εr

d . The other is a fast system under the change of variables z dz

= εξ , which is

r ̇ = s, { { { { { 1 1 { { { ṡ = −ε(c − ν − (α + m)r 2 )s − r(1 − r 2 − r 4 − ϕ2 ) + εr 3 (q − n)ϕ, { { 2 2 { { θ̇ = εϕ, { { { { { { { {ϕ̇ = −ε(c − ν − (α + m)r 2 + 2 s )ϕ + ε(σ + δr 2 + γr 4 ) − ε(q + n)rs, εr { where ̇ =

d . dξ

(3.2.9)

(3.2.10)

When ε = 0, there exists an invariant manifold M0 = {(r, s, θ, ϕ) : r 2 + r 4 + 2ϕ2 = 2, s = 0, θ ∈ ℝ}.

(3.2.11)

212 | 3 Structures of small dissipative dynamical systems or ϕ2 < 35−64 73 , M0 is Upon linearizing (3.2.10) about this manifold, when r 2 > 73−3 8 normally hyperbolic. Thus there exists an invariant manifold Mε for (3.2.9) which is O(ε) close to M0 (see Hale et al. [65]). The manifold Mε is given by √

√

Mε = {(r, s, θ, ϕ) : r = M r (ϕ, ε), s = M s (ϕ, ε), θ ∈ ℝ, ϕ2
0 such that if 0 < ε < ε0 , then there exists a domain in the parameter space, for which a heteroclinic orbit to (3.2.9) exists on Mε . This orbit connects any two different stable plane wave solutions to (3.2.8). Consider the stability of singular orbits. Let the solution be denoted by z

r(z) = ρ(z),

θ(z) = ∫ ϕ(s)ds, 0

(3.2.15)

3.2 The derivative Ginzburg–Landau equations |

213

where limz→±∞ ϕ(z) = ϕ± . We discuss perturbations z

r(z, t) = ρ(z) + R(z, t),

θ(z, t) = ∫ ϕ(s)ds + Θ(z, t).

(3.2.16)

0

Linearizing (3.2.8) about the traveling waves (3.2.15), 2ρϕ ρ3 2 { { R = R + [c − ν − (α + m)ρ ]R − [ + (q − n)]Θz { t zz z { ε ε2 { { { { { 3 5 1 { { { − 2 [1 − ρ2 − ρ4 − ϕ2 − 2ε2 (α + m)ρρz − 3ερ2 (q − n)ϕ]R, { { 2 2 ε { { ρ ϕ { { Θt = Θzz + [c − ν − (α − m)ρ2 + 2 z ]Θz + [2 + ε(q + n)ρ]Rz { { { ρ ρ { { { { { ρ { { + [ε(q + n)ρz − 2(α − m)ϕρ − 2 z ϖ − 2δρ − 4γρ3 ]R. ρ {

(3.2.17)

Setting R(z, t) = R(z)eλt , Θ(z, t) = Θ(z)eλt into (3.2.17) yields the eigenvalue problem 2ρϕ 3 { { εR󸀠󸀠 + ε2 [c − ν − (α + m)ρ2 ]R󸀠 − [ 2 + ρ3 ε(q − n)]Θ󸀠 − [1 − ρ2 { { 2 ε { { { { { { { − 5 ρ4 − ϕ2 − 2ε2 (α + m)ρρ − 3ερ2 (q − n)ϕ]R = ε2 λR, { z { { 2 { { { {Θ󸀠󸀠 + [c − ν − (α − m)ρ2 + 2 ρz ]Θ󸀠 + [2 ϕ + ε(q + n)ρ]R󸀠 { { { ρ ρ { { { { { ρ { + [ε(q + n)ρ − 2(α − m)ϕρ − 2 z ϖ − 2δρ − 4γρ3 ]R = λΘ, { z ρ {

(3.2.18)

d and λ ∈ ℂ is an eigenvalue. If for λ ∈ ℂ with positive real part, there exists where 󸀠 = dz a nontrivial solution (R(z), Θ(z)) to (3.2.18), then singular orbits are unstable. System (3.2.18) has a fast–slow structure which is inherited from the ODE (3.2.9)–(3.2.10). The slow equations are given by 󸀠

εR { { { { { { { { εs󸀠 { { { { { { { { { { { { { { { 󸀠 Θ { { { { { { { Φ󸀠 { { { { { { { { { { { { { { { { 󸀠 { τ

= s, = −ε[c − ν − (α + m)ρ2 ]s + [

2ρϕ 3 + ρ3 ε(q − n)]Φ − [1 − ρ2 2 ε2

5 − ρ4 − ϕ2 − 2ε2 (α + m)ρρz − 3ερ2 (q − n)ϕ − ε2 λ]R, 2

(3.2.19)

= Φ, ρ ϕ s󸀠 = −[c − ν − (α − m)ρ2 + 2 z ]Φ + [2 + ε(q + n)ρ] ρ ρ ε − [ε(q + n)ρz − 2(α − m)ϕρ − 2 = k(1 − τ2 ).

ρz ϖ − 2δρ − 4γρ3 ]R + λΘ. ρ

214 | 3 Structures of small dissipative dynamical systems Define the relation z=

1 1+τ ln( ), 2k 1−τ

(3.2.20)

where k > 0 is sufficiently small so that the solution (R, s, Θ, τ) of (3.2.19) is C 1 on ℂ4 × [−1, 1] (see Alexander et al. [1], Lemma 3.1). Note that τ → ±1. Because (3.2.19) is independent of τ if and only if z → ±∞, (3.2.19) is self-consistent. Setting z = εξ , we get the fast equations Ṙ = s, { { { { { 2ρϕ 3 { { { ṡ = −ε[c − ν − (α + m)ρ2 ]s + [ 2 + ρ3 ε(q − n)]Φ − [1 − ρ2 { { 2 ε { { { { { 5 { { − ρ4 − ϕ2 − 2ε2 (α + m)ρρz − 3ερ2 (q − n)ϕ − ε2 λ]R, { { { 2 { ̇ { Θ = εΦ, { { { { { { ̇ ϕ ρ̇ ρ̇ { { Φ = −ε[c − ν − (α − m)ρ2 ]Φ + 2 2 ϕR − 2 Φ − 2 s − ε(q + n)ρs { { ρ ρ ρ { { { { { { + ελΘ − ε([q + n)ρ̇ − 2(α − m)ϕρ − 2δρ − 4γρ3 ]R. { { { { 2 { τ̇ = εk(1 − τ ).

(3.2.21)

Define Σ to be the open set in the complex plane containing the right-half plane and Γ as its boundary. Let Σ denote its closure. Clearly, {0} ⊂ σc (L) ∩ Σ. By setting Y = (R, s, Θ, Φ)T , equation (3.2.19) can be written as Y 󸀠 = M(λ, τ, ε)Y,

{

τ󸀠 = k(1 − τ2 ).

(3.2.22)

Define the asymptotic matrices M ± (λ, ε) by M ± (λ, ε) = lim M(λ, τ, ε). τ→±1

(3.2.23)

This gives the asymptotic systems for (3.2.22) as Y 󸀠 = M ± (λ, ε)Y,

{

τ󸀠 = 0.

(3.2.24)

By [92], for λ ∈ Σ, there exist solutions Yj (λ, τ, ε), j = 1, 2, 3, 4, such that lim Yj (λ, τ, ε) = 0, {z→−∞ { lim Y (λ, τ, ε) = 0, {z→+∞ j

j = 1, 2, j = 3, 4.

(3.2.25)

3.2 The derivative Ginzburg–Landau equations |

215

The sets {Y1 , Y2 } and {Y3 , Y4 } are linearly independent, because for λ ∈ Σ matrices M ± (λ, ε) have two eigenvalues with positive real part and two eigenvalues with negative real part. For λ ∈ Σ, Evans function D(λ, ε) is given by t

D(λ, ε) = e− ∫0 tr M(λ,s,ε)ds |Y1 ⋅ ⋅ ⋅ Y4 |(λ, z, ε), where | ⋅ | represents the determinant of the 4 × 4 matrix. Lemma 3.2.1. Fix ε and λ ∈ Σ. Then D(λ, ε) is analytic in λ ∈ Σ and D(λ, ε) = 0 if and only if λ is an eigenvalue. Proof. See Alexander et al. [1]. Lemma 3.2.2. For λ ∈ Σ, D(λ, ε) = 0 implies that there exists an eigenfunction of L which decays exponentially fast as |z| → ∞. The operator L given by (3.2.17) and (3.2.17) can be conveniently written as R R ( ) = L( ). Θ t Θ Actually, a bit more can be said about the Evans function. Recall that the wave is constructed via restricting the ODE (3.2.9) to a two-dimensional invariant manifold Mε . This manifold is normally hyperbolic, with a one-dimensional unstable manifold and a one-dimensional stable manifold. On Mε the wave is realized as the intersection of a one-dimensional unstable manifold and a one-dimensional stable manifold. As such, in the four-dimensional phase space the wave is realized as the intersection of a twodimensional unstable manifold with a two-dimensional stable manifold. By Kapitula [92], the following lemma is realized. Lemma 3.2.3. For each ε > 0 the Evans function D(λ, ε) is analytic at λ = 0. Remark. When ε = 0, D(λ, 0) is not analytic at λ = 0. This is due to the fact that the wave is a singular solution. Lemma 3.2.4. There exists an ε0 > 0 such that if 0 < ε < ε0 , then D(0, ε) ≠ 0. Let K ⊂ Ω be a simple closed curve which contains no eigenvalues. Let ε(K) be a k-bundle over S2 which is called the augmented unstable bundle depending on K. The topological invariant of interest is a certain Chern number of the bundle ε(K) over S2 . For any complex bundle E over S2 , let c1 [E] ∈ H 2 (S2 ; ℤ) be the first Chern class of E. A natural way to achieve this is to evaluate c1 [E] on a specific generator of H 2 (s2 ; ℤ), say the fundamental class of S2 , denoted [S2 ]. The first Chern number of E is then c1 [E] = ⟨c1 [E], [S2 ]⟩ ∈ ℤ. We know that for the Evans function D(λ) in Ω, its zeros are exactly the eigenvalues of L which are inside Ω, counting multiplicities. Let K be a simple closed curve in Ω.

216 | 3 Structures of small dissipative dynamical systems Consider the curve D(K) in ℂ. If there are no eigenvalues of L on K then D(K) ∈ ℂ \ {0} and the winding number W(D(K)) counts the number of zeros inside K, and hence the number of eigenvalues of L. Theorem 3.2.2 ([1]). The following three numbers are equal: (1) W(D(K)); (2) c1 (ε(K)); (3) the number of eigenvalues of L inside K, counting algebraic multiplicities. Letting ε > 0, the Evans function is analytic at the origin, and then there exists an β0 (ε) > 0, such that D(λ, ε) is analytic on the domain Σε = Σ ∪ B(0, β0 (ε)). Let U ± (λ) be the two-dimensional unstable subspaces of M ± (λ, ε), respectively. Starting with these subspaces, a complex two-dimensional bundle over S2 , the augmented bundle will be constructed relative to a given simple closed curve K in Σε . The Grassmannian Gk (C 4 ) is the set of k-dimensional subspaces of ℂ4 . The linear differential equation (3.2.22) induces an equation on Gk (ℂ4 ) × [−1, +1]. If the equation is such that τ󸀠 = 0, then each k-dimensional subspace would become a fixed point in Gk (ℂ4 ) for each τ. There is a close relationship between bundles and Grassmannians. Let ε be a k-bundle over S2 which is a subbundle of S2 × ℂ4 . Then there exists a natural map F : S2 → ℂ4 which sends each point to the fiber above that point when considered as a subspace of ℂ4 . The bundle ε is the pullback of the canonical bundle over Gk (ℂ4 ). There is a one-to-one correspondence between isomorphism classes of such k-bundles and the homotopy classes of these maps. A key property is that if the classifying map is homotopic to a constant, i. e., if the bundle is trivial, then the first Chern number of the bundle is zero. For each fixed λ ∈ Σε , a two-bundle over τ ∈ [−1, +1], say U ∗ (λ, τ), can be constructed. It is realized as the unstable manifold of (Y, τ) = (0, −1) for (3.2.22). As in [1], if λ is not an eigenvalue then the bundle can be extended to τ = ±1, with the fibres over τ = ±1 being U ± (λ). Let K ∈ Σε be a curve homeomorphic to S1 which contains no eigenvalues of L. Setting K0 to be the region bounded by K, the above construction renders a bundle over the set K0 × {−1} ∪ K × [−1, +1] ∪ K0 × {+1},

(3.2.26)

which is topologically S2 ; see Figure 3.16. The augmented unstable bundle, which is denoted by ε(K), has the fibres given by U ± (λ),

{

∗

U (λ, τ),

τ = ±1, τ ∈ (−1, +1).

We will show that there exists a λ0 > 0 such that if |λ| > λ0 , then λ cannot be an eigenvalue for (3.2.18). Set Ψ(z) =

ρ󸀠 (z) . ρ(z)

3.2 The derivative Ginzburg–Landau equations |

217

Figure 3.16

Consider (3.2.19) for ε ≠ 0 with the scaling 1

y = z|λ| 2 ,

s̃ =

s |λ|

1 2

Φ Φ̃ = . 1 |λ| 2

,

(3.2.27)

Using this scaling, (3.2.19) becomes 󸀠

εR { { { { { { { { εs󸀠̃ { { { { { { { { { { { { { { { { { { { { { { { { { { { { Θ󸀠 { { { { { ̃󸀠 { {Φ { { { { { { { { { { { { { { { { { { { { { { { { { { { 󸀠 { { τ { where 󸀠 =

= s,̃ ε

=−

1 2

[c − ν − (α + m)ρ2 ]s̃ +

ρ

1 2

[

2ρϕ + ρ3 ε(q − n)]Φ̃ ε2

|λ| |λ| 1 3 5 − {[1 − ρ2 − ρ4 − ϕ2 − 2ε2 (α + m)ρ2 Ψ |λ| 2 2 − 3ερ2 (q − n)ϕ2 ] − ε2 ei arg λ }R,

= Φ,̃ s̃ 1 2ϕ + ε(q + n)ρ] = − 1 [c − ν − (α + m)ρ + zΨ]Φ̃ − [ 1 ε |λ| 2 |λ| 2 ρ 1 2Ψ − [ε(q + n)ρΨ − 2(α − m)ϕρ − ϕ − 2δρ |λ| ρ 1

2

(3.2.28)

− 4γρ3 ]R + εei arg λ Θ, =

d . dy

k |λ|

1 2

(1 − τ2 ),

The limit as |λ| → ∞ of (3.2.28) is the equation 󸀠 {εR { { { { { εs󸀠̃ { { { 󸀠 Θ { { { { { { Φ̃ 󸀠 { { { 󸀠 { τ

= s,̃ = ε2 ei arg λ R, = Φ,̃ = εe = 0.

i arg λ

Θ,

(3.2.29)

218 | 3 Structures of small dissipative dynamical systems Setting ε = 1, the above equation can be rewritten as 󸀠

R { { { { { { s󸀠̃ { { { 󸀠 Θ { { { { { { Φ̃ 󸀠 { { { 󸀠 {τ

= s,̃ = ei arg λ R, = Φ,̃

(3.2.30)

= ei arg λ Θ, = 0.

Since τ󸀠 = 0, each τ-slice for the above equation is invariant. For each τ ∈ [−1, +1] the eigenvalues of the right-hand side of (3.2.30) are given by ±ei arg λ/2 . Thus, when |arg λ| < π, for each τ-slice there are two eigenvalues with positive real part, and two eigenvalues with negative real part. For each fixed τ the flow induced by (3.2.30) on the Grassmannian G2 (ℂ4 ) then has an attractor, which is the unstable subspace. When considered over the entire range τ ∈ [−1, +1], there then exists a curve in G2 (ℂ4 ) which is an attractor. Proceeding as in the rest of the proof of Proposition 2.2 in [92] yields the following result. Lemma 3.2.5. There exists a λ0 > 0 such that if |λ| > λ0 , then λ is not an eigenvalue for (3.2.18), the constant λ0 is independent of ε. We will consider the construction of the fast unstable bundle εf (K) and the slow unstable bundle εs (K). Using the fast–slow structure of (3.2.19), the two-dimensional bundle ε(K) will be decomposed into the direct sum of two one-dimensional bundles, εf (K) and εs (K). First, the bundle εf (K) will be constructed. Recall that when ε = 0 d both ρ̇ = 0 and ρ2 + ρ4 + 2ϕ2 = 2, where ̇ = dξ . When ε = 0 the fast system (3.2.21) becomes Ṙ = s, { { { { { { ṡ = (4 − ρ2 − 4ϕ2 )R + 2ϕρs, { { { { { 2ϕ Φ̇ = − s, { { ρ { { { { { ̇ Θ = 0, { { { { { τ̇ = 0.

(3.2.31)

The eigenvalues of the right-hand side of the above equation are given by ±√4 − ρ2 − 8ϕ2 and 0. Let ζ (λ, τ) be the unstable eigenvector for (3.2.31) for each τ ∈ [−1, +1]. By projecting, a trajectory ζ ̂ ∈ ℂP 3 × [−1, +1] is obtained. Since this trajec-

tory is formed by taking the fast unstable eigenvector, it is an attractor in ℂP 3 ×[−1, +1]; therefore, it perturbs to a nearby attractor for small ε. This trajectory forms a onedimensional subbundle of ε(K), say εf (K). The subbundle is clearly trivial when ε = 0, and hence continues to be trivial for small ε. This yields the following:

3.3 The perturbed nonlinear Schrödinger equation

| 219

Lemma 3.2.6. There exists an ε1 > 0 such that if 0 ≤ ε < ε1 , then the bundle εf (K) is such that c1 (εf (K)) = 0. The slow unstable bundle εs (R) will now be constructed. The system with ε = 0 in (3.2.31) has an invariant manifold MoL given by MoL = {(R, s, Θ, Φ, τ) : s = 0, R =

2ρϕ , Θ ∈ ℝ, τ ∈ [−1, 1]}. 4 − ρ2 − 4ϕ2

(3.2.32)

By (3.2.12), we know that, when 4 − ρ2 − 4ϕ2 > 0, this manifold is normally hyperbolic. But since this was a condition for the existence of the wave, it is trivially satisfied. Using only slightly different arguments as those in [92], we get Lemma 3.2.7. There exists an ε2 > 0 such that if 0 < ε < ε2 , then the bundle εs (K) is such that c1 (εs (K)) = 0. Then we can get the main results. The full bundle ε(K) is given as the Whitney sum τ(K) = εs (K) ⊕ εf (K). Let ε3 = min{ε0 , ε1 , ε2 }. Since the first Chern number of a vector bundle is additive over subbundles, c1 (ε(K)) = c1 (εs (K)) + c1 (εf (K)) = 0. By the results of Alexander et al. [1], the following linear stability result is now at hand: Theorem 3.2.3. If 0 < ε < ε3 , then there exist no eigenvalues to (3.2.18) in K0 , which is the region bounded by K. The nonlinear stability result follows immediately from Theorem 1.2 in Kapitula [92]. We have supposed that the traveling wave is realized as the intersection of a two-dimensional unstable manifold with a two-dimensional stable manifold in a four-dimensional phase space. Furthermore, suppose that the Evans function is nonzero in the closed right-half side of the complex plane. Then the wave is stable. Thus we have the following result: Theorem 3.2.4. Let the parameters c, ν, β, σ, δ, γ, α, q, m, n be chosen such that p(ϕ) = 0 √ has two solutions, say ϕ± , such that ϕ2± < 35−64 73 . There exists an ε0 > 0 such that if 0 < ε < ε0 , then there exists a traveling wave solution to (3.2.8) which is O(ε) close to the solution of the formal equations (3.2.12). Call this solution as in (3.2.15). There exists an 0 < ε1 < ε0 such that if 0 < ε < ε1 , the wave is stable.

3.3 The perturbed nonlinear Schrödinger equation We will establish the existence and invariance property of a nontrivial symmetric pair of homoclinic orbits for a perturbed nonlinear Schrödinger equation. So we need to establish the existence of normally hyperbolic manifolds and global integrable theory, etc.

220 | 3 Structures of small dissipative dynamical systems 3.3.1 Preliminary results Consider a perturbed nonlinear Schrödinger equation (see [74]) iqt = qxx + 2[|q|2 − ω2 ]q + iε[̂ Dq − 1],

(3.3.1)

D is a bounded diswhere the constant ω ∈ ( 21 , 1), ε is a small positive constant, and ̂ 1 sipative linear operator on the Sobolev space He,p of even, 2π-periodic functions that are square integrable with square integrable first derivative. The PDE is well posed in 1 He,p as the following theorem states: 1 Theorem 3.3.1 (Cauchy problem). For all q0 ∈ He,p and for all t ∈ (−∞, +∞), there 1 exists a unique solution q(t, q0 ; ε), continuous in t with values in He,p of equation (3.3.1) such that q|t=t0 = q0 ; moreover, q(t, q0 ; ε) depends smoothly on q0 and ε.

Thus the flow F t (q0 ; ε) = q(t, q0 ; ε) can be viewed as a smooth dynamical system 1 on He,p . The plane of constants Πc , def

Πc := {q(x, t) : 𝜕x q(x, t) = 0}, is an invariant plane for equation (3.3.1), and on Πc the equation takes the form iqt = 2[qq̄ − ω2 ]q − iε[αq + 1],

(3.3.2)

where it is assumed that the dissipation operator ̂ D acts invariantly on Πc as ̂ Dq = −αq for α a positive constant. Equivalently, in terms of polar coordinates, q = √I exp iθ, these equations take the form √ {It = −2ε[αI + I cos θ], {θ = −2(I − ω2 ) + ε sin θ. { t √I

(3.3.3)

When ε = 0, the unperturbed orbits on Πc are nested circles with Sω being a circle of fixed points given by I = ω2 . For positive ε the perturbed orbits on Πc are very different, see Figure 3.17. First, only three fixed points exist: O, which is a deformation of the origin; Q, a saddle that deforms from the circle Sω ; and P, a spiral sink that also deforms from the

3.3 The perturbed nonlinear Schrödinger equation

| 221

Figure 3.17: Phase-plane diagram of the ODE.

circle Sω . Formulas for these fixed points, together with their associated growth rates, are given by 1 { I0 = ε2 [ 4 ] + O(ε4 ), { { { 4ω { { { π α { { { θ0 = − − ε 2 + O(ε2 ), { { 2 2ω { { { ε √ { 2 { { I =ω + 1 − α2 ω2 + O(ε2 ), { {p 2ω { √1 − α2 ω2 { { { θp = − arctan − π + O(ε2 ), { { { αω { { { {I = ω2 − ε √1 − α2 ω2 + O(ε2 ), { { q { 2ω { { { { { √1 − α2 ω2 { θ = arctan − π + O(ε2 ), { q αω σ0 = ±i − εα + O(ε2 ), { { { 1 { 3 σp = ±2i√εω[1 − α2 ω2 ] 4 − εα + O(ε 2 ), { { { 1 { 3 2 2 {σq = ±2√εω[1 − α ω ] 4 − εα + O(ε 2 ).

(3.3.4)

(3.3.5)

By introducing the variable J, J = I − ω2 , equation (3.3.3) can be written as 2 2 { {Jt = −2ε[α(J + ω ) + √J + ω cos θ], ε { { θt = −2J + sin θ. √J + ω2 {

(3.3.6)

In order to describe the flow close to the stable manifold of the point Q, we rescale the coordinates as τ = νt,

{

J = νj,

(3.3.7)

222 | 3 Structures of small dissipative dynamical systems where ν = √ε. In these scaled coordinates, equations (3.3.3) on the plane Πc take the form 1

2 2 {jτ = −2[α(ω + νj) + (ω + νj) 2 cos θ], { − 21 2 {θτ = −2j + ν(ω + νj) sin θ,

(3.3.8)

which can be written in terms of the variable y = (j, θ)T as yτ = Y(y; ν),

(3.3.9)

where Y = (Y1 , Y2 )T is defined by equation (3.3.8). In terms of these variables, the point Q has coordinates yq = (jq , θq ) where √1 − α2 ω2

+ O(ν3 ), 2ω √1 − α2 ω2 θq = arctan( ) − π + O(ν2 ). αω

jq = −ν

Linearizing the equation around yq and writing ỹ = y − yq , we obtain ỹτ = Y 󸀠 (yq , ν)ỹ + O(ỹ 2 ), where Y 󸀠 is the 2 × 2 matrix whose entries are −ν(2α + (ω2 + νjq )−1/2 cos θq ) [ −2 − 21 ε(ω2 + νjq )−3/2 sin θq

2(ω2 + νjq )1/2 sin θq ]. ν(ω2 + νjq )−1/2 cos θq

Thus we see that the eigenvalues of Y 󸀠 (yq , ν) are 1

2 2 2 {λ = −2√ω(1 − α ω ) 4 − να + O(ν ), 1 { 2 2 2 4 {μ = 2√ω(1 − α ω ) − να + O(ν ),

(3.3.10)

with eigenvectors e1 (ν) and e1 (ν), respectively. The eigenvectors depend smoothly on ν and T

σ e1 (0) = (− , 1) , 2 T

σ e2 (0) = ( , 1) , 2

1

σ = 2√ω(1 − α2 ω2 ) 4 . Referring to [65], from regular perturbation theory and the stable manifold theorem we have, for ε sufficiently small, an open neighborhood U of Q, independent of ε, such

3.3 The perturbed nonlinear Schrödinger equation

| 223

that the stable manifold of Q in U is a smooth function of ν. Therefore a portion of the local stable manifold can be parameterized in terms of y = y∗ (s; ν),

(3.3.11)

s = exp[λτ],

for 0 ≤ s ≤ s∗ = exp[λτ∗ ], with s∗ small and independent of ε. To understand the stable manifold away from Q, we note that Q is an order ν perturbation of the point Q0 j0 = 0, θ0 = arctan(

√1 − α2 ω2 ) − π, αω

and that equations (3.3.11) are an O(ν) perturbation of the conservative system jτ = −2(αω2 + ω cos θ), θτ = −2j.

Thus, it is useful to introduce the energy of the above system, def

E(j, θ) =

1 2 j − ω(sin θ + αωθ), 2

(3.3.12)

where the curve E(j, θ) = E(j0 , θ0 ) is the stable manifold of the conservative system, which we denote by Cεs : y0 (τ) = (j0 (τ), θ0 (τ)). If we fix a τ0 < τ∗ , we have from regular perturbation theory that for τ0 < τ < τ∗ , the stable manifold of Q is given by y = y0 (τ) + O(ν). Therefore, if we denote by Cεs the portion of the stable manifold corresponding to τ ∈ [τ0 , ∞), then Cεs can be parameterized by s ∈ [0, s0 ] y = y∗ (s; ν), s = exp[λτ], where y∗ is a smooth function of s and ν. The curve y∗ is order O(ν) close to the y0 for τ0 < τ and satisfies the equation (λs)y∗,s = Y∗ , where Y∗ = Y(y∗ ; ν), see Figure 3.18. Since |y0,s | is bounded away from 0, we have def 󵄨 󵄨 m(s; ν) = 󵄨󵄨󵄨y∗,s (s; ν)󵄨󵄨󵄨 ≥ m0 ,

0 ≤ s ≤ s0 ,

224 | 3 Structures of small dissipative dynamical systems

Figure 3.18: Phase-plane diagram of the ODE in the (j, θ)-coordinates.

where m0 is a positive constant independent of ε and the unit tangent vector to the curve is given by t=

y∗,s Y∗ = . m (λs)m

(3.3.13)

To describe the flow near Cεs , it is convenient to introduce coordinates (r, s) where r is a measure of the distance in the normal direction to the curve Cεs . These coordinates are given by y = y∗ (s; ν) + rn(s; ν),

(3.3.14)

where n(s; ν) is the unit normal vector to the curve y∗ (s; ν). The unit normal vector and the unit tangent vector are related by the equation ns = k(s, ν)t, where k is a smooth function. We can rewrite equation (3.3.9) in terms of (r, s) by observing that yτ = (m + kr)tsτ + rτ n, which leads to the equations rτ = Y ⋅ n,

sτ =

Y ⋅t . m + kr

Expanding the above equation in r, we obtain Y ⋅ n = [(Y∗󸀠 ⋅ n) ⋅ n]r + O(r 2 ),

(Y 󸀠 ⋅ n) ⋅ t k|Y∗ | Y ⋅t Y ⋅t = ∗ +[ ∗ − ]r + O(r 2 ). m + kr m m m2

Note that from equation (3.3.13) we have Y∗ ⋅ t = λs m

3.3 The perturbed nonlinear Schrödinger equation

| 225

and the equations for (r, s) are rτ = a(s; ν)r + O(r 2 ),

sτ = λs + b(s, ν)r + +O(r 2 ),

(3.3.15)

where a and b are smooth functions in (s, ν) for 0 < s < s0 . In these coordinates Cεs corresponds to r = 0, and the flow on Cεs is given by sτ = λs. Moreover, since e1 is tangent to Cεs at Q, we have from (3.3.10) a(0, ν) = (Y 󸀠 (yq , ν)n) ⋅ n = μ. Although the (r, s) equations are simpler than the original equations (3.3.6), they are still not in a form in which the flow near Cεs can be described well. The reason is the presence of the linear term b(s, ν)r. Let s = β + (h0 + h1 β)r,

(3.3.16)

where the constants h0 and h1 are chosen to be h0 =

b0 , μ−λ

h1 =

b1 − h0 a1 , μ

and where a = μ + a1 s + O(s2 ), b = b0 + b1 s + O(s2 ). In terms of (r, β) the equations are rτ = a(β, ν)r + O(r 2 ),

βτ = λβ + c(β, ν)r + O(r 2 ),

(3.3.17)

where |c(β, ν)| ≤ c0 |β|2 on Cεs ; that is, 0 ≤ β ≤ s0 . Equation (3.3.17) has the advantage that the linearized flow around Cεs , that is, r = 0 and β∗ = β0 eλτ , δrτ = a∗ δr,

δβτ = λδβ + c∗ δr,

has expansion and contraction rates similar to those obtained by linearizing the equation around Q. 3.3.2 The equations in a neighborhood of Sω In order to study the dynamics of solutions to the nonlinear problem in a neighborhood of the circle of fixed points Sω , we write (3.3.1) in terms of coordinates that are

226 | 3 Structures of small dissipative dynamical systems suited for this purpose. This entails introducing coordinates (J, θ, f ), where J is a measure of distance from Sω on the plane Πc , θ is the angle on Sω , and f is in the orthogonal complement of Πc . These coordinates are determined in the following manner: First, write q as def

q = [ρ(t) + f (x, t)] exp iθ(t),

(3.3.18)

where ρ and θ are polar coordinates on the plane Πc , and f ∈ Π⊥ c ; that is, f has spatial mean zero. The L2 -norm is a constant of motion for the unperturbed (ε = 0) flow; therefore, it will be used as a coordinate instead of ρ: def

I =

2π

1 ̄ = ρ2 + ⟨f f ̄⟩. ∫ qqdx 2π

(3.3.19)

0

Finally, since we are working in a neighborhood of the circle of fixed points Sω that corresponds to I = ω2 , it will be convenient to introduce the variable J defined by J = I − ω2 .

(3.3.20)

In terms of these variables, equation (3.3.1) takes the form Jt = −2ε[α(ω2 + J) + ρ cos θ] + εQ1 (f ), 1 1 θt = −2J − ε sin θ + Q2 (f ) + C2 (f ), ρ ρ 1 ift = Lε f + Wε f + ρQ3 (f ) + C2 (f )f , ρ where ρ = √J + ω2 − ⟨f f ̄⟩, 2

Lε f = fxx + iεDf̂ + 2ω (f + f ̄),

Wε f = 2J(f + f ̄) +

ε sin θ f, ρ

and where Q1 (f ) = 2⟨f ̄(D̂ − α)f ⟩, Q (f ) = −⟨(f + f ̄)2 ⟩, 2

Q3 (f ) = 4(f f ̄ − ⟨f f ̄⟩) + 2(f 2 − ⟨f 2 ⟩), C (f ) = −⟨f f ̄(f + f ̄)⟩, 2

C3 (f ) = 2(f f ̄f − ⟨f f ̄f ⟩) − ⟨f 2 + f ̄2 + 6f f ̄⟩f + 2⟨f f ̄⟩f ̄.

(3.3.21)

3.3 The perturbed nonlinear Schrödinger equation

| 227

In these equations, ⟨⋅⟩ denotes the spatial mean over one period. For J and f in a small but otherwise fixed neighborhood of 0, equation (3.3.21) can be considered as a perturbation of (3.3.6): Jt = −2ε[α(J + ω2 ) + √J + ω2 cos θ] + ε1 (J, θ, f ; ε), θt = −2J + ε(J + ω2 )

− 21

sin θ + ε2 (J, θ, f ; ε),

(3.3.22)

ift = Lε f + Wε f + ωQ3 (f ) + ε3 (J, θ, f ; ε),

where Wε f = 2J(f + f ̄) +

ε sin θ f, √J + ω2

and where εk are 2π-periodic functions in θ of order ε1 (J, θ, f ; ε) = O(εf 2 ), ε2 (J, θ, f ; ε) = O(f 2 ), 2

(3.3.23) 3

ε3 (J, θ, f ; ε) = O(Jf + f ), for small J and f . The point Q is a critical point for equation (3.3.21). In constructing a homoclinic orbit to Q, we need to estimate the size of the local stable manifold of Q. The size of the variables (J, θ) is determined from the curve Cεs , which is the intersection of the stable manifold and the plane of constants Πc . To estimate the size of f , we have to use equation (3.3.21), which has a troublesome quadratic term Q3 (f ). However, this term is nonresonant and can be removed using a normal form transformation, as will be demonstrated below. In order to eliminate the quadratic term Q3 (f ), we must analyze the quadratic resonances of the linear equation ift = fxx + 2ω2 (f + f ̄),

(3.3.24)

which corresponds to the linear part of the f equation (3.3.21) evaluated at ε = 0. Substituting f = ei(kx+λt) , we find that the dispersion relation of the linear equation is given by: λ = ±ik √k 2 − 4ω2 ,

k = 1, 2, . . .

Quadratic resonance means that there exist nonzero integers k1 , k2 , and k3 such that k1 + k2 = k3 ,

λ1 ± λ2 = ±λ3 .

228 | 3 Structures of small dissipative dynamical systems We check the above resonance conditions by squaring the λ-equation and substituting the k-equation to obtain [(k1 + k2 )2 + k12 + k22 − 6ω2 ](k1 + k2 )2 = 0, which is only possible if k1 + k2 = k3 = 0, since ω ∈ ( 21 , 1). However, since we are considering the equation in the space Π⊥ c , we have that k3 ≠ 0. Therefore the linear equation (3.3.24) has no quadratic resonance in the space Π⊥ c . To compute the normal form transformation, we start by writing f in terms of a Fourier expansion: f (x) = ∑ f ̂(k)eikx . k =0 ̸

The quadratic term Q3 (f ) is made up of two terms: f 2 (x) − ⟨f 2 ⟩ = ∑ f ̂(k)f ̂(l)ei(k+l)x , k+l=0 ̸

󵄨󵄨 󵄨2 2 i(k+l)x . 󵄨󵄨f (x)󵄨󵄨󵄨 − ⟨|f | ⟩ = ∑ f ̂(k)f ̂(l)e k+l=0 ̸

Now a general quadratic near-identity map that is translation-invariant with respect to x can be written as g = f + K(f , f ), { { def ̄ ̄ ̄ ̄ {K(f , h) = K11 (f , h) + K11̄ (f , h) + K11̄ (f , h) + K11̄ ̄ (f , h),

(3.3.25)

⊥ ⊥ where K are bounded bilinear maps K : Π⊥ c × Πc → Πc ,

K11 (f , h) = ∬ K11 (x − y1 , x − y2 )f (y1 )h(y2 )dy1 dy2 , ̄ )dy dy , K11̄ (f , h)̄ = ∬ K11̄ (x − y1 , x − y2 )f (y1 )h(y 2 1 2 with similar expressions for K11̄ and K11̄ ̄ . In terms of Fourier expansions these bilinear maps can be written as i(k+l)x ̂ K11 (f , h) = ∑ K̂ 11 (k, l)f ̂(k)h(l)e , k+l=0 ̸

i(k+l)x ̄̂ K11̄ (f , h)̄ = ∑ K̂ 11̄ (k, l)f ̂(k)h(−l)e . k+l=0 ̸

Proposition 3.3.1. There exists a near-identity quadratic map of the form (3.3.25) that transforms the equation i𝜕t f = fxx + 2ω2 (f + f ̄) + ωQ3 (f ) into an equation with a cubic nonlinearity i𝜕t g = gxx + 2ω2 (g + g)̄ + O(g 3 ).

3.3 The perturbed nonlinear Schrödinger equation

| 229

Proof. Compute def

̄ Sg = i𝜕t g − 𝜕x2 g − 2ω2 (g + g), where g is given by (3.3.25). We obtain Sg = Sf + H11 (f , f ) + H11̄ (f , f ̄) + H11̄ (f ̄, f ) + H11̄ ̄ (f ̄, f ̄) + C(f ), where Ĥ 11 = 2(kl + ω2 )K̂ 11 − 2ω2 K̂ 11̄ − 2ω2 K̂ 11̄ − 2ω2 K̂ 11̄ ̄ ,

Ĥ 11̄ = 2ω2 K̂ 11 + 2(l2 + kl − ω2 )K̂ 11̄ − 2ω2 K̂ 11̄ − 2ω2 K̂ 11̄ ̄ ,

Ĥ 11̄ = 2ω2 K̂ 11 − 2ω2 K̂ 11̄ + 2(k 2 + kl − ω2 )K̂ 11̄ − 2ω2 K̂ 11̄ ̄ ,

Ĥ 11̄ ̄ = −2ω2 K̂ 11 + 2ω2 K̂ 11̄ + 2ω2 K̂ 11̄ + 2(k 2 + l2 + kl − 3ω2 )K̂ 11̄ ̄ ,

and where C(f ) consists of terms of the form K(f , Sf ). Substituting the equation for f in the above Sf = ωQ3 (f ), we obtain Sg = ωQ3 (f ) + H11 (f , f ) + H11̄ (f , f ̄) + H11̄ (f ̄, f ) + H11̄ ̄ (f ̄, f ̄) + C(f ), where C(f ) is a cubic term. Therefore, to eliminate the Q3 term from the g equation, we need Ĥ 11 = −2ω,

Ĥ 11̄ = −2ω,

Ĥ 11̄ = −2ω,

Ĥ 11̄ ̄ = 0,

for all the k and l such that k + l ≠ 0. Since Hab = Ĥ ab , we deduce that Kab = K̂ ab for a ̄ To find the four-vector K, we have to solve the linear equation and b ∈ {1, 1}. UK = H, for the given vector H = (−2ω, −2ω, −2ω, 0)T . Since there are no quadratic resonances, det U ≠ 0 and these equations have a unique solution ω ω , K̂ 11 (k, l) = − , K̂ 11̄ (k, l) = − kl l(k + l) ω K̂ 11̄ (k, l) = − , K̂ 11̄ ̄ (k, l) = 0. k(k + l) Note that k ≠ 0, l ≠ 0, and k + l = 0 since we are in the space Π⊥ c . Moreover, since 󵄨2 󵄨 ∑ 󵄨󵄨󵄨K̂ ab (k, l)󵄨󵄨󵄨 < ∞, we have K ∈ L2 (S1 × S2 ), which implies that K is a bounded bilinear map on Π⊥ c, 󵄩󵄩 󵄩 2 󵄩󵄩K(f , f )󵄩󵄩󵄩H 1 ≤ C‖f ‖H 1 ,

230 | 3 Structures of small dissipative dynamical systems for all f ∈ Π⊥ c . Finally, we can invert the equation g = f + K(f , f ), for f in a neighborhood of the zero to obtain f = g + K (g), where K is of order O(g 2 ). Therefore C(f ) is a cubic term in g. The equations in a neighborhood of Sω were given in (3.3.22) in terms of the variables (J, θ, f ): Jt = −2ε[α(J + ω2 ) + √J + ω2 cos θ] + ε1 (J, θ, f ; ε), θt = −2J + ε(J + ω2 )

− 21

sin θ + ε2 (J, θ, f ; ε),

ift = Lε f + Wε f + ωQ3 (f ) + ε3 (J, θ, f ; ε).

We apply the transformation g = f + K(f , f ). This procedure will eliminate the Q3 term, but it will introduce new quadratic terms in the equation for g that have ε coefficients, such as ̂ εDK(f , f ). Therefore, using (J, θ, f ) as coordinates, the equations near Sω are written as Jt = −2ε[α(J + ω2 ) + √(J + ω2 ) cos θ] + N1 (J, θ, g; ε), − 21

θt = −2J + ε(J + ω2 )

sin θ + N2 (J, θ, g; ε),

(3.3.26)

igt = Lε g + Wε g + N3 (J, θ, g; ε),

where, for J and g in a neighborhood of 0, we have N1 (J, θ, g; ε) = O(εg 2 ), N2 (J, θ, g; ε) = O(g 2 ),

N3 (J, θ, g; ε) = O(Jg 2 + εg 2 + g 3 ).

Since we will be working with invariant real manifolds in a neighborhood of the circle of fixed points Sω , it will be convenient to introduce a real coordinate system: T

u = (Re(g), Im(g)) .

3.3 The perturbed nonlinear Schrödinger equation

| 231

In terms of these variables the above equation takes the form Jt = −2ε[α(J + ω2 ) + √J + ω2 cos θ] + N1 (J, θ, u; ε), − 21

θt = −2J + ε(J + ω2 )

sin θ + N2 (J, θ, u; ε),

(3.3.27)

ut = Lε u + Vε u + N3 (J, θ, u; ε).

Here N3 is interpreted as a two-vector, Lε = J 𝜕x2 − 4ω2 S + εD,̂ ε sin θ Vε = −4Js + J, √J + ω2 where 0 −1

J =[

1 ], 0

S=[

0 1

0 ] 0

3.3.3 Existence of local invariant manifolds We prove the persistence of invariant manifolds in a neighborhood of Sω , and fiber these manifolds in the strong contracting or expanding directions. We also estimate the size of the stable manifold at the point Q by using equations (3.3.27) derived above. In a neighborhood of Sω , equations (3.3.27) can be viewed as a perturbation of the linear system Jt = 0, { { { θt = −2J, { { { {ut = Lε u.

(3.3.28)

In order to study the local behavior of solutions to the nonlinear equations (3.3.27), we have to analyze the spectrum of the operator Lε; therefore, we consider the eigenvalue problem Lε e = λe, for the eigenpairs {e(x), λ}. Using Fourier expansions, one finds a quadratic expression for the eigenvalue λ: 2

(λ − εd(j)) + j2 (j2 − 4ω2 ) = 0,

j = 1, 2, . . . ,

where d(j) denotes the symbol of −D.̂ Since ω ∈ ( 21 , 1), we have for j = 1, 1 { (1, ∓σ)T cos x, {es,u = 2√πω { { ε {σs,u = ±σ − εd(1),

(3.3.29)

232 | 3 Structures of small dissipative dynamical systems where σ = √4ω2 − 1.

(3.3.30)

For j ≥ 2, the eigenvalues come in complex conjugate pairs with negative real part: λj = iΩj − εd(j), where Ωj = j√j2 − 4ω2 > 0.

(3.3.31)

In terms of this eigenbasis, the mean zero function u can be written as u(x) = vu eu (x) + vs es (x) + v0 (x),

(3.3.32)

where vu and vs are real scalars and where v0 (x) ∈ [span{Πc , eu , es }]⊥ . In terms of these variables, linear equations (3.3.28) split into Jt = 0, { { { { { { {θt = −2J, { { v = σuε u, { { u,t { { { vs,t = −σsε u, { { { { {v0,t = Lε v0 .

(3.3.33)

Thus, we explicitly see that, for ε = 0, the linear equations have one unstable direction (εu ), one stable direction (es ), and an infinite number of center directions (J, θ, v0 ). Combining these center variables as vc = (J, θ, v0 )T , equation (3.3.33) can be written as vu,t = σuε u, { { { vs,t = −σsε u, { { { {vc,t = Avc ,

(3.3.34)

where A is defined from equations (3.3.33). In a δ-neighborhood of the circle of fixed points Sω the nonlinear equation (3.3.27) can be viewed as a perturbation of the linear equation (3.3.28). Under the flow of this linear equation and for ε = 0, Sω has one-dimensional stable and unstable manifolds, together with a codimension 2 center manifold. We focus our attention on the center manifold E c (Sω ), together with the center-stable E cs (Sω ) and center-unstable E cu (Sω ) manifolds: E cs (Sω ) = span{eu }⊥ , E cu (Sω ) = span{es }⊥ ,

3.3 The perturbed nonlinear Schrödinger equation

| 233

E c (Sω ) = span{eu , es }⊥ . An important feature of the linear equation (3.3.33) is that the growth rates on the invariant manifolds are separated by a wide gap. To see this, we note that for ε = 0 the σ spectrum of the operator has real part ±σ and 0. Thus for any integer n and for ε < 4n we have σ|t| 󵄩 󵄩󵄩 ], 󵄩󵄩exp[AT]󵄩󵄩󵄩 ≤ nC exp[ n and the invariant manifolds E cs , E cu , and E c can be described by solutions whose growth rates are bounded by exp[ σt ] for t > 0, exp[− σt ] for t < 0, and exp[ σ|t| ] for n n n all t, respectively. We fix an integer n0 sufficiently large and a localization parameter δ = na2 where 0

a is a constant that is independent of ε that will be specified later. We introduce a localization function ψδ s ψδ (s) = ψ( ), δ

ψδ : ℝ → ℝ, where ψ is C ∞ and satisfies

1,

|s| ≤ 1,

0,

|s| ≥ 2.

ψ(s) = {

The localization function localizes equations (3.3.27), namely Jt = −2ε[α(Jδ + ω2 ) + √Jδ + ω2 cos θ] + N1 (Jδ , θ, uδ ; ε), { { { { −1 { θt = −2J + ε[(Jδ + ω2 ) 2 sin θ] + N2 (Jδ , θ, uδ ; ε), { { { {ut = Lε u + Vε uδ + N3 (Jδ , θ, uδ ; ε),

(3.3.35)

where for any variable s we denote sδ = sψ( δs ). Note that we do not cut off the variable θ; thus, the function ψδ (J, f ) has the effect of cutting off the right-hand sides whenever the phase point lies outside a neighborhood Uδ of the circle Sω . Moreover, all the nonlinear terms in equation (3.3.35) are either multiplied by c or are at least quadratic in (J, u) and localized in a δ-neighborhood of 0. This implies that equation (3.3.35) has a global Lipschitz constant of order O(ε + δ). def

Using v = (vu , vs , vc )T as variables and the operator A defined in equation (3.3.34), we can write equations (3.3.35) as vu,t = σuε u + Rδu (v; ε), { { { { v = −σsε u + Rδs (v; ε), { { s,t { { δ {vc,t = Avc + Rc (v; ε), where Rδ (v; ε) and its first derivatives are of order O(ε + δ).

(3.3.36)

234 | 3 Structures of small dissipative dynamical systems We will show that the localized equation has C l -invariant manifolds. For the original equations, these manifolds will be locally invariant in a δ-neighborhood of Sω . Definition 3.3.1. Given an open set O, a manifold ℳ is called locally invariant (in O) under a flow F t if, for every open interval I such that F l (q) ⊂ O, F t∗ (q) ∈ ℳ for some t∗ ∈ I ⇒ F t (q) ∈ ℳ, ∀t ∈ I. Theorem 3.3.2. There exist a δ-neighborhood Uδ of Sω , ε0 (S) > 0, and an integer l > 3 such that ∀ε ∈ [0, ε0 ), equation (3.3.27) has a locally invariant in Uδ manifold of codimension 1, Wεcs = {v ∈ H 1 , vu = hu (vs , vc ; ε)},

(3.3.37)

where hu is C l in all of its arguments and 2π-periodic in θ. Moreover, for ε = 0, W0cs intersects E cs tangentially along Sω . Similarly, we have a locally invariant manifold given by Wεcu = {v ∈ H 1 , vs = hs (vu , vc ; ε)},

(3.3.38)

where the function hs is C l in all of its arguments and 2π-periodic in θ. Moreover, for ε = 0, W0cu intersects E cu tangentially along Sω . The existence of a codimension 2 slow manifold ℳε is then given by the following: Corollary 3.3.1. Let ℳε denote the intersection cs

cu

ℳε = Wε ∩ Wε ,

then ℳε is a locally invariant in Uδ manifold of codimension 2, 1

c

c

ℳε = {v ∈ H , vu = hu (vc ; ε), vs = hs (vc ; ε)},

(3.3.39)

where the functions hcu,s are C l in their arguments and 2π-periodic in θ. Moreover, for ε = 0, ℳε intersects E c tangentially along Sω . Remark. The flow on ℳε is given by the equations ̃1 (Jδ , θ, v0δ ; ε), Jt = −2ε[α(Jδ + ω2 ) + √(Jδ + ω2 ) cos θ] + N { { { { 1 − ̃2 (Jδ , θ, v0δ ; ε), { θt = −2J + ε(Jδ + ω2 ) 2 sin θ + N { { { ̃3 (Jδ , θ, v0δ ; ε), {v0,t = Lε v0 + Vε v0δ + N ̃ are the restrictions of N to ℳε , given by the functions hc and hc . where N u s

(3.3.40)

3.3 The perturbed nonlinear Schrödinger equation

| 235

Proof of Theorem 3.3.2. Once the integral equations are set up, this proof is just a standard application of a fixed-point argument. We include it here for the sake of completeness. First, we rewrite equation (3.3.36) in integral form: t

{ { { vu (t) = exp[σuε (t − tu )]vu (tu ) + ∫ exp[σuε (t − s)]Rδu (v(s); ε)ds, { { { { { tu { { { { t { { { ε v (t) = exp[−σ (t − t )]v (t ) + exp[−σsε (t − s)]Rδs (v(s); ε)ds, ∫ s u s s s { { { { ts { { { { { t { { { δ { { {vc (t) = exp[At]vc (0) + ∫ exp[A(t − s)]Rs (v(s); ε)ds. { 0 Because of the gap in the growth rates, we will characterize the invariant manifolds Wεcs and Wεcu by Wεcs = {v ∈ H 1 : sup(exp[− t≥0

Wεcu = {v ∈ H 1 : sup(exp[ t≤0

σ 󵄩󵄩 t 󵄩 t]󵄩F (v;̄ ε)󵄩󵄩󵄩H 1 ) < ∞}, n0 󵄩

σ 󵄩󵄩 t 󵄩 t]󵄩F (v;̄ ε)󵄩󵄩󵄩H 1 ) < ∞}, n0 󵄩

(3.3.41) (3.3.42)

where F t (v;̄ ε) is the flow of equations (3.3.36). Focusing our attention upon Wεcs , for v̄ ∈ B(0, ρ) in a ball B of arbitrary radius ρ, we introduce the norm ‖v‖λ =

sup

0≤t,σ≤‖v‖H 1

exp{−

σt 󵄩󵄩 󵄩 󵄩v(t)󵄩󵄩󵄩H 1 }. λ󵄩

From the definition of Wεcs we have for v ∈ Wεcs 󵄨 󵄨 exp[−σu tu ]󵄨󵄨󵄨vu (tu )󵄨󵄨󵄨 → 0,

tu → ∞.

Therefore for solutions on Wεcs the integral equation can be written as t

{ { { { vu (t) = ∫ exp[σuε (t − s)]Rδu (v(s); ε)ds, { { { { +∞ { { { { t { { { ε v (t) = exp[−σ t]v + exp[−σsε (t − s)]Rδs (v(s); ε)ds, ∫ s s { { s { { 0 { { { { t { { { δ { { { {vc (t) = exp[At]vc + ∫ exp[A(t − s)]Rs (v(s); ε)ds. 0 {

(3.3.43)

236 | 3 Structures of small dissipative dynamical systems To show existence of Wεcs we use Newton’s iterations: Let v0 = 0 and t

{ { { { vuk+1 (t) = ∫ exp[σuε (t − s)]Rδu (vk (s); ε)ds, { { { { +∞ { { { { t { { { k+1 ε v (t) = exp[−σ t]v + exp[−σsε (t − s)]Rδs (vk (s); ε)ds, ∫ s s s { { { { 0 { { { { t { { { { {vck+1 (t) = exp[At]vc + ∫ exp[A(t − s)]Rδs (vk (s); ε)ds. { { 0 {

(3.3.44)

This generates a well-defined sequence of functions. For if ‖vk ‖n0 ≤ C, then σ 󵄩󵄩 k+1 󵄩󵄩 󵄩󵄩v 󵄩󵄩H 1 ≤ n0 C exp[ t](‖vs ‖H 1 + ‖vc ‖H 1 ) n0 ∞

σ 󵄩 󵄩 + ∫ exp[ (t − s)]󵄩󵄩󵄩Rδu (vk (s); ε)󵄩󵄩󵄩H 1 ds 2 t

t

σ 󵄩 󵄩 + ∫ exp[− (t − s)]󵄩󵄩󵄩Rδs (vk (s); ε)󵄩󵄩󵄩H 1 ds 2 0

t

+ ∫ n0 C exp[ 0

σ 󵄩 󵄩 (t − s)]󵄩󵄩󵄩Rδc (vk (s); ε)󵄩󵄩󵄩H 1 ds. 2n0

By (3.3.35) we note that Rδ (ω, ε) is a smooth function whose terms are either linear with coefficient ε or nonlinear and localized in a δ-neighborhood of Sω . Therefore if we let R󸀠 be the derivative of Rδ , we have the estimate 󵄩󵄩 δ 󵄩 󵄩 󸀠󵄩 󵄩󵄩R (ω, ε)󵄩󵄩󵄩H 1 ≤ 󵄩󵄩󵄩R 󵄩󵄩󵄩‖W‖H 1 + ε,

(3.3.45)

where ‖R󸀠 ‖ is the supremum of the magnitude of R󸀠 , which is equal to C(δ + ε) because of the localization function. This implies that σ 󵄩󵄩 k+1 󵄩󵄩 󵄩󵄩v (t)󵄩󵄩H 1 ≤ n0 C exp[ t](‖vs ‖H 1 + ‖vc ‖H 1 + ε) n ∞

0

σ 󵄩 󵄩 + ∫ exp[ (t − s)]C(ε + δ)󵄩󵄩󵄩vk (s)󵄩󵄩󵄩H 1 ds 2 t

t

+ ∫ exp[ 0

σ 󵄩 󵄩 (t − s)]n0 C(ε + δ)󵄩󵄩󵄩vk (s)󵄩󵄩󵄩H 1 ds. 2n0

3.3 The perturbed nonlinear Schrödinger equation

| 237

By using the bound on vk , we obtain 󵄩󵄩 k+1 󵄩󵄩 󵄩󵄩v (t)󵄩󵄩H 1 ≤ C[‖vs ‖H 1 + ‖vc ‖H 1 + ε

σ 󵄩 󵄩 + n20 (ε + δ)󵄩󵄩󵄩vk (s)󵄩󵄩󵄩n ] exp[ ], 0 n0 t

where the constant C is independent of n0 , ε, and δ. Now by fixing δ = a=

1 c, 4

we obtain for all ε
2. Passing to the

hu (vs , vc ; ε) = vu (0) = ∫ exp[−σuε s]Rδu (v(s); ε)ds,

(3.3.49)

repeated to obtain bounds on {Dj vk } in the ‖ ⋅ ‖ n -norm provided limit, we obtain v ∈ C l for l ≤

n [ 20 ]

j

− 1. With these estimates we define 0

∞

which is a C l functional with ‖Dhu ‖ ≤ 21 , and Wεcs = {v ∈ H 1 : vu = hu (vs , vc ; ε)} is a C l manifold. The invariance of Wεcs follows from the definition of hu and from the invariance of the equation under time translation. For Wεcs to be a locally invariant manifold for equations (3.3.1), the function hu has to be 2π-periodic in θ. This is obviously so since the integral equations are 2π-periodic in θ and have a unique solution for every θ ∈ ℝ. Finally, the tangency of Wεcs to Sω , follows from observing that for ε = 0, Rδ is at least quadratic in u and J for all θ ∈ [0, 2π]. An identical argument establishes the existence of a C l function hs , such that ‖Dhs ‖ < 21 and of a C l locally invariant manifold for equation (3.3.1) given by 󵄨 Wεcu {v ∈ H 1 󵄨󵄨󵄨 vs = hs (vu , vc ; ε)}.

3.3 The perturbed nonlinear Schrödinger equation

| 239

Proof of Corollary 3.3.1. The intersection of Wεcu and Wεcs can be described by the solution of the system of equations vu = hu (vs , vc ; ε), vs = hs (vu , vc ; ε).

Note that ‖Dhu ‖, and ‖Dhs ‖ ≤ 21 . By the implicit function theorem, the above system has a unique solution given by vu = hcu (vc ; ε), vs = hcs (vc ; ε),

where hcu,s are C l functions and 1

c

c

ℳε = {v ∈ H , vu = hu (vc ; ε), vs = hs (vc ; ε)}.

Because the linearized problem has small growth rates, the global construction of homoclinic orbits for t ∈ (−∞, ∞) is a singular perturbation problem. The equations under consideration basically have the following model structure: η̇ = [−1 + εΩ(η, v)]η, v̇ = ε[v + S(η, v)],

(3.3.50)

where Ω is of order O(|η|+|v|) and S of order O(η2 +v2 ) for small (η, v). Note that for equation (3.3.50) η = 0 is an invariant manifold ℳ on which the motion is slow. This slow motion is very different when ε > 0 as opposed to ε = 0. This difference is the origin of the singular nature of the problem. If the system were completely uncoupled, then the long-time behavior of the initial value problem would be completely described by the solution of the v equation. For the coupled system, it is unclear which particular solution of the v equation is approached asymptotically. Assume there exists a smooth change of variables (η, v) → (η, ηc ),

v = f (η, ηc ; ε),

f (0, ηc ; ε) = ηε ,

which partially decouples the flow by placing equation (3.3.50) into the following form: η̇ = [−1 + εΩ(η, v)]η, ̄ )]. v̇ = ε[η + εS(η c

c

With this form of the equations, it is clear that as t → ∞ (η(t; η0 ), ηc (t; η0 )) → (0, ηc (t; η̄ 0 )),

(3.3.51)

240 | 3 Structures of small dissipative dynamical systems exponentially fast as exp(−t), where η0 = (η(0), ηc (0)) and η̄ 0 = (0, ηc (0)) Thus longtime motion through an arbitrary point (η(0), ηc (0)) can be tracked by following motion through the point (0, ηc (0)). Following the work of [24], it has become the practice in the literature to describe this partial decoupling in a coordinate-free manner. One introduces a family of curves indexed by the slow manifold ℳ: cs

Fv : [−1, 1] → Wε (ℳ),

∀v ∈ ℳ,

which satisfy Fv (0) = v. These curves are C 2 in (η, v; ε) and are characterized by the requirement that a point (η, v)̄ lies on the curve Fvs if and only if 󵄩󵄩 t 󵄩 t 󵄩󵄩F (η, v;̄ ε) − F (0, v; ε)󵄩󵄩󵄩 → 0 at a fast rate as t → ∞. The curve Fv is called a Fenichel stable fiber through base point v ∈ ℳ. Clearly, in our partially decoupled coordinate system, the fiber Fv is given by Fv = {(η, v)̄ : v̄ = f (η, v; ε)},

where f (η, v; ε) is the coordinate transformation. Theorem 3.3.3. For all ε ∈ [0, ε0 ] the C l manifold Wεcu admits a C l−2 coordinate system vu = ηu , u

ηu ∈ [−η0 , η0 ],

vc = f (ηu , ηc ; ε),

ηu ∈ E c ,

such that the submanifold ℳε corresponds to ηu = 0, and the flow on Wεcu decouples in the following manner: η̇ u = [σuε + Γδ (ηu , ηc ; ε)]ηu , η̇ c = Aηc + S̄cδ (ηc ; ε),

where η, Γδ , S̄cδ , and their first derivatives are of order O(ε + δ). A similar statement holds for Wεcs . The manifold ℳε is a codimension 1 submanifold of Wεcu , and this implies that locally Wεcu can be viewed as a product ℳε × (−z0 , z0 ) with local coordinates (vc , ηu ), here vc are coordinates on ℳε , and ηu is the coordinate on an interval (−z0 , z0 ). The setup to prove the fibration of Wεcu over the slow manifold ℳε begins by writing the flow on Wεcu in terms of these local coordinates. The manifold Wεcu is given as a graph over (vc , ηu ), vs = hs (vu , vc ; ε),

3.3 The perturbed nonlinear Schrödinger equation

| 241

and the flow on Wεcu is given by restricting equations (3.3.36) to this graph: v̇u = σuε vu + Suδ (vu , vc ; ε),

{ v̇c = Avc + Scδ (vu , vc ; ε),

(3.3.52)

def

δ where Su,c (vu , vc ; ε) = Rδu,c (vu , hs (vu , vc ; ε), vc ; ε) are the restrictions of Rδu,c to the graph. Similarly, using (vu , vc ) as local coordinates on Wεcu , the submanifold ℳε is given as a graph over vc ,

vc = hcu (vc ; ε), and the flow on ℳε is given by v̇c = Avc + S̄cδ (vc ; ε),

(3.3.53)

def

where S̄cδ (vc ; ε) = Scδ (hcu (vc ; ε), vc ; ε) is the restriction of Scδ to the graph. Since ℳε is an invariant submanifold of Wεcu , we introduce coordinates given by (vc , ηu ) where vc are coordinates on ℳε and ηu = vu − hcu (vc ; ε).

(3.3.54)

In terms of these coordinates the submanifold ℳε is given by ηu = 0. To derive the equations for the flow on Wεcu in terms of (vc , ηu ), we have to differentiate ηu in equation (3.3.54) along solutions of (3.3.52). This gives rise to a technical difficulty, since solutions are only continuous in time with values in H 1 not C 1 . (Solutions are C 1 in the sense of distributions.) However, if we start with initial data in H 3 , we have solutions that are C 1 in time with values in H 1 . We also have that solutions to equation (3.3.52) have continuous dependence on the initial data in the space H 1 . Therefore we can derive the equation for the flow on Wεcu by assuming that our initial data are in H 3 and then using continuous dependence on the initial data to conclude that the equations hold in the sense of distributions for initial data in H 1 . Proceeding with the derivation, differentiate (3.3.54) with respect to t, η̇ u = v̇u − Dhcu (vc ; ε)v̇c , where Dhcu denotes the derivative of hcu with respect to vc . By equation (3.3.52) we can write the above equation as η̇ u = σuε vu + Suδ − Dhcu (Avc + Scδ ).

(3.3.55)

We consider the flow on ℳε given by (hcu (vc (t)), vc (t)). Differentiate with respect to t and substitute into equation (3.3.52) to obtain Dhcu v̇u = σuε hcu + Suδ (hcu , vc ; ε),

v̇c = Avc + Scδ (hcu , vc ; ε).

242 | 3 Structures of small dissipative dynamical systems These equations imply that hcu satisfies Dhcu (Avc + S̄cδ ) = σuε hcu + S̄uδ ,

(3.3.56)

where S̄uδ (vc ; ε) = Scδ (hcu , vc ; ε). Using this identity for hcu we can simplify equation (3.3.55) and describe the flow on Wεcu in terms of (vc , ηu ) equations: η̇ u = σuε ηu + Ωu (ηu , vc ; ε), v̇c = Avc + Ωc (ηu , vc ; ε),

(3.3.57)

where Ωu (ηu , vc ; ε) = Suδ (ηu + hcu , vc ; ε) − S̄uδ − Dhcu [Scδ (ηu + hcu ; ε) − S̄cδ ],

{

Ωu (ηu , vc ; ε) = Scδ (ηu + hcu , vc ; ε)

(3.3.58)

are C l−1 . Note that Ωu (0, vc ; ε) = 0, Ωc (0, vc ; ε) = S̄uδ (vc ; ε) and that all the terms in Ωu,c and their first derivatives are either of order ε or δ due to the localization. This implies that equations (3.3.57) can be written as η̇ u = [σuε + Ω̄ u (ηu , vc ; ε)]ηu , { v̇c = Avc + Ωc (ηu , vc ; ε),

(3.3.59)

where Ω̄ u , Ωc and first derivatives of Ωc are of order O(ε + δ). The above equations will be used to prove the fibration of Wεcu over ℳε , which is equivalent to finding another coordinate system (ηc , ηu ) where the above system decouples in the following manner: η̇ u = [σuε + Γu (ηu , ηc ; ε)]ηu , { η̇ c = Aηc + Scδ (ηu ; ε). def

(3.3.60)

Here Γ = Ω̄ u is evaluated in the new coordinates. If this change of variable exists, then ηu (t) approaches 0 at a fast rate, at least σt e 2 as t → ∞. This implies that the equation for vc approaches the equation for ηc exponentially fast. Thus, we let γc = vc − ηc and have γ̇c = Aγc + Ωc (ηu , γc + ηc ; ε) − Ωc (0, ηc ; ε), which can be integrated from −∞ to t t

vc (t) − ηc (t) = ∫ exp[−A(t − s)](Ωc (ηu , γc + ηc ; ε) − Ωc (0, ηc ; ε))ds. −∞

3.3 The perturbed nonlinear Schrödinger equation

| 243

The coordinate change is given by vc (0) 0

def

f u (ηu (0), ηc (0); ε) = ηc (0) + ∫ exp[−As](Ωc − Scδ )ds,

(3.3.61)

−∞

where the right-hand side depends implicitly on f u . Since the terms under the integral sign are of O(ε + δ), we can find f u as the fixed point of the above equation with the property that f u (ηu , ηc ; ε) = ηc + f ̃u (ηu , ηc ; ε),

(3.3.62)

where f ̃u is a C l−2 function whose first derivatives are of order O(δ) and f ̃u (0, ηc ; ε) = 0. This is the basic idea of the proof that will be presented below. Proof of Theorem 3.3.3. Fix an orbital ηc on ℳε and consider the system η̇ u = [σuε + Ωu (ηu , γc + ηc ; ε)]ηu , { γ̇c = Aγc + Ωc (ηu , γc + ηc ; ε) − Ωc (0, ηc ; ε), where γc = vc − ηc . Set a(ηc ; ε) = Ω̄ u (0, ηc ; ε); then the above equation can be written as η̇ u = [σuε + a(ηc ; ε) + Ψ(ηu , γc , ηc ; ε)]ηu , { γ̇c = Aγc + Φ(ηu , γc , ηc ; ε),

(3.3.63)

where Ψ(0, 0, ηc ; ε) = 0, Φ(0, 0, ηc ; ε) = 0 and a, Ψ, and Φ are of order O(ε + δ). The proof now proceeds exactly as in the proof of Theorem 3.3.2, where we defined a norm on continuous functions with values in H 1 : β = (γc , ηu ),

󵄩 󵄩 ‖β‖λ = sup{exp[−σt/λ]󵄩󵄩󵄩β(t)󵄩󵄩󵄩H 1 }, t≤0

and we set up a Newton’s iteration given by ηk+1 u (t)

t

= G(t, 0)ηu (0) + ∫ G(t, s)Ψ(ηku , γck , ηc ; ε)ηku ds, 0

0

γck+1 = ∫ exp[A(t − s)]Φ(ηku , γck , ηc ; ε)ds, −∞ t

where G(t, s) = exp[∫s (σuε + a)].

The convergence and smoothness of the sequence βk = (γck , ηku ) follows, as in the proof of Theorem 3.3.2, from the following:

244 | 3 Structures of small dissipative dynamical systems For |ηu (0)| ≤ δ we have a well-defined sequence. Assume that for some k we have ‖βk ‖m0 ≤ cδ where m0 = n0 /l; then the equation for βk+1 implies 0

σt σ σ 󵄩󵄩 k 󵄩󵄩 󵄨󵄨 k+1 󵄨󵄨 s]󵄩β 󵄩 ds + Cδ exp[ ], 󵄨󵄨ηu (t)󵄨󵄨 ≤ C(δ + ε) ∫ exp[ (t − s) + 2 m0 󵄩 󵄩m0 2 t

t

σ σ 󵄩󵄩 k 󵄩󵄩 󵄩󵄩 k+1 󵄩󵄩 s]󵄩β 󵄩 ds, 󵄩󵄩γc (t)󵄩󵄩H 1 ≤ C(δ + ε) ∫ m0 exp[ (t − s) + n0 m0 󵄩 󵄩m0 −∞

and for ε ≤ ε0 , we have 󵄩󵄩 k+1 󵄩󵄩 󵄩 k󵄩 󵄩󵄩β 󵄩󵄩m0 ≤ Cδ + Cδm0 n0 󵄩󵄩󵄩β 󵄩󵄩󵄩m0 . Since δ was chosen to be δ = 1/(Cn20 ), where C is a large constant, we have 󵄩󵄩 k+1 󵄩󵄩 󵄩󵄩β 󵄩󵄩m0 ≤ Cδ +

1 󵄩󵄩 k 󵄩󵄩 󵄩β 󵄩 . 2 󵄩 󵄩m0

The sequence {βk } converges. The functions Φ and Ψ are C l−1 , and all the terms in these expressions are either localized in a region of size δ or have a linear growth with a coefficient of order ε. Therefore, the above argument implies that 1󵄩 k 󵄩󵄩 k+1 k−1 󵄩 k󵄩 󵄩󵄩β − β 󵄩󵄩󵄩m0 ≤ 󵄩󵄩󵄩β − β 󵄩󵄩󵄩m0 , 2 which gives the convergence of βk → β, a continuous function in t with values in H 1 , and ‖β‖m0 ≤ Cδ. We also can prove that β is smooth with respect to ηc (0), ηu (0), and ε. The difficulty here is that ‖ηc (t)‖H 1 can grow for t < 0 at a rate exp[−σt/n0 ] and that a, Φ, and Ψ depend on ηc (t). If we differentiate βk+1 with respect to ηc (0), we obtain k+1 η󸀠 u (t)

t

t

t

0

s

= ηu (0)G(t, 0) ∫ a ds + ∫ G(t, s)(∫ a󸀠 )Ψηku ds t

0

󸀠

k

k

k

+ ∫ G(t, s)[(Ψ󸀠 η󸀠 c + Ψ󸀠 η󸀠 u + Ψ󸀠 γ 󸀠 c )ηku + Ψη󸀠 u ]ds, k+1

t

0

k

k

γ 󸀠 c (t) = ∫ exp[A(t − s)](Φ󸀠 η󸀠 c + Φ󸀠 η󸀠 u + Φ󸀠 γ 󸀠 c )ds, −∞

where the primes denote various derivatives. The above equation cannot be estimated with the ‖ ⋅ ‖m0 -norm because of a󸀠 and η󸀠 c , which grow at a rate exp[−σt/n0 ] for t < 0. However, we can estimate the above equation with respect to the ‖ ⋅ ‖2m0 -norm, which

3.3 The perturbed nonlinear Schrödinger equation

| 245

has a slower rate of decay, 0

σ σ σ(t − s) 󵄩󵄩 󸀠 k+1 󵄩󵄩 (t)󵄩󵄩H 1 ≤ Cδ exp[( − )t] + Cδ ∫ exp[ ] 󵄩󵄩β 2 n0 2 t

σ σ 󵄩 k+1 󵄩 × [exp[( − )s] + 󵄩󵄩󵄩β󸀠 (s)󵄩󵄩󵄩]ds m0 n0 t

+ Cδ ∫ exp[ −∞

× [exp[(

σ(t − s) ] n0

σ σ 󵄩 k 󵄩 − )s] + 󵄩󵄩󵄩β󸀠 (s)󵄩󵄩󵄩]ds. m0 n0

Furthermore, since 2m0 > n0 , the integrals converge and give σt 󵄩 k󵄩 󵄩󵄩 󸀠 k+1 󵄩󵄩 ](1 + 󵄩󵄩󵄩β󸀠 󵄩󵄩󵄩2m ), (t)󵄩󵄩H 1 ≤ Cδ exp[ 󵄩󵄩β 0 2m 0

󸀠k

which implies the boundedness of ‖β ‖2m0 . Similarly, we can estimate the (j − 1)th derivative of βk in the ‖ ⋅ ‖jm0 -norm, and all the integrals will converge provided jm0 < n0 . Since all the terms in equation (3.3.63) are C l−1 , this procedure can be continued to obtain β in C l−2 . Now we can define the fibration of the manifold Wεcu by setting f u (ηc (0), ηu (0); ε) = ηc (0) + γc (0), 0

vc (0) = ηc (0) + ∫ exp[−As][Ωc (ηu , γc + ηc ; ε) − Ωc (0, ηc ; ε)]ds. −∞

The invariance of the function f u under the flow follows from the definition of f and from reparameterizing time t → t + τ. Therefore vc (t) = f u (ηc (t), ηu (t); ε), which is well-defined provided the initial data ηu (0) is sufficiently small. This invariance implies that if we use (ηu , ηc ) as a coordinate system instead of (vc , ηu ), we’ll obtain a decoupled system of equations (3.3.60) that describe the flow on Wεcu , η̇ u = [σuε + Γu (ηu , ηc ; ε)]ηu , { η̇ c = Aηc + Scδ (ηu ; ε). Remark. The fibers are given by vu = ηu ,

{

vc = f u (ηu , ηc ; ε).

Now we consider stable manifold to Q in ℳε .

246 | 3 Structures of small dissipative dynamical systems For α < 1/ω, the point Q is given in terms (J, θ, u) by ε √ 1 − α2 ω2 + O(ε2 ), 2ω √1 − α2 ω2 θq = arctan( ) − π + O(ε), αω Jq = −

u = 0.

By linearizing equation (3.3.35) around Q we obtain that Q is a saddle point with a twodimensional unstable manifold and a codimension 2 stable manifold. The unstable manifold intersects the plane of constants Πc along the curve Cεs and intersects Wεcu along a curve tangent to the vu -direction. The local stable manifold of Q intersects Πc along the curve and therefore intersects ℳε in a submanifold of codimension 1. We are interested in the size of 𝒲 = W s (Q) ∩ ℳε . Recall that the stable manifold of Q in the plane Πc is parameterized by Cεs = {y = (j, θ) : y = y∗ (s; ν)}, where ν = √ε and y∗ is given in equation (3.3.11), y∗,τ = Y1 (j∗ , θ∗ ; ν),

θ∗,τ = Y2 (j∗ , θ∗ ; ν), and where s = exp[λτ]. The flow on ℳε in terms of (J, θ, v0 ) is derived from equations (3.3.40) by substituting J = νj: jt = νY1 (j, θ; ν) + N̄ 1 (j, θ, v0 ; ν), { { { θt = νY2 (j, θ; ν) + N̄ 2 (j, θ, v0 ; ν), { { { ̄ {v0t = Lε v0 + Vε v0 + N3 (j, θ, v0 ; ν).

(3.3.64)

To estimate the size of 𝒲 = W s (Q) ∩ ℳε , we use the coordinates (β, γ) on the plane Πc defined in (3.3.14) and (3.3.16). In terms of these variables, the flow on ℳε in a neighborhood of Cεs is given by the equations r ̇ = νa(β, jε )r + O(νr 2 + v02 ), { { { β̇ = νλβ + νc(β, ν)r + O(νr 2 + v02 ), { { { 2 3 {v̇0 = Lε v0 + V∗ v0 + O(νrv0 + νv0 + v0 ), where V∗ = −4νj∗ (s, ν)S +

ν2 sin(θ∗ (s; ν)) √ω2 + νj∗ (s; ν)

J,

(3.3.65)

3.3 The perturbed nonlinear Schrödinger equation

| 247

and where a and c are smooth functions in (β, ν). Finally, to construct a local stable manifold to Q that contains Cεs , we linearize the flow along the flow on Cεs , β∗ (t; ν) = β0 exp[νλt], where 0 ≤ β0 ≤ s0 and t ≥ 0. Introducing the variable γ = β − β∗ (t, ν), equations (3.3.65) can be written as r ̇ = νa∗ (t, ν, β0 )r + N∗1 (t, r, γ, v0 ; ν), { { { β̇ = νλγ + νc∗ (t, ν)r + N∗2 (t, r, γ, v0 ; ν), { { { {v̇0 = Lε γ + V∗ (t, ν, β0 )v0 + N∗3 (t, r, γ, v0 ; ν),

(3.3.66)

where N∗1 (t, r, γ, v0 ; ν) = O(νr 2 + νγ 2 + v02 ),

N∗2 (t, r, γ, v0 ; ν) = O(νr 2 + νγ 2 + v02 ),

N∗3 (t, r, γ, v0 ; ν) = O(νr 2 + νγ 2 + νv02 + v03 ).

The linear part of equation (3.3.66) consists of a coupled system of ODEs and a PDE with a time-dependent coefficient. ODE Estimates. The fundamental solution of the ODEs is given by the 2 × 2 matrix [

A∗ (t, s; β0 , ν) Γ(t, s; β0 , ν)

0 ], exp[νλ(t − s)]

where t

A∗ (t, s; β0 , ν) = exp[∫ νa∗ ds󸀠 ], t

s

Γ(t, s; β0 , ν) = ν ∫ exp[νλ(t − α)]c∗ (α)A∗ (α, s)dα. s

To estimate A∗ , we note that a∗ can be written as a∗ = μ + a,̄ |a|̄ ≤ Cβ0 exp[νλt] ≤ Cs0 exp[νλt], where μ is given in equation (3.3.10). This implies the following estimate on A∗ for t, s ≥ 0: C1 exp[νμ(t − s)] ≤ A∗ ≤ C2 exp[νμ(t − s)],

248 | 3 Structures of small dissipative dynamical systems where C1 and C2 are constants independent of ε. To obtain a bound on Γ, we recall that |c∗ | ≤ Cs󸀠0 exp[2νλt], which implies 󵄨󵄨 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 |Γ| ≤ Cν󵄨󵄨󵄨∫ exp ν[λ(t − x) + 2λx + μ(x − s)]dx 󵄨󵄨󵄨, 󵄨󵄨 󵄨󵄨 󵄨 󵄨s for t, s ≥ 0. Since λ + μ = −2αν + O(ν2 ) < 0, we can bound the above integral for t, s ≥ 0 as follows: |Γ| ≤ Cν|t − s| exp[νλ(t − s)], where C is independent of ε. PDE Estimates. The difficulty in this estimate is the time dependence of the linear operator. To estimate the growth rate of the fundamental solution of the PDE, it is easiest to represent the linear operator in terms of its Fourier coefficients L (k) given by A=[

−εd(k) k − 4ω2 − εα1 − να2 2

−k 2 + εα1 ] −εd(k)

̂ where α1 and α2 are smooth functions of exp[νλt] and ν, d(k) ≥ α (the symbol of D), and where k = 2, 3, . . . The operator L (k) has eigenvalues given by λ1,2 = −εd(k) ± iD(k), D(k) = √(k 2 − εα1 )(k 2 − 4ω2 − εα1 − να2 ) and can be diagonalized by the matrix U(k) as [

1

iD(k) −k 2 +εα1

1

iD(k) ] . k 2 −εα1

Note that U is a bounded operator on the space [span{Πc , eu , es }]⊥ . Thus if we change ̂ variables from v0 to ω where v̂0 = U(k)ω(k), we obtain ̇̂ ̂ ̂ ω(k) = Λω(k) − U −1 U̇ ω(k). The term U −1 U̇ can be bounded as follows: 󵄨󵄨 −1 ̇ 󵄨󵄨 C 󵄨󵄨U U 󵄨󵄨 ≤ 2 ε exp[νλt], k

3.3 The perturbed nonlinear Schrödinger equation

| 249

̂ where C is independent of k and ε. Also ω(k) is given by the integral equation t

̂ ̂ 󸀠, ω(k) = F(t, s; k)ω̂ 0 (k) + ∫ F(t, s󸀠 ; k)U −1 U̇ ωds s

and can be estimated for t, s ≥ 0 by 󵄨󵄨 ̂ 󵄨󵄨 󵄨 󵄨 󵄨󵄨ω(k)󵄨󵄨 ≤ C exp[−εd(k)(t − s)]󵄨󵄨󵄨ω̂ 0 (k)󵄨󵄨󵄨 t

󵄨 ̂ 󵄨󵄨 󸀠 + C ∫ exp[−εd(k)(t − s󸀠 ) + νλs󸀠 ]󵄨󵄨󵄨ω(k) 󵄨󵄨ds . s

Since λ < 0 this implies that for t ≥ s ≥ 0, 󵄨󵄨 ̂ 󵄨󵄨 󵄨 󵄨 󵄨󵄨ω(k)󵄨󵄨 ≤ C exp[−εd(k)(t − s)]󵄨󵄨󵄨ω̂ 0 (k)󵄨󵄨󵄨. Finally, if we denote the fundamental solution of the PDE by U(t, s), we have the estimate 󵄩󵄩 󵄩 󵄩󵄩U(t, s)vi0 󵄩󵄩󵄩H 1 ≤ C exp[−εd(k)(t − s)]‖vi0 ‖H 1 , since d(k) ≥ α. Theorem 3.3.4. The point Q has a C 1 local stable manifold in ℳε that can be parameterized by (β, vi0 ), 𝒲 = {(r, β, vi0 ) : r = f (β, vi0 )},

for all β ∈ [0, s0 ] and ‖vi0 ‖H 1 ∈ [0, ε3/4 ]. Moreover, f (β, 0) = 0 and |r| ≤ Cε. Proof. Given the linear estimates and the normal form transformation, the proof of this theorem becomes standard. From equations (3.3.66) we set up integral equations for t ≥ 0 as ∞

{ { { r = ∫ A∗ (t, s)N∗1 ds, { { { { { t { { { { t { { { γ = [exp[νλ(t − s)]N∗2 + Γ(t, s)N∗1 ]ds, ∫ { { { { 0 { { { { t { { { {v = U(t, 0)v + ∫ U(t, s)N ds. { { i0 ∗3 { 0 0 {

250 | 3 Structures of small dissipative dynamical systems Using the linear estimates on A∗ , Γ and U and the order of magnitude of N∗ , we have: ∞

|r| ≤ C ∫ exp[νλ(t − s)][νr 2 + νγ 2 + ‖v0 ‖2H 1 ]ds, t

t

|γ| ≤ C ∫ exp[νλ(t − s)][1 + νλ(t − s)][νr 2 + νγ 2 + ‖v0 ‖2H 1 ]ds, 0

t

‖v0 ‖H 1 ≤ C ∫ exp[−εα(t − s)][ν(r 2 + γ 2 + ‖v0 ‖2H 1 ) + ‖v0 ‖3H 1 ]ds + e−εαt ‖vi0 ‖H 1 . 0

Scaling (r, γ, v0 ) by √ε yields O(√ε) a priori estimates on solutions. Since we only need O(εμ ), μ < 1, estimates, these inequalities imply the following for t ≥ 0 |r| ≤ Cε exp[−εαt], |γ| ≤ Cε exp[−εαt], ‖v0 ‖H 1 ≤ Cε3/4 exp[−εαt], provided ‖vi0 ‖ ≤ Cε3/4 . This a priori estimate allows us to apply Newton’s iteration to the integral equations to show that they have a unique solution. Definition 3.3.2. ∞

f (β0 , vi0 ) = ∫ A∗ (0, s)N∗1 ds,

(3.3.67)

0

where the dependence on β0 is implicit in A∗ , as well as in N∗1 . The differentiability of the function f follows from the differentiability of A∗ and N∗ .

3.3.4 Global integrability theory The unperturbed (ε = 0) NLS equation is a Hamiltonian system on the function 1 space He,p , − iqt =

δ H, δq̄

with the Hamiltonian H given by 2π

̄ H = ∫ [qx q̄ x − (qq)̄ 2 + 2ω2 qq]dx. 0

(3.3.68)

3.3 The perturbed nonlinear Schrödinger equation

| 251

It is well-known that this is a completely integrable Hamiltonian system, a fact whose verification begins with the Lax pair φx = U (λ) φ,

(3.3.69)

φt = V (λ) φ, where 0 U (λ) φ = iλσ3 + i [ q̄

q ], 0

V (λ) φ = i[2λ2 − (qq̄ − ω2 )]σ3 + [

0 2iλq̄ − q̄ x

2iλq + qx ], 0

def

and σ3 denotes the third Pauli matrix σ 3 = diag(1, −1). This overdetermined system is compatible (𝜕t φx = 𝜕x φt ) if and only if the coefficient q satisfies the NLS equation. Consequently, one can use this linear system to develop representations of solutions q(x, t) of the NLS. This observation is the starting point for the inverse spectral representation of q, which forms the basis of completely integrable soliton mathematics. We now focus on the “spatial flow” (3.3.69). The integration of the NLS equation is ̂ accomplished by employing the spectral theory of the differential operator L̂ = L(q), 0 d −[ L̂ = −iσ3 −q̄ dx

q ], 0

which is viewed as an operator on L2 (ℝ) with dense domain H 1 . In this L2 (ℝ) setting, the spectrum σ(L)̂ is defined as the closure of the set of complex λ for which there exists a solution of ̂ = λψ, Lψ

(3.3.70)

which is bounded for all x ∈ (−∞, ∞). Since the coefficient q is a periodic function of x, Floquet theory can be used to characterize this spectrum. Floquet theory begins from the fundamental matrix M = M(x; λ; q) which is defined as the 2 × 2 matrix valued solution of the linear problem (3.3.70) whose initial value at x = 0 is the identity matrix. Next, one introduces the transfer matrix T, T(λ; q) = M(2π; λ; q). Then the spectrum σ(L)̂ can be characterized as the set of all λ for which the 2×2 matrix T has eigenvalues on the unit circle. Since det T = 1, this is in turn determined by a single scalar function called the Floquet discriminant: 1 Δ : ℂ × He,p → ℂ by Δ(λ; q) = tr[T(λ; q)].

In terms of Δ, the spectrum is given by ̂ σ(L(q)) = {λ ∈ ℂ : Δ(λ; q) is real and − 2 ≤ Δ ≤ +2}.

252 | 3 Structures of small dissipative dynamical systems ̄ Proposition 3.3.2. (i) The Floquet discriminant Δ(λ; q, q), 1 1 Δ : ℂ × He,p × He,p → ℂ,

is entire in λ, q and q.̄ (ii) The first variation admits the following representation: 2π

δΔ(λ; q, q)̄ = ∫ [ 0

δΔ δΔ ̄ δq(x) + δq(x)]dx, ̄ δq(x) δq(x)

where 0 i δ Δ(λ; q, q)̄ = − tr [M −1 (x) [ 0 δq(x) 2

1 ] M(x + 2π)] , 0

0 δ i Δ(λ; q, q)̄ = − tr [M −1 (x) [ ̄ 1 δq(x) 2

0 ] M(x + 2π)] . 0

Here M(x) = M(x; λ, q, q)̄ denotes the fundamental matrix. A similar representation d exists for dλ Δ. To prove part (i) of this proposition, one writes the linear differential equation (3.3.70) for M as the integral equation M(x) = exp(iσ3 λx) x

+ ∫ exp[iσ3 λ(x − y)] [ 0

0 ̄ iq(y)

iq(y) ] M(y)dy. 0

Iteration produces a formal series representation, each term of which consists of polynomials in q and q̄ of the form ̄ n+1 ) ⋅ ⋅ ⋅ q(y ̄ m ). q(y1 ) ⋅ ⋅ ⋅ q(yn )q(y The series is shown to converge uniformly, hence, Δ is entire in q and q.̄ Analyticity in λ is established similarly. The first variation in part (ii) of the theorem is computed as follows: (L̂ − λ)M = 0, (L̂ − λ)δM = [

M(0) = I,

0 −δq̄

δq ] M, 0

δM = (0) = 0.

One then solves for δM(x) by variation of parameters, which together with the definition δΔ = tr δM(1), produces the representation.

3.3 The perturbed nonlinear Schrödinger equation

| 253

Proposition 3.3.3. (i) Floquet discriminants Poisson-commute: ̄ Δ(λ󸀠 ; q, q)} ̄ = 0, {Δ(λ; q, q),

∀λ, λ󸀠 ,

where the Poisson bracket is defined as 2π

{F, G} = ∫ i( 0

δF δG δF δG − )dx; δq δq̄ δq̄ δq

(ii) Δ(λ; q, q)̄ is a constant of the motion for the NLS equation since its Poisson bracket with the Hamiltonian vanishes: ̄ H(q, q)} ̄ = 0, {Δ(λ; q, q),

∀λ.

Thus, Δ(λ, q)̄ generates an infinite family of NLS constants of motion, one for each λ. Proof. The proof of this proposition can be found in the survey [79]. Operator L̂ is not self-adjoint. The spectrum occurs in bands, not necessarily real, which terminate at periodic or antiperiodic eigenvalues λj for which Δ(λj ) = ±2. Now we define critical points and multiple points: First, critical points are defined by the condition dΔ(λ; q) 󵄨󵄨󵄨󵄨 = 0, 󵄨 dλ 󵄨󵄨󵄨λc (q) while a multiple point, denoted λm , is a critical point for which Δ(λm ; q) = ±2. The algebraic multiplicity of λm is defined as the order of the zero of Δ(λ) ∓ 2. Usually it is 2, but it can exceed 2; when it does equal 2, we call the multiple point a double point and denote it by λd . The geometric multiplicity of λm is defined as the dimension of the eigenspace of L̂ at λm and is either 1 or 2. The real axis is a subset of the spectrum, ̂ Turning to properties of the spectrum of L̂ that are rather directly related to ℝ ⊂ σ(L). the non-self-adjointness of L,̂ we consider a critical point λc at which −2 < Δ(λc ) < 2. Such a critical point is a point of bifurcation of the spectrum. Many spectral figures may be found in [79]. Consider the example of q(x, t), constant and independent of x: q(x, t) = c exp{−i[2(c2 − ω2 )t − γ]}. In this case, two linearly independent solutions of the Lax pair are given by ψ(±) 1 ] = exp{±i[κ(λ)(x + 2λt)]} ψ(±) 2

[

×[

c exp{−i[2(c2 − ω2 )t − γ]/2} ], (±κ(λ) − λ) exp{i[2(c2 − ω2 )t − γ]/2}

(3.3.71)

254 | 3 Structures of small dissipative dynamical systems in these formulas, κ(λ) is given by κ(λ) = √λ2 + c2 . The spectrum of the linear operator L̂ for coefficients independent of x is easily computed from the Floquet discriminant: 1

Δ[λ; q(⋅, t; c, γ)] = 2 cos 2πκ(λ) = 2 cos[2π(λ2 + c2 ) 2 ], from which we see that λj is given by j κ(λj ) = . 2 Notice that the continuous spectrum consists of the real axis together with one band of spectrum on the imaginary axis. All critical points except the origin are double points. For c ≃ ω, one double point lies in the upper half-plane on the band of spectrum. Its conjugate is also a double point. All other double points (countable in number) reside on the real axis. Using Bäcklund (Darboux) transformations, one can exponentiate these linearized solutions to obtain global solutions (homoclinic orbits) of the NLS equation. Fix a periodic solution q of an NLS that is quasi-periodic in t and for which the linear operator L̂ has a complex double point ν of geometric multiplicity 2 associated with an NLS instability. We denote two linearly independent solutions of the Lax pair at λ = ν by (ϕ+ , ϕ− ). Thus, a general solution of the linear system at (q, ν) is given by ϕ(x, t; ν; c+ , c− ) = c+ ϕ+ + c− ϕ− .

(3.3.72)

We use ϕ to define a transformation matrix G by λ−ν G = G(λ; ν; ϕ) = N [ 0

0 ] N −1 , λ−ν

(3.3.73)

where ϕ N=[ 1 ϕ2

−ϕ̄ 2 ]. ϕ̄ 1

(3.3.74)

Then we define Q and Ψ by def

Q(x, t) = q(x, t) + 2(ν, ν)̄

ϕ1 ϕ̄ 2 , ϕ1 ϕ̄ 1 + ϕ2 ϕ̄ 2

(3.3.75)

and Ψ(x, t; λ) ≡ G(λ; ν; ϕ)ψ(x, t; λ),

(3.3.76)

where ψ solves the Lax pair at (q, λ). Formulas (3.3.75) and (3.3.76) are the Bäcklund transformations.

3.3 The perturbed nonlinear Schrödinger equation

| 255

Theorem 3.3.5. Let q(x, t) denote a periodic solution of NLS which is linearly unstable ̂ with an exponential instability associated with a complex double point ν in σ(L(q)). Furthermore, assume that the complex double point ν has geometric multiplicity 2. Let (ϕ+ , ϕ− ) denote an eigenbasis for the Lax pair at (q, ν), and define Q(x, t) and Ψ(x, t; λ) by (3.3.75) and (3.3.76). Then (i) Q(x, t) is a solution of NLS with spatial period 2π; ̂ ̂ (ii) σ(L(Q)) = σ(L(q)); (iii) Q(x, t) is homoclinic to q(x, t) in the sense that Q(x, t) → qθ± exponentially as e−σν |t| as t → ±∞. Here qθ± is a “torus translate” of q, σν is the nonvanishing growth rate associated with the complex double point ν, and explicit formulas can be developed for this growth rate and for the translation parameters θ±; (iv) Ψ(x, t; λ) solves the linear system (3.3.69) at (Q, λ). For the above example q(x, t) = c exp{−i[2(c2 − ω2 )t − γ]} = ceiθ , Δ(x; q) = 2 cos[2πκ(λ)], κ(λ) = √λ2 + c2 ; 1 i κ(ν) = ⇒ ν = √4c2 − 1. 2 2 Furthermore, from (3.3.28) we know that q is unstable, with linearized growth rate given by σ = √4c2 − 1 = 2|ν|. Using these formulas, as well as the eigenfunctions (3.3.71), specializes the general formula for the homoclinic orbit QH to qh± − {

cos 2p − i sin 2p tanh τ ± sin p sech τ cos x }q, 1 ∓ sin p sech τ cos x

where τ = σ(t + t0 ),

eip =

1 + iσ . 2c

Here ± denotes the two lobes of ∞ in the trouser diagram. Notice that − cos x = cos(x + π) which shows that one lobe (+) represents an excitation centered at x = 0, while the other (−) an excitation centered at x = π. Thus, (3.3.72) provides an explicit representation of a “whiskered circle”, while from another viewpoint, it provides an explicit representation of the unstable manifold W u (S) = W s (S) =

⋃

γ,t0 →±∞

qh± (t; γ, t0 , c).

256 | 3 Structures of small dissipative dynamical systems 1 Fix a potential q0 ∈ He,p that has either a purely real or a purely imaginary critical c point λ ,

󵄨󵄨 𝜕 󵄨 Δ(λ; q0 )󵄨󵄨󵄨 = 0. 󵄨󵄨λc 𝜕λ 1 Let Nb = N − b(q0 ) denote a small neighborhood of q0 ∈ He,p , and consider the critical c c point as a functional on this neighborhood, λ = λ (q):

󵄨󵄨 𝜕 󵄨 Δ(λ; q0 )󵄨󵄨󵄨 = 0; 󵄨󵄨λc (q) 𝜕λ

λc (q0 ) = λc .

In terms of this purely real (or purely imaginary) critical point, we introduce an important invariant F : Nb → ℝ given by def

F = Δ(λc (q); q). Proposition 3.3.4. F : Nb → ℝ is smooth, provided

d2 Δ(λ, q) dλ2

≠ 0 for all q ∈ Nb .

Proof. To prove this, one calculates δF δ δλc δΔ = Δ(λc (q); q) = Δ󸀠 (λc (q); q) + δq δq δq δq 󵄨 󵄨 δΔ 󵄨󵄨 = . 󵄨 δq 󵄨󵄨󵄨λ=λc (q) Here λc (q) is smooth, as the following calculation shows: Δ󸀠 (λc (q); q) = 0, Δ󸀠󸀠 (λc (q); q)

δλc δΔ󸀠 + = 0, δq δq

and if Δ󸀠󸀠 (λc (q); q) ≠ 0, one can continue to differentiate to any order. Remark. The specific eigenfunctions (3.3.71) used in the example of a plane wave, independent of x, were Bloch functions. Let Ψ± (x, λ) denote Bloch functions, that is, solutions of the Lax pair at [q, λ]. These functions are defined by the transfer condition across one period: Ψ(x + 2π, λ) = ρ(λ)Ψ(x, λ).

(3.3.77)

Here ρ(λ) denotes the Floquet multiplier, which is related to the Floquet discriminant by 1 ρ(λ) = [Δ(λ) + √Δ2 (λ) − 4]. 2

(3.3.78)

3.3 The perturbed nonlinear Schrödinger equation

| 257

On the Riemann surface for (λ, √Δ2 (λ) − 4), the functions ρ and Ψ are well-defined, and Ψ± (x, λ) denote the values of Ψ on the two sheets over λ. At branch points (simple periodic or antiperiodic points), the two sheets touch and the Ψ± become linearly dependent. In any case, for fixed λ, these Bloch eigenfunctions can be represented explicitly in terms of the columns of the fundamental matrix M(x; λ) = column{Y (1) (x, λ), Y (2) (x, λ)} Ψ± (x; λ) = α± {M21 (1; λ)Y (1) (x, λ) + [M22 (1; λ) − ρ± (λ)]Y (2) (x, λ)},

(3.3.79)

where α± denotes normalization constants. The gradient of the Floquet discriminant admits a beautiful representation in terms of these Bloch functions: Corollary 3.3.2. For λ ≠ a branch point (i. e., λ ≠ a periodic or antiperiodic eigenvalue), √Δ2 (λc ) − 4 ψ+2 (x, λ)ψ−2 (x, λ) δ Δ(λ; q, q)̄ = i [ ], δq W[ψ+ , ψ− ] −ψ+1 (x, λ)ψ−1 (x, λ)

(3.3.80)

where q = (q, q)̄ T and W[ψ+ , ψ− ] denotes the Wronskian of ψ+ and ψ− . The representation is extended to the periodic or antiperiodic eigenvalues by continuity. With this representation one has the following proposition: Proposition 3.3.5. grad F(q, q)̄ = i

√Δ2 (λc ) − 4 ψ+2 (x, λ)ψ−2 (x, λ) [ ]. W[ψ+ , ψ− ] −ψ+1 (x, λ)ψ−1 (x, λ)

(3.3.81)

From critical points of F, one obtains the following theorem: Theorem 3.3.6. The potential q is a critical point of the functional F if and only if λc (q) is a multiple point with geometric multiplicity 2. We begin from equation (3.3.81) for the grad F, √Δ2 (λc ) − 4 Ψ2+ ⋅ Ψ2− 𝜕F = lim [ ], 𝜕q λ→ν W[Ψ+ , Ψ− ] −Ψ1+ ⋅ Ψ1−

(3.3.82)

where Ψ± (x, λ) are a Hoquet basis at (QH , ν). We compute this limit using the Bäcklund formulas. The result is C C W[Ψ+ , Ψ− ] ϕ̄ 21 𝜕F = Cν + − [ ̄ 2] , −ϕ2 𝜕q |ϕ|4 where the constant Cν is given by Cν ≡ i(ν − ν)̄ √Δ(ν)Δ󸀠󸀠 (ν).

(3.3.83)

258 | 3 Structures of small dissipative dynamical systems Choosing q∗ = c exp{−i[2(c2 − ω2 )t − γ]} = ceiθ , then δF [(∓ sin p cosh τ ± i cos p sinh τ) cos x + 1] iθ { =a ce , { { { δq [1 ∓ sin p sech τ cos x]2 { { { { δF = δF , { δq̄ δq

(3.3.84)

where τ = σ(t − t0 ), tan p = σ, σ = √4c2 − 1 and a = 2π sin2 p sech2 τ. From this representation, we see explicitly that F󸀠 (qh (t)) ≠ 0,

lim F󸀠 (qh (t)) → 0.

t→±∞

3.3.5 Persistent homoclinic orbit (ε ≥ 0) We will combine the local analysis given so far with explicit global information from the unperturbed integrable system to establish the existence of a homoclinic orbit to the saddle point Q for the perturbed NLS equation, qt = iH 󸀠 (q) + εG(q),

(3.3.85)

2

̂ −1 with B̂ a bounded dissipative where H (q) = −qxx −2(qq̄ −ω )q and G(q) = −αq −βBq operator. More specifically, we combine geometric singular perturbation with the construction of a Melnikov function to prove the existence of such orbits. The argument proceeds in two steps, which we shall call the “first measurement” and the “second measurement.” In the first measurement we will construct a distance function Δ (not the Floquet discriminant of integrability theory) whose zeros correspond to orbits that do not lie in the invariant plane Πc and are asymptotic to the saddle point Q in backward time and asymptotic to ℳε in forward time. The second measurement consists of constructing a function d whose zeros correspond to one of these orbits intersecting a fiber whose base point is in the stable manifold of Q. Therefore, from the definition of fibers, the simultaneous vanishing of A and d ensures the existence of a homoclinic orbit to the point Q. We first construct the first measurement. Recall that the invariant plane Πc ⊂ ℳ ∩ ℳε and that Q has a one-dimensional unstable manifold in Πc given by 󸀠

1/2

q = (ω2 + √εju (s)) eiθ(s) . Let qb be a point on the above curve corresponding to s = sb . The unperturbed flow 2 2 has an orbit that passes through qb at t = 0, q = rb e−i(2(rb −ω )t−θb ) , and an orbit qh that is asymptotic to the above orbit as t → −∞, qh (t) = (

cos 2p − i sin 2p tanh τ + sin p sech τ cos x ) 1 − sin p sech τ cos x

× rb exp{−i[2(rb2 − ω2 )t − θb + 2p]},

3.3 The perturbed nonlinear Schrödinger equation

| 259

where tan p = √4rb2 − 1,

τ = (tan p)(t + t0 ), 1

rb eiθb = (ω2 + √εju (sb )) 2 eiθ(sb ) . The asymptotic behavior of qh as t → ∞ is given by 2

2

qh (t) → rb e−i(2(rb −ω )t−θb +4p) , which implies that we have a phase shift of −4p, e−4ip = [

1 − i√4rb2 − 1 2rb

4

] .

Choose t0 so that the orbit is a distance of order δ from qb (0), δ 󵄩 󵄩󵄩 󵄩󵄩qh (0) − qb 󵄩󵄩󵄩H 1 = . 4 Using the explicit expression of the unperturbed orbit qh , we see that there exists a T∗ (δ) such that dist(qh (t), Sω ) ≤ δ/4 for t ≥ T∗ provided ε is small. The point qb is the base point of fibers of length δ. The v-coordinates of qb are (0, 0, vc = ηc ), where ηc = (√εju (sb ), θu (sb ), v0 = 0), and the fibers through can be parameterized by 0 ≤ ηu ≤ δ as follows: vc = f u (ηc , ηu ; ε),

vu = ηu + hcu (vc ; ε), vs = hs (vu , vc ; ε).

From equation (3.3.85), qh (0) belongs to an unperturbed fiber; that is, ε = 0 with ηu = η̃ u ≤ δ/2. Let qδ be the point on the perturbed fiber corresponding to ηu = η̃ u (see Figure 3.19). Since the functions f , h ∈ C 2 (ε), we have 󵄩󵄩 󵄩 󵄩󵄩qδ − qh (0)󵄩󵄩󵄩H 1 ≤ Cε. Denote by qε (t) the solution of the perturbed equation with initial value qε (t) = qδ . From the fiber construction, we have that the orbit qε (t) is asymptotic to Q as t → −∞,

Figure 3.19: The initial points on the Wεr fibers.

260 | 3 Structures of small dissipative dynamical systems and by continuous dependence on the initial data we have that for any finite time T and for 0 ≤ t ≤ T, 󵄩 󵄩󵄩 󵄩󵄩qε (t) − qh (t)󵄩󵄩󵄩H 1 ≤ C(T)ε.

(3.3.86)

Define q0 = qh (T∗ ),

ql = qε (T∗ ).

The point q0 belongs to W0cs , since as t → ∞ we have qh → Πc , and q0 is at a distance less then δ/2 from the plane Πc : 2

2

q0 = rb e−i(2(rb −ω )t−θb +4p) + δq0 , where ‖δq0 ‖H 1 ≤ δ2 . By (3.3.86) we have ‖q0 − ql ‖H 1 ≤ C(δ)ε < dist(ql , Sω ) ≤

δ 2

δ , 4

for small ε.

The distance between ql and Wεcs will be measured along the normal to W0cs at q0 . This is accomplished in the following manner: The manifold W0cs is given as a graph vu = hu (vs , vc ; 0), where the function hu has small derivative. This implies that the vector in the vu -direction, V = (1, 0, 0), is transversal to W0cs . This vector V is the eigenfunction eu (x) = 1 (1 + iσ) cos x given in (3.3.29). W0cs is also characterized by {q ∈ H 1 : F(q) + τ = 0}. 2√πω Therefore the transversality of eu translates into ⟨F󸀠 (q0 ), eu ⟩ ≠ 0,

(3.3.87)

where the above is a shorthand for the duality pairing ⟨

δF δF ̄ , δq⟩ + ⟨ , δq⟩. δq δq̄

Since hu is C 2 in all of its arguments, then for every q in an ε-neighborhood of q0 , the straight line through q in the direction of eu intersects Wεcs at a point of distance ε from q. Let qs be the intersection of the line through ql with the manifold Wεcs . By (3.3.87), we can define Δ = ⟨F󸀠 (q0 ), ql − qε ⟩ as a measure of distance between ql and qs , see Figure 3.20.

3.3 The perturbed nonlinear Schrödinger equation

| 261

Figure 3.20: Schematic diagram of the first measurement.

To actually calculate Δ, we define, for t ≤ 0, the orbits q∗ (t) = qh (t + T∗ ), qu (t) = qε (t + T∗ ),

while, for t ≥ 0, we define qs (t) to be the solution of the cutoff flow with initial data qs and q∗ (t) = qh (t + T∗ ). Note that since qh remains in a δ-neighborhood of Sω for t ≥ T∗ , q∗ is also a solution of the cutoff equation, and at t = 0 the initial data of all orbits, ql , qs , and q∗ , are order ε apart. For t ≤ −T∗ both orbits qu and q∗ remain in a δ-neighborhood of Sω . Therefore by Gronwall inequality applied to equation (3.3.35) and from (3.3.86), we have for t < 0 ‖qu − q∗ ‖H 1 ≤ C(δ)e−δt ε.

(3.3.88)

For t ≥ 0 both orbits, qs , and q∗ , are solutions of the cutoff equations. Again by Gronwall inequality we have ‖qs − q∗ ‖H 1 ≤ Ceδt ε.

(3.3.89)

These orbits allow us to introduce the measurements Δ− (t) = ⟨F󸀠 (q∗ (t)), qu (t) − q∗ (t)⟩, +

󸀠

Δ (t) = ⟨F (q∗ (t)), qs (t) − q∗ (t)⟩, −

t ≤ 0, t ≥ 0,

+

Δ = Δ (0) − Δ (0). Proposition 3.3.6. The distance Δ is given by ∞

Δ = ε ∫ ⟨F󸀠 (q∗ (t)), G(q∗ (t))⟩dt + O(ε2 ). −∞

Proof. Note that q∗ is a smooth function in (x, t), F󸀠 (q∗ ) is smooth in q∗ , and by Proposition 3.3.4 F󸀠 (q∗ ) is a C 1 function in t with values in H 1 . Since qu and qs are C 1 functions in t with values in H −1 , we conclude that Δ− and Δ+ are C 1 functions of t. We start by computing the time derivative of Δ− (t) using the notation of equation (3.3.85): Δ̇ − (t) = ⟨F󸀠󸀠 (q∗ )q̇ ∗ (t), qu − q∗ ⟩ + ⟨F󸀠 (q∗ ), q̇ u − q̇ ∗ ⟩ = ⟨F󸀠󸀠 (q∗ )iH 󸀠 (q∗ ), qu − q∗ ⟩

+ ⟨F󸀠 (q∗ ), iH 󸀠 (qu ) − iH 󸀠 (q∗ ) + εG⟩.

(3.3.90)

262 | 3 Structures of small dissipative dynamical systems By expanding the nonlinear part of H 󸀠 (q) around q∗ and using the fact that both orbits qu , and q∗ are bounded for t ≤ 0, we obtain H 󸀠 (qu ) − H 󸀠 (q∗ ) = H 󸀠󸀠 (q∗ )(qu − q∗ ) + R(qu , q∗ ) ‖R‖H 1 ≤ C‖qu − q∗ ‖2H 1 .

(3.3.91)

Equation (3.3.85) can be written as Δ̇ − = ⟨F󸀠󸀠 (q∗ )iH 󸀠 (q∗ ), qu − q∗ ⟩ + ⟨F󸀠 (q∗ ), iH 󸀠󸀠 (q∗ )(qu − q∗ )⟩ + ε⟨F󸀠 (q∗ ), G(qu )⟩ + ⟨F󸀠 (q∗ ), iR⟩.

(3.3.92)

We have {F(q), H(q)} = 0, which implies that ⟨F󸀠󸀠 (q∗ )iH 󸀠󸀠 (q∗ ), qu − q∗ ⟩ + ⟨F󸀠 (q∗ ), iH 󸀠 (q∗ )(qu − q∗ )⟩ = 0, and the equation for Δ̇ − simplifies to Δ̇ − = ε⟨F󸀠 (q∗ ), G(qu )⟩ + ⟨F󸀠 (q∗ ), iR⟩. Finally, from (3.3.84), (3.3.88), and (3.3.91) we have for t < 0 󵄩󵄩 󸀠 󵄩 σt 󵄩󵄩F (q∗ )󵄩󵄩󵄩H 1 ≤ Ce , ‖qu − q∗ ‖H 1 ≤ Ce−δt ε, 󵄩󵄩 󵄩󵄩 −2δt 2 󵄩󵄩R(t)󵄩󵄩H 1 ≤ Ce ε , with 2δ < σ and Δ− (t) → 0 as t → −∞. This implies 0

Δ (0) = ε ∫ ⟨F󸀠 (q∗ ), G(q∗ )⟩dt + O(ε2 ). −

−∞

To obtain a similar expression for Δ+ , we repeat the same argument as above, thus expanding the cutoff equations around q∗ and using the fact that {F(q), H(q)} = 0, we obtain ̃ Δ̇ + = ⟨F󸀠 (q∗ ), Gδ (qs )⟩dt + ⟨F󸀠 (q∗ ), iR⟩, where R̃ is the remainder from expanding the cutoff flow Hδ󸀠 (qs ) around q∗ 󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩R(t)󵄩󵄩H 1 ≤ C(δ)‖qs − q∗ ‖H 1 , and Gδ (qs ) is the cutoff perturbation evaluated at qs . Again from (3.3.84), (3.3.89), and (3.3.91), we have for t ≥ 0 󵄩󵄩 󸀠 󵄩 −σt 󵄩󵄩F (q∗ )󵄩󵄩󵄩H 1 ≤ Ce ,

3.3 The perturbed nonlinear Schrödinger equation

| 263

‖qs − q∗ ‖H 1 ≤ Ceδt ε, 󵄩󵄩 ̃ 󵄩󵄩 2δt 2 󵄩󵄩R(t)󵄩󵄩H 1 ≤ Ce ε . Therefore the derivative of Δ+ can be written as 0

Δ (0) = −ε ∫ ⟨F󸀠 (q∗ ), Gδ (q∗ )⟩dt + O(ε2 ) +

−∞ 0

= −ε ∫ ⟨F󸀠 (q∗ ), G(q∗ )⟩dt + O(ε2 ), −∞

and the distance Δ has an expansion in t given by 0

Δ = Δ (0) − Δ (0) = ε ∫ ⟨F󸀠 (q∗ ), G(q∗ )⟩dt + O(ε2 ). −

+

−∞

Since the base point of q∗ depends on ε, 1

rb eiθb = (ω2 + √εju (sb )) 2 eiθ(sb ) , we can simplify the expression for Δ further by using the homoclinic orbit qω (t) whose base point is ωeiθb and which is order O(√ε) away from q∗ (t). Thus, using the explicit form of G we obtain the following: Corollary 3.3.3. The distance Δ has an expansion in ε given by Δ = εM(α, β, θb ) + O(ε3/2 ), { { { { ∞ { { { 󸀠 {M(α, β, θb ) = ∫ ⟨F (qω (t)), G(qω (t))⟩dt { { { −∞ { { { = −[αMα + βMβ + M(θb )], { where ∞

Mα = ∫ ⟨F󸀠 (qω (t)), qω (t)⟩dt, −∞ ∞

̂ ω (t)⟩dt, Mβ = ∫ ⟨F󸀠 (qω (t)), Bq −∞ ∞

M(θb ) = ∫ ⟨F󸀠 (qω (t)), 1⟩dt. −∞

(3.3.93)

264 | 3 Structures of small dissipative dynamical systems From the above corollary we conclude that to find a zero for Δ it is sufficient to find a nondegenerate zero of M(α, β, θb ) and then use the implicit function theorem. The dependence of M on θb can be computed by using the explicit formula for F󸀠 given in (3.3.84). [(− sin p cosh τ + i cos p sinh τ) cos x + 1] iθ δF = 2π sin2 p sech2 τ ce , δq (1 − sin p sech τ cos x)2 to obtain 2π

∞

M(θb ) = cos(θb − 2p0 ) ∫ dτ ∫ dx 0

−∞

4πω sin2 p0 sech2 τ σA2

× (sech τ − sin p0 cos x), where p0 = arctan √ω2 − 1 and A = 1 − sin p0 sech τ cos x. Assuming that Mα or Mβ is nonzero, the function M(α, β, θb ) = αMα + βMβ + M0 cos(θb − 2p0 )

(3.3.94)

has a nondegenerate zero. This implies that we can choose our parameters so that Δ = 0; that is, ql = qs ∈ Wεcs . We consider the second measurement. First, recall that the unperturbed orbit q∗ (t) = qh (t + T ∗ ) is asymptotic as t → ∞ to the orbit 2

2

rb e−i(2(rb −ω )(t+T

∗

)−θb +4p)

,

which implies that q0 = qh (T ∗ ) belongs to an unperturbed fiber whose base point is 2

2

q0,b = rb e−i(2(rb −ω )T

∗

−θb +4p)

.

The point ql ∈ Wεcs belongs to a perturbed fiber whose base point ql,b ∈ ℳε does not necessarily lie on the plane Πc . The distance between q0,b and ql,b is of order ε, which can be proved as follows, see Figure 3.21. The point q0 has v-coordinates v0,c = f s (η0,s , η0,c ; 0),

v0,s = η0,s + hcs (v0,c ; 0), v0,u = hu (v0,s , v0,c ; 0),

where η0,s ∈ [−δ, δ], and η0,c = (√εju (sb ), 2(rb2 − ω2 )T ∗ − θb + 4p, 0). The base point q0,b corresponds to ηs = 0 in the above equations. The functions f s and hsc are C 1

3.3 The perturbed nonlinear Schrödinger equation

| 265

Figure 3.21: The base points of the fibers.

functions with well-defined inverses, and the points q0 and ql are at a distance of order O(ε); therefore, by the inverse function theorem, the point ql has parameters (ηl,c , ηl,s ), which differ from (η0,c , η0,s ) by order ε. The base point ql,b has v-coordinates vl,c = f s (ηl,s , 0; ε), vl,s = hcs (vl,c ; ε),

vl,u = hu (vl,s , vl,c ; ε),

which differ from the v-coordinates of ql,b ∈ ℳε by order ε. This difference implies that ‖q0,b − ql,b ‖H 1 = O(ε).

(3.3.95)

To construct a distance function from ql,b ∈ ℳε to W, the stable manifold of Q, we recall that the curve Cεs ⊂ W ∩ Πc , expressed in the y = (j, θ)-coordinates, is given by 1 y∗ (s, √ε) for s ∈ [0, s0 ]. In terms of z, where z = (ω2 + √εj) 2 eiθ , Cεs can be represented by 1

z∗ (s; √ε) = (ω2 + √εj0 (s)) 2 eiθ0 (s) + O(ε), where (j0 (s), θ0 (s)) is the planar homoclinic orbit to the unperturbed (j, θ) equations. The manifolds ℳε and W are given by the following: 1

c

c

ℳε = {v ∈ H : vu = hε (v0 ; ε), vs = hs (v0 ; ε)},

W = {v ∈ ℳε : r = g(s, v0 ; ε)}, 3

where s ∈ [0, s0 ] and ‖v0 ‖H 1 ∈ [0, ε 4 ]. Here r is the signed Euclidean distance on the plane Πc from a point y to the curve Cεs . For every point q ∈ ℳε that is a distance of order O(ε) from Πc and a distance of order O(√ε) from Sω , we associate a point q̃ ∈ W as follows, see Figure 3.22. Here q can be represented as (zq , vq,0 ) where zq = (ω2 + √εjq )eiθq . In the y = (j, θ)-coordinates, let y∗ (sq ) ∈ Cεs be the point where the distance from yq = (jq , θq )

266 | 3 Structures of small dissipative dynamical systems

Figure 3.22: The map from q to q.̃

to Cεs is achieved. By the implicit function theorem, such a point exists and is unique provided (jq , θq ) is in a fixed neighborhood O of C0s . On the line joining yq to y∗ (sq ), let def

yq̃ to be the point that is at a distance rq̃ = g(sq , vq ; √ε) from y∗ (sq ). In the plane Πc this point has coordinates zq̃ = (ω2 + √εjq̃ )eiθq̃ . Define q̃ ∈ W to be the point corresponding to (zq̃ , vq,0 ). To measure the distance from q to q,̃ we introduce the function d(q) = E(yq ) − E(yq̃ ), where E is the Hamiltonian of the unperturbed ODE given in (3.3.12): 1 E(j, θ) = j2 − ω(sin θ + αωθ). 2

Proposition 3.3.7. The map q → q̃ has a fixed point if and only if d(q) = 0. Moreover, d(q) has an expansion given by d(q) = E(jq , θq ) − E0 + O(√ε). Proof. To show that the zeros of d correspond to fixed points of q → q,̃ we note that in a neighborhood of C0s we have that the level curves of E intersect the normal to Cεs in exactly one point. Therefore on the line joining yq to y∗ (sq ), which is normal to Cεs , we can use E as a measure of distance, and E(yq ) = E(yq̃ ) would imply yq = yq̃ or, equivalently, zq = zq̃ . Since the coordinates of q and q̃ are the same by definition, we conclude that the zeros of d(q) = E(yq ) − E(yq̃ ) correspond to fixed points of q → q.̃ To expand d(q), we note that 󵄨󵄨 󵄨 󵄨󵄨yq̃ − y∗ (sq )󵄨󵄨󵄨 = |rq̃ | = O(ε), since from equation (3.3.67) |rq̃ | = O(ε). Moreover, y∗ (sq ) is order √ε away from C0s , the unperturbed stable manifold; therefore E(yq̃ ) = E(y∗ (sq )) + O(ε) = E0 + O(√ε). This completes the proof of the proposition.

3.3 The perturbed nonlinear Schrödinger equation

| 267

We will use the function d to measure the distance from ql,b to W. From (3.3.95) we have ql,b and q0,b are an order O(ε) apart, which implies that d(ql,b ) = d(q0,b ) + O(ε). Moreover, since the base point of q0,b is 1

rb ei(θb −4p) = (ω2 + √εju (sb )) 2 ei(θb −4p) , we have the following corollary: Corollary 3.3.4. The distance from ql,b to W can be measured by d(ql,b ) = ω[2 sin 2p0 cos(θb − 2p0 ) + 4αωp0 ] + O(√ε). Proof. From the definition of d we have q0,b = E(ju (sb ), θb − 4p) − E0 , and since q0,b ∈ Cεu which is an order O(√ε) perturbation of C0u , we also have E(ju (sb ), θb ) − E0 = O(√ε). This fact implies that d(q0,b ) = E(ju (sb ), θb − 4p) − E(ju (sb ), θb ) + O(√ε)

= −ω[sin(θb − 4p) − sin θb + 4αωp] + O(√ε) = ω[2 sin 2p cos(θb − 2p) + 4αωp] + O(√ε).

Finally, observing that 1

1

p = arctan(rb2 − 1) 2 = arctan(ω2 − 1) 2 + O(√ε) = p0 + O(√ε). Concludes the proof of the corollary. Fix α ∈ (0, ω1 ) and consider M and d̃ as functions of θb and β, M(α, β, θb ) = αMα + βMβ + M0 cos(θb − 2p0 ), ̃ , α) = 2 sin 2p cos(θ − 2p ) + 4αωp . d(θ b

0

b

0

0

In order to prove the existence of a homoclinic orbit to Q, it is sufficient to show that M and d̃ vanish in a nondegenerate manner for some β > 0 and θb ∈ (θmin , θ0 ), which

268 | 3 Structures of small dissipative dynamical systems is the range of 0 for the unperturbed homoclinic orbit in the plane: √1 − α2 ω2

) − π, αω sin(θmin ) + αωθmin = sin θ0 + αωθ0 ,

θ0 = arctan(

where θmin < θ0 . For d̃ to vanish, the range of α has to be restricted further to be in the interval def

(0, α∗ ) = (0,

2ωp0 1 ) ∩ (0, ). ω sin 2p0

Moreover, if Mβ is not equal to 0, then we can solve d̃ = 0 and M = 0 at the point cos(θb − 2p0 ) = β=−

α[Mα +

2αωp0 , sin 2p0

(3.3.96)

.

(3.3.97)

2αωp0 M ] sin 2p0 0

Mβ

By the implicit function theorem, for ε small we can solve d = Δ = 0 in a small neighborhood of the point given by equations (3.3.96) and (3.3.97). Theorem 3.3.7. Fix ω ∈ ( √12 , 1) and α ∈ (0, α∗ ). If θb and β given by equations (3.3.96) satisfy θb ∈ (θmin , θ0 ) and β > 0, then for small ε, equation (3.3.85) has a symmetric pair of homoclinic orbits. Proof. By the implicit function theorem, for ε small we can solve d = Δ = 0 in a small neighborhood of the point given by equations (3.3.96). Since d = 0, the point ql lies on a fiber whose base point ql,b ∈ W the stable manifold of Q. Therefore the orbit qs (t) for t ≥ 0 remains in a small neighborhood of Sω . This fact implies that qs (t) solves the original equation, and the orbit q(t) = {

qu (t) t ≤ 0, qs (t) t ≥ 0

is homoclinic to Q. Since the unperturbed system has two homoclinic orbits qh± , this argument establishes the existence of a symmetric pair.

3.4 Center manifold theory in infinite dimensions Center manifold theory forms one of the cornerstones of the theory of dynamical systems. In its simplest form, the center manifold theory reduces the study of a system near a (non-hyperbolic) equilibrium point to that of an ordinary differential equation on a low-dimensional invariant center manifold. For finite-dimensional systems this

3.4 Center manifold theory in infinite dimensions | 269

means a (sometimes considerable) reduction of the dimension, leading to simpler calculations and a better geometric insight. But in the infinite-dimensional case, how do we construct center manifold theory? And how many differences are there with the finite-dimensional case? We introduce the work of Vanderbauwhede and Iooss et al. [95], who give general results, and show how the hypothesis can be verified for some examples. Let X, Y and Z be Banach spaces, with X continuously embedded in Y, and Y continuously embedded in Z. In most applications the embeddings will be dense, but (except when explicitly stated) we do not need this for our theory. Let A ∈ L (X, Z) and g ∈ C k (X, Y) for some k ≥ 1. We will consider differential equations of the form ẋ = Ax + g(x).

(3.4.1)

By a solution of (3.4.1) we mean a continuously differentiable mapping x : I → Z, where I is an open interval, and such that the following properties hold: (i) x(t) ∈ X, ∀t ∈ I, and x : I → Z is continuous; ̇ = Ax(t) + g(x(t)), ∀t ∈ I. (ii) x(t) Let E and F be Banach spaces, V ∈ E an open subset, k ∈ ℕ and η ≥ 0. Then we define 󵄨󵄨 Cbk (V; F) = {w ∈ C k (V; F) 󵄨󵄨󵄨 󵄨 def 󵄩 󵄩 |w|j,V = sup 󵄩󵄩󵄩Dj w(x)󵄩󵄩󵄩 < ∞, 0 ≤ j ≤ k}, x∈V

and 󵄨󵄨 󵄨 Cb0,1 (E; F) = {w ∈ C 0,1 (E; F) 󵄨󵄨󵄨 󵄨󵄨 def

|w|Lip =

sup

x,y∈E,x =y̸

‖w(x) − w(y)‖ < ∞}. ‖x − y‖

In case V = E we write |w|j for |w|j,E . We also define 󵄨󵄨 def def BC η (ℝ; E) = {w ∈ C 0 (ℝ; E) 󵄨󵄨󵄨 ‖w‖η = sup e−η|t| ‖w‖E < ∞}. 󵄨 t∈ℝ We will impose the following basic hypothesis on A. (H) There exists a continuous projection Πc ∈ L (Z; X) onto a finite-dimensional subspace Zc = Xc ⊂ X such that AΠc x = Πc Ax,

∀x ∈ X,

and such that if we set def

Zh = (I − Πc )(Z),

def

Xh = (I − Πc )(X),

def

Yh = (I − Πc )(Y),

270 | 3 Structures of small dissipative dynamical systems def

Ac = A|Xc ∈ L (Xc ),

def

Ah = A|Xh ∈ L (Xh , Zh ),

then the following hold: (i) σ(Ac ) ⊂ iℝ (where σ(A) denotes the spectrum of A); (ii) there exists some β > 0 such that for each η ∈ [0, β) and for each f ∈ BC η (ℝ; Yh ) the linear problem ẋh = Ah Xh + f (t),

xh ∈ BC η (ℝ; Xh )

(3.4.2)

has a unique solution xh = Kh f , where Kh ∈ L (BC η (ℝ; Yh ); BC η (ℝ; Xh )) for each η ∈ [0, β), and ‖Kh ‖η ≤ r(η),

∀η ∈ [0, β),

(3.4.3)

for some continuous function r : η ∈ [0, β) → ℝ+ . Using the equivalent integral equation, Vanderbauwhede and Iooss [95] proved the following theorem: Theorem 3.4.1. Assuming (H), there exists a δ0 > 0 such that for all g ∈ Cb0,1 (X, Y) satisfying |g|Lip < δ0

(3.4.4)

there exists a unique ψ ∈ Cb0,1 (Xc Xh ) with the property that for all x̃ : ℝ → X the following statements are equivalent: (i) x̃ is a solution of (3.4.1) and x̃ ∈ BC η (ℝ; X) for some η ∈ [0, β); ̃ ̃ for all t ∈ ℝ, and Πc x̃ : ℝ → Xc is a solution of the ordinary = ψ(Πc x(t)) (ii) Πh x(t) differential equation ẋh = Ac xc + Πc g(xc + ψ(xc )).

(3.4.5)

Definition 3.4.1. Under the foregoing hypothesis, we call def 󵄨 Mc = {xc + ψ(xc ) 󵄨󵄨󵄨 xc ∈ Xc } ⊂ X

(3.4.6)

the unique global center manifold of (3.4.1). Theorem 3.4.2. Assume (H). Then there exists for each k ≥ 1 a number δk > 0 such that if g ∈ Cb0,1 (X, Y) ∩ Cbk (Vρ , Y), with Vρ = {x ∈ X | ‖Πh x‖ < ρ} and ρ > ‖Kh ‖0 |Πh g|0 , and if moreover |g|Lip < δk ,

(3.4.7)

then the mapping ψ given by Theorem 3.4.1 belongs to the space Cbk (Xc ; Xh ). Moreover, if g(0) = 0 and Dg(0) = 0, then also ψ(0) = 0 and Dψ(0) = 0.

3.4 Center manifold theory in infinite dimensions | 271

Theorem 3.4.3. Assume (H), and let g ∈ C k (X; Y) for some k ≥ 1, with g(0) = 0 and Dg(0) = 0. Then there exist a neighborhood Ω of the origin in X and a mapping ψ ∈ Cbk (Xc ; Xh ), with ψ(0) = 0 and Dψ(0) = 0, and such that the following properties hold: ̃ := x̃c (t) + ψ(x̃c (t)) ∈ Ω for all t ∈ I, (i) if x̃c : I → Xc is a solution of (3.4.5) such that x(t) then x̃ : I → X is a solution of (3.4.1); ̃ ∈ Ω for all t ∈ ℝ, then (ii) if x̃ : ℝ → X is a solution of (3.4.1) such that x(t) ̃ = ψ(Πc x(t)), ̃ Πh x(t) and Πc x̃ : ℝ → Xc is a solution of (3.4.5). In order to use spectral theory, we state some quite general spectral hypothesis on the linear operator A appearing in (3.4.1) and to show that these spectral hypothesis imply the hypothesis (H). We make the following assumptions (Σ): (i) σ(A) ∩ iℝ consists of a finite number of isolated eigenvalues, each with a finitedimensional generalized eigenspace; (ii) there exist constants ω0 > 0, c > 0 and a α ∈ [0, 1) such that for all ω ∈ ℝ with |ω| ≥ ω0 we have iω ∈ ρ(A), c 󵄩󵄩 −1 󵄩 󵄩󵄩(iω − A) 󵄩󵄩󵄩L (Z) ≤ |ω|

(3.4.8)

c 󵄩󵄩 −1 󵄩 . 󵄩󵄩(iω − A) 󵄩󵄩󵄩L (Y;X) ≤ |ω|1−α

(3.4.9)

and

Theorem 3.4.4. Assume (Σ) and let η ∈ [0, β). Then the equation ẋh = Ah xh + f (t)

(3.4.10)

has for each f ∈ BC η (ℝ; Xh ) a unique solution x̃h ∈ BC η (ℝ; Xh ), given by def

t

∞

x̃h = (Kh f )(t) = ∫ S+ (t − s)f (s)ds − ∫ S− (t − s)f (s)ds,

(3.4.11)

t

−∞

where S± (t) =

1 ∫ eλt (λ − Ah )−1 dλ, 2πi

(3.4.12)

Γ±

def

and Γ± = {∓δ|ω| + iω | ω ∈ ℝ}, δ > 0. Moreover, there exists a continuous function, r : [0, β) → ℝ+ such that ‖K − h‖L (BCη (ℝ;Yh );BCη (ℝ;Xh )) ≤ r(η),

∀η ∈ [0, β).

272 | 3 Structures of small dissipative dynamical systems Choosing suitable space Y, we can prove that hypothesis (Σ) implies hypothesis (H). We consider in detail some simple examples, on which we show how our hypothesis can be verified. Example 1. Consider the parabolic equation 𝜕u 𝜕u 𝜕2 u { { = 2 + u + g(x, ), 𝜕t 𝜕x 𝜕x { { {u(0, t) = u(π, t) = 0, (x, t) ∈ (0, π) × ℝ.

(3.4.13)

We suppose that g ∈ C k+1 (ℝ2 ; ℝ) for some k ≥ 1, and that g(u, v) = O(|u|2 + |v|2 ) as (u, v) → (0, 0). We can rewrite (3.4.13) in the form (3.4.1) by introducing the following spaces and operators. We set Z = L2 ((0, π)), X = H 2 ((0, π)) ∩ H01 ((0, π)), and define A ∈ L (X; Z) by Au = def

where D =

d . dx

d2 u + u = (D2 + 1)u, dx2

(3.4.14)

Since H 1 (0, π) ⊂ C 0 ([0, π]) we have for each u ∈ X that g(u, Du),

( 𝜕g )(u, Du) and ( 𝜕g )(u, Du) are in C 0 ([0, π]). We give the following map: 𝜕u 𝜕v u 󳨃→ g(u, Du) ∈ C k : X → Y = H 1 ((0, π)).

We want to show now that the operator A satisfies the hypothesis (Σ).

def

In fact, the spectrum of A is well known; it consists of the simple eigenvalues λn = def

1 − n2 , with n = 1, 2, . . . , corresponding to the eigenfunctions un (x) = sin nx. Hence we have just one simple eigenvalue on the imaginary axis, namely λ1 = 0. Next let def

λ ∈ ρ(A), v ∈ Z = L2 ((0, π)) and u = (λ − A)−1 ∈ X. Then we have −D2 u + (λ − 1)u = v.

(3.4.15)

Multiplying (3.4.15) by ū and integrating over (0, π) shows that π

̄ ‖Du‖2L2 + (λ − 1)‖u‖2L2 = ∫ vudx.

(3.4.16)

0

Taking λ = iw, with w ∈ ℝ \ {0}, and considering the imaginary part of (3.4.16) gives |w|‖u‖2L2 ≤ ‖v‖L2 ‖u‖L2 , and hence 󵄩󵄩 −1 󵄩 −1 󵄩󵄩(iw − A) 󵄩󵄩󵄩L (Z) ≤ |w| ,

∀w ∈ ℝ \ {0}.

(3.4.17)

3.4 Center manifold theory in infinite dimensions | 273

We consider the equation −D2 u + s2 u = v,

s ∈ ℝ,

(3.4.18)

and throughout the following discussion we restrict attention to the case |s| > 1. Using Fourier series it is easy to show that the equation (3.4.18) has for each v ∈ H m ((0, π)) m ≥ −1 a unique solution u ∈ H m+2 ((0, π)) ∩ H01 ((0, π)). Let us use the def

notation ‖u‖m = ‖u‖H m ((0,π)) and |u|m = ‖Dm u‖L2 ((0,π)) . We take v ∈ H −1 ((0, π)) and u ∈ H01 ((0, π)); then (3.4.18) is an equality in H −1 ((0, π)) = (H01 ((0, π)))∗ which we can apply to u to find |u|21 + s2 ‖u‖20 ≤ ‖v‖−1 ‖u‖1 ; this implies and s‖u‖0 ≤ ‖v‖−1 .

‖u‖1 ≤ ‖v‖−1

(3.4.19)

Next we take v ∈ L2 ((0, π)) and u ∈ H 2 ((0, π)) ∩ H01 ((0, π)). Taking the inner product in L2 ((0, π)) of (3.4.18) with u gives |u|21 + s2 ‖u‖20 ≤ ‖v‖0 ‖u‖0 , from which we find |s|2 ‖u‖0 ≤ ‖v‖0

and |s|‖u‖1 ≤ ‖v‖0 .

(3.4.20)

Taking the inner product in L2 ((0, π)) of (3.4.18) with D2 u gives |u|22 + s2 ‖u‖21 ≤ ‖v‖0 |u|2 ,

(3.4.21)

and hence |u|2 ≤ ‖v‖0 , which in combination with (3.4.20) gives ‖u‖2 ≤ C‖v‖0 ,

(3.4.22)

for some appropriate C > 0. If v ∈ H 1 ((0, π)) then we can rewrite (3.4.21) as 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 |u|22 + s2 ‖u‖21 ≤ |v|1 |u|1 + 󵄨󵄨󵄨v(0)󵄨󵄨󵄨󵄨󵄨󵄨Du(0)󵄨󵄨󵄨 + 󵄨󵄨󵄨v(π)󵄨󵄨󵄨󵄨󵄨󵄨Du(π)󵄨󵄨󵄨. Let θ ∈ C ∞ ([0, π], ℝ) be such that θ(0) = 1 and θ(π) = 0, then π

v(0) = − ∫(Dθ ⋅ v + θ ⋅ Dv)dx, 0

(3.4.23)

274 | 3 Structures of small dissipative dynamical systems and hence |v(0)| ≤ C1 ‖v‖1 . In a similar way we have |v(π)| ≤ C1 ‖v‖1 , and then (3.4.20) and (3.4.23) imply 󵄨 󵄨 󵄨 󵄨 |u|22 ≤ C1 (󵄨󵄨󵄨Du(0)󵄨󵄨󵄨 + 󵄨󵄨󵄨Du(π)󵄨󵄨󵄨)‖v‖1 + s−1 ‖v‖21 .

(3.4.24)

In order to estimate Du(0) and Du(π) we take the inner product of (3.4.18) with θu ; considering the real part and after some integration by parts, we find π

π

0

0

󵄨󵄨 󵄨2 2 2 2 󵄨󵄨Du(0)󵄨󵄨󵄨 = − ∫ Dθ|Du| du + s ∫ Dθ|u| dx π

̄ + D(θv)u]dx; ̄ − ∫[D(θv)u 0

using (3.4.20) it follows that 󵄨󵄨 󵄨2 −2 2 󵄨󵄨Du(0)󵄨󵄨󵄨 ≤ C2 s ‖v‖1 . In a similar way one estimates |Du(π)|; bringing these estimates in (3.4.24) and combining with (3.4.20) then gives 1

‖u‖2 ≤ C3 s− 2 ‖v‖1 .

(3.4.25)

Now let us return to the equation (3.4.15), in which we take λ = μ ∈ ℝ with μ ≥ 2. Comparing with (3.4.18) and using (3.4.25) then proves that 󵄩󵄩 −1 󵄩 −1 󵄩󵄩(μ − A) 󵄩󵄩󵄩L (Y;X) ≤ C3 μ 4 ,

∀μ ∈ ℝ, μ ≥ 2.

(3.4.26)

Finally, using the identity (iw − A)−1 − (μ − A)−1 = (μ − iw)(iw − A)−1 (μ − A)−1 , following from (3.4.17), we find by taking μ = |w| that 󵄩󵄩 −1 󵄩 −1 󵄩󵄩(iw − A) 󵄩󵄩󵄩L (Y,X) ≤ C|w| 4 ,

∀w ∈ ℝ, |w| ≥ 2.

(3.4.27)

We conclude that the operator A defined by (3.4.14) satisfies the hypothesis (Σ), with α = 43 . Example 2. An elliptic problem on a strip: 𝜕2 u 𝜕2 u 𝜕u 𝜕u { { 2 + 2 + μu + g(u, , ) = 0, 𝜕x 𝜕y 𝜕y { 𝜕x { u(x, 0) = u(x, π) = 0, ∀x ∈ ℝ; (x, y) ∈ ℝ × (0, π). {

(3.4.28)

3.4 Center manifold theory in infinite dimensions | 275

We suppose that g ∈ C k+1 (ℝ3 , ℝ) (k ≥ 1), with g(u, v, w) = O(|u|2 + |v|2 + |w|2 ) as (u, v, w) → 0. Fix some μ0 ∈ ℝ and let μ = μ0 + ν. We then rewrite (3.4.28) as u1 0 d u1 ], [ ] = A[ ] + [ −νu − g(u dx u2 u2 1 1 , u2 , Du1 )

(3.4.29)

where A=[

0 −D − μ0 2

1 ], 0

D=

d dx

def

def

We have A ∈ L (X; Z), where Z = H01 ((0, π)) × L2 ((0, π)), and X = H 2 ((0, π)) ∩ H01 ((0, π)) × H01 ((0, π)). The same argument as for the Example 1 proves that the mapdef

ping (u1 , u2 ) 󳨃→ (0, −νu1 − g(u1 , u2 , Du1 )) is of class C k from X into the space Y = H 2 ((0, π)) ∩ H01 ((0, π)) × H01 ((0, π)). To determine the spectrum of A fix some λ ∈ ℂ and some v = (v1 , v2 ) ∈ Z, and consider the equation Au = λu + v,

(3.4.30)

or more explicitly with u = (u1 , u2 ): u2 = λu1 + v1 ,

{

−D2 u1 − μ0 u1 = λu2 + v2 .

(3.4.31)

Eliminating u2 we find −D2 u1 − (λ2 + μ0 )u1 = λv1 + v2 .

(3.4.32)

If this equation has for each v ∈ Z a unique solution u1 ∈ H 1 ((0, π)) ∩ H01 ((0, π)), then also (3.4.28) has a unique solution u ∈ X, and λ is in the resolvent set of A. It follows that 󵄨 σ(A) = {λ ∈ ℂ 󵄨󵄨󵄨 λ2 + μ0 = n2 , n = 1, 2, . . . } def 󵄨 = {λ±n = ±√n2 − μ0 󵄨󵄨󵄨 n = 1, 2, . . . }.

(3.4.33)

The eigenfunctions corresponding to the eigenvalues λ±n are given by Φ±n (y) = [

sin ny ], λ±n sin ny

n ≥ 1.

If n2 ≠ μ0 then λ±n are simple eigenvalues; however, if n2 = μ0 , then λ±n = 0 is non-semisimple, since the equation (3.4.31) with (v1 , v2 ) = (sin ny, 0) has a solution (u1 , u2 ) = (0, sin ny) in that case λ±n = 0 has algebraic multiplicity two.

276 | 3 Structures of small dissipative dynamical systems Suppose that μ0 ∈ [m2 , (m + 1)2 ) for some m ≥ 1. Then Re λ±n = 0 for 1 ≤ n ≤ m and Re λ±n ≠ 0 for n > m; more precisely, |Re λ±n | ≥ β = √(m + 1)2 − μ0 ,

∀n ≥ m + 1.

We conclude that the center subspace Xc is 2m-dimensional. To verify that A satisfies (Σ) we will use again the estimates (3.4.19)–(3.4.25) satisfied by the solutions of the equation (3.4.18). We consider (3.4.30), or equivalently (3.4.31), for λ = iw, with w ∈ ℝ Rand w2 ≥ μ0 + 1. We first take v1 = 0 and v2 ∈ L2 (0, π), then (3.4.32) in combination with (3.4.20) gives (w2 − μ0 )‖u1 ‖0 ≤ ‖v2 ‖0

and

1

(w2 − μ0 ) 2 ‖u1 ‖1 ≤ ‖v2 ‖0 .

Since u2 = iwu1 it follows that ‖u2 ‖0 ≤

C |w| ‖v2 ‖0 ≤ 1 ‖v2 ‖0 |w| − μ0

w2

and ‖u1 ‖1 ≤

1 (w2

− μ0 )

1 2

‖v2 ‖0 ≤

C2 ‖v ‖ , |w| 2 0

where C1 and C2 are constants independent of w. Next we take v2 = 0 and v1 ∈ H01 (0, π); eliminating u1 from (3.4.31) we find the following equation in H −1 ((0, π)): −D2 u2 + (w2 − μ0 )u2 = −D2 v1 − μ0 v1 .

(3.4.34)

Using (3.4.19) we conclude that 󵄩 󵄩 ‖u2 ‖1 ≤ 󵄩󵄩󵄩D2 v1 + μ0 v1 󵄩󵄩󵄩−1 ≤ C3 ‖v1 ‖1 and 1

(w2 − μ0 ) 2 ‖u2 ‖0 ≤ C3 ‖v1 ‖1 ; since u1 = (iw)−1 (u2 − v1 ) it follows that ‖u1 ‖1 ≤

C4 ‖v ‖ |w| 1 1

and ‖u1 ‖0 ≤

C4 ‖v ‖ . |w| 1 1

These estimates in combination with the linearity of (iw − A)−1 then imply that C 󵄩󵄩 −1 󵄩 , 󵄩󵄩(iw − A) 󵄩󵄩󵄩L (Z) ≤ |w|

∀w ∈ ℝ, w2 ≥ μ0 + 1.

(3.4.35)

3.4 Center manifold theory in infinite dimensions | 277

Next suppose that (v1 , v2 ) ∈ Y = H 2 ((0, π)) ∩ H01 ((0, π)) × H01 ((0, π)). If v1 = 0 then it follows from (3.4.32), (3.4.20) and (3.4.25) that − 43

‖u1 ‖1 ≤ C5 (w2 − μ0 )

‖v2 ‖1

and ‖u1 ‖2 ≤ C5 (w2 − μ0 )

− 41

‖v2 ‖1 ,

since u2 = iwu1 it follows that 1

‖u1 ‖2 ≤ C6 |w|− 2 ‖v2 ‖1

1

and ‖u2 ‖1 ≤ C6 |w|− 2 ‖v2 ‖1 .

(3.4.36)

From the other side, if v2 = 0 then (v1 , 0) ∈ X, and we can use the estimate ‖(iw − C A)−1 ‖L (Z) ≤ |w| which follows from (3.4.35). In combination with (3.4.36) and the linearity of (iw − A)−1 this proves that

C󸀠 󵄩󵄩 −1 󵄩 , 󵄩󵄩(iw − A) 󵄩󵄩󵄩L (Y;X) ≤ 1 |w| 2

∀w ∈ ℝ, w2 ≥ μ0 + 1.

(3.4.37)

We conclude that the operator A defined by (3.4.30) satisfies (Σ). Example 3. Consider the Navier–Stokes equations: 𝜕V { + (V ⋅ ∇)V + ∇p = νΔV + f (x), { { 𝜕t { { { {∇ ⋅ V = 0, for x ∈ Ω, { { { { { { {V|𝜕Ω = a, ∫ a ⋅ ndσ = 0. { 𝜕Ω

(3.4.38)

Here Ω is a bounded domain in ℝ3 (or ℝ2 ), with smooth boundary an and exterior normal unit vector n : 𝜕Ω → ℝ3 , V = V(t, x) ∈ ℝ3 , p = p(t, x) ∈ ℝ, ν is a dimensionless positive number related to the Reynolds number, while f : Ω → ℝ3 and a : Ω → ℝ3 are given vector fields. As we will see the problem (3.4.38) splits into two equations, one for V and one which gives ∇p in function of V, and hence determines p up to a constant once V is known. So the Cauchy problem associated to (3.4.38) consists in finding solutions (V, p) : ℝ+ × Ω → ℝ3 × ℝ of (3.4.38) such that V(0, x) = V(x) for some given V(x) : Ω → ℝ3 satisfying ∇ ⋅ V = 0. ̃ μ̃ ∈ ℝm and suppose that f and a are depend on μ,̃ and for We set μ = (ν, μ), (0) (0) each μ we have a stationary solution (Vμ(0) , p(0) μ ) = (Vμ (x), pμ (x)) of (3.4.38). Setting V = Vμ(0) + U, p = p(0) μ + ν p̃ and performing a time rescale with scaling factor ν reduces (3.4.38) to the system 𝜕U ̃ μ U + N(U)] ̃ ̃, = ΔU + ν−1 [B − ∇p { { 𝜕t {∇ ⋅ U = 0, U|𝜕Ω = 0,

(3.4.39)

with ̃ μ U def B = −((U ⋅ ∇)Vμ(0) + (Vμ(0) ⋅ U)U), def ̃ N(U) = −(U ⋅ ∇)U.

(3.4.40)

278 | 3 Structures of small dissipative dynamical systems Now let W ∈ (L2 (Ω))3 be such that ∇ ⋅ W ∈ L2 (Ω). It follows then from the identity ∫ ∇ψ ⋅ Wdx + ∫ ψ(∇ ⋅ W)dx = ∫ ψW ⋅ ndx, Ω

Ω

(3.4.41)

𝜕Ω 1

we can define W ⋅ n|𝜕Ω as an element of the dual space of the space H 2 (𝜕Ω) of the 1 traces ψ|𝜕Ω of functions ψ ∈ H 1 (Ω), and W ⋅ n ∈ H − 2 (Ω). Define the basic space 3󵄨 def Z = {U ∈ (L2 (Ω)) 󵄨󵄨󵄨 ∇ ⋅ U = 0, U ⋅ n|𝜕Ω = 0}.

(3.4.42)

Let Π0 be the orthogonal projection on Z in (L2 (Ω))3 ; then one can show that (I − ̃ disapΠ0 )(L2 (Ω))3 = {∇ψ | ψ ∈ H 1 (Ω)}, and projecting with Π0 makes the term ∇p pear. This projection gives us the equation dU = Aμ U + ν−1 N(U), dt

(3.4.43)

def

where Aμ : D(Aμ ) = X = {U ∈ Z | U ∈ (H 2 (Ω))3 , U|𝜕Ω = 0} → Z is a densely defined closed linear operator given by def

Aμ U = TU + ν−1 Bμ U,

U = Π0 ΔU,

def

̃μU BU = Π0 B

(3.4.44)

while def

̃ N(U) = Π0 N(U).

(3.4.45)

We have Aμ ∈ L (X, Z) when we put on X the standard scalar product of (H 1 (Ω))3 . Let us show first that N ∈ C ∞ (X; Y), where Y := {W ∈ Z | W ∈ (H 1 (Ω))3 }. The Sobolev imbedding theorem gives us the continuous imbeddings H 2 (Ω) 󳨅→ C 0 (Ω) and H 1 (Ω) 󳨅→ L4 (Ω). From this it follows easily that the mapping (U1 , U2 ) ∈ X 2 󳨃→ V := (U1 ⋅ ∇)U2 ̃ ∈ C ∞ (X; defines a bounded bilinear operator from X into (H 1 (Ω))3 hence we have N 1 3 1 3 (H (Ω)) ). Next take any V ∈ (H (Ω)) and consider the Neumann problem 2 {Δϕ = ∇ ⋅ V ∈ L (Ω), 1 { 𝜕ϕ 2 { 𝜕n = V ⋅ n ∈ H (Ω).

(3.4.46)

Let ϕ ∈ H 2 (Ω) be any solution of (3.4.46), and set W = V∇ϕ then one easily verifies that W = Π0 V ∈ Y, and that ‖W‖H 1 ≤ C‖V‖H 1 . This proves that Y = Π0 ((H 1 (Ω))3 ), and ̃ ∈ C ∞ (X; Y), with hence we have N = Π0 N 󵄩󵄩 󵄩 2 󵄩󵄩N(U)󵄩󵄩󵄩Y ≤ C‖U‖X .

(3.4.47)

3.4 Center manifold theory in infinite dimensions | 279

Now we turn to the principal part of the operator Aμ which is the so-called Stokes operator T ⊂ L (X; Z). Solving the equation TU = g for U ∈ X and for given g ∈ Z is equivalent to finding solutions (U, ψ) ∈ (H 2 (Ω))3 × H 1 (Ω) of the system ΔU + ∇ψ = g,

{

∇ ⋅ U = 0 on Ω,

U|𝜕Ω = 0.

(3.4.48)

It is well known that (3.4.48) has a unique solution, and hence T has a bounded inverse T −1 ∈ L (X; Z). Moreover, we have (TU, V) = (U, TV),

∀U, V ∈ X

(3.4.49)

and 󵄨󵄨 𝜕U 󵄨󵄨2 󵄨 󵄨 (TU, U) = − ∫ ∑󵄨󵄨󵄨 i 󵄨󵄨󵄨 dx ≤ 0, 󵄨󵄨 𝜕xj 󵄨󵄨 i,j

∀U ∈ X.

(3.4.50)

Ω

It follows that T is self-adjoint and negative; moreover, T has a compact resolvent, since the imbedding X 󳨅→ Z is compact. It results that 1 , |λ| 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩L (Z) ≤ { 1

|Im λ|

Re λ > 0, ,

Im λ ≠ 0.

(3.4.51)

Using techniques similar to those explained before one also proves that M

{ |λ|1/4 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩L (Y;X) ≤ { 1 { |Im λ|1/4

if Re λ > 0 and |λ| sufficiently large, if Re λ ≤ 0 and |Im λ| sufficiently large.

(3.4.52)

Rewriting λ − Aμ = λ − (T + ν−1 Bμ ) = [IZ − ν−1 Bμ (λ − T)−1 ](λ − T) = (λ − T)[IX − ν−1 (λ − T)−1 Bμ ],

(3.4.53)

one then easily deduces from (3.4.51) and (3.4.52) that Aμ satisfies the hypothesis (Σ) and (S).

4 Existence and stability of solitary waves We know that the local traveling waves of nonlinear evolution equations are called solitary waves. The so-called “local” means that a solitary wave tends to zero in some convenient sense as the space variable |x| → ∞, or the convergence should have the property that physical quantities, such as the energy and charge are finite. A solitary wave is called a “soliton”, which interacts with other solitons, and emerges from the collision unchanged, except for a phase shift. Whether there exist solitary waves of a nonlinear evolution equation and the solitary wave is a soliton need to be checked and proved. In general, the existence of a solitary wave is easy to be proved in a one-dimensional space, because the problem involves proving the existence of the solution of the indefinite boundary value problem for an ordinary differential equation. We have some good methods to solve these problems, in some special cases, we can analytically obtain the exact solution, such as for KdV equation, nonlinear Schrödiner equation, and Sine–Gordon equation, etc. For multidimensional spaces, the problem of existence of solitary waves becomes complex, in general, the problem involves showing of existence of the solution for a multidimensional semilinear elliptic equation in unbounded domains. Because of non-compactness in the embedding theorem on unbounded domains, we use the new theoretical framework, such as the concentration–compactness principle to prove the existence of solitary wave solutions, but the solitary wave solutions are not unique in some cases. Whether the solitary wave is a soliton needs to be checked theoretically or using numerical simulation; one of the most important factors is that we must consider whether the solitary wave is “stable.” We know that if it is stable when t → ∞, then it never vanishes. For the unstable traveling waves, how will they evolve? Do they vanish or collapse, or blow-up, or tend to a singular self-similar solution? These problems are our main concern. For the stability of the solitary waves there are many different notations of “stability” which have been used in the literature. The stability generally falls into two types: One is linear stability, namely, we linearize the system around a solitary wave under weak perturbations and consider its Lyapunov stability; another one is nonlinear stability, namely, we argue in favor of the stability of solitary wave solutions with some functionals, such as whether there exists the least energy solitary wave for the system. Recently, weak stability was proposed, namely, orbital stability, and strong stability was also considered, such as asymptotic stability (when temporal variable t → ∞) and exponential stability. The theories of stability of solitary waves have been developed rapidly over the last two decades. Strauss, Weinstein, and Bona, among others, have given insight into stability and instability of the solitary waves, and have obtained abundant results, see [7, 34, 35, 80, 88, 98]. In particular, Grillakis et al. [34, 35] have investigated the theory of orbital stability for the abstract Hamiltonian system. Recently, Weinstein et al. https://doi.org/10.1515/9783110587265-004

282 | 4 Existence and stability of solitary waves have studied asymptotic stability of solitary waves for KdV, BBM, and other equations, and using the Evans function method, they obtained many detailed results. Results of the existence and stability of multidimensional solitary waves have been improved substantially. Saut and other authors [14, 18, 19, 76, 83, 97] have obtained many good results for KP, DS, and other equations. We have studied orbital stability of solitary waves of one-dimensional nonlinear derivative Schrödinger, LS and KdV equations, see [2–6, 9, 10, 12–17, 21, 22, 24, 27, 30, 32, 36, 40, 42, 43, 45–47, 49–62, 64–68, 73, 75, 78, 79, 81–87, 89, 93–96, 99]. The existence and non-existence of solitary waves for two-dimensional nonlinear coupled Schrödinger–Kadomtsev–Petviashvili equations have also been discussed, see [21].

4.1 Orbital stability Consider abstract Hamiltonian systems of the form [34, 35] du = JE 󸀠 (u(t)), dt

u(t) ∈ X.

(4.1.1)

Here E is a functional (the “energy”) and J is a skew-symmetric linear operator, 󸀠 is the Fréchet derivative. Let X be a real Hilbert space with inner product (⋅, ⋅). If X ∗ is its dual, there is a natural isomorphism I : X → X ∗ defined by ⟨Iu, v⟩ = (u, v), where ⟨⋅, ⋅⟩ denotes the paring between X and X ∗ . Here J is a closed linear operator from X ∗ to X with dense domain D(J) ∈ X ∗ . We assume that J is skew-symmetric, that is, ⟨Ju, v⟩ = −⟨u, Jv⟩,

u, v ∈ D(J),

(4.1.2)

and also that J

is onto.

(4.1.3)

Let E : X → ℝ be a C 2 functional defined on all of X. We write its derivative as ⟨E (u), v⟩, where E 󸀠 : X → X ∗ , and its second derivative as ⟨E 󸀠󸀠 (u)w, v⟩. Let T be a one parameter group of unitary operators on X. Thus T(s) is a unitary operator from X onto X for each s ∈ ℝ, that is, ‖t(s)u‖ = ‖u‖, which is strongly continuous and satisfies T(s)T(r) = T(s + r) for all real s and r. Let T 󸀠 (0) denote the infinitesimal generator, an operator X → X, which is skew-adjoint with respect to the inner product (⋅, ⋅) with dense domain. Using our definition of adjointness, the unitarity of T can be expressed as 󸀠

T ∗ (s)I = IT(−s) where T ∗ (s) : X ∗ → X ∗ .

for s ∈ ℝ,

4.1 Orbital stability |

283

We assume that E is invariant under T. That is, E(T(s)u) = E(u)

for s ∈ ℝ, u ∈ X.

(4.1.4)

Differentiating (4.1.4) with respect to u, we get T ∗ (s)E 󸀠 (T(s)u) = E 󸀠 (u).

(4.1.5)

T ∗ (s)E 󸀠󸀠 (T(s)u)T(s) = E 󸀠󸀠 (u).

(4.1.6)

Differentiating again, we get

Differentiating (4.1.4) with respect to s at s = 0, we get ⟨E 󸀠 (u), T 󸀠 (0)u⟩ = 0

for u ∈ D(T 󸀠 (0)).

(4.1.7)

We assume that J commutes with T in the sense that T(s)J = JT ∗ (−s).

(4.1.8)

This can be written equivalently as T(s)JT ∗ (s) = J or as JIT(s) = T(s)JI. In particular, (4.1.8) implies that T ∗ (s)(D(J)) = D(J). Formally, if we differentiate (4.1.8) with respect to s at s = 0, we get T 󸀠 (0)J = −J(T 󸀠 (0))∗ . Hence J −1 T(0) = −(T 󸀠 (0))∗ J −1 = (J −1 T 󸀠 (0))∗ . In order to make this precise, we make the further assumption that There is a bounded linear operator B : X → X ∗ such that B∗ = B, and the operator JB is an extension of T 󸀠 (0).

(4.1.9)

We define another functional Q : X → ℝ by 1 Q(u) = ⟨Bu, u⟩. 2

(4.1.10)

Equations (4.1.9) and (4.1.10) imply that Q is also invariant under T: Q(T(s)u) = Q(u) for s ∈ ℝ, u ∈ X.

(4.1.11)

In fact, let u ∈ D(T 󸀠 (0)). Then T(s)u ∈ D(T 󸀠 (0)) ⊂ D(JB) and d Q(T(s)) = ⟨Q󸀠 (T(s)u), T 󸀠 (0)T(s)u⟩ ds = ⟨BT(s)u, JBT(s)u⟩ = 0 Differentiating (4.1.10) and (4.1.11), we have Q󸀠 (u) = Bu for all u ∈ X. Furthermore, (a) { { { { { {(b) { { (c) { { { { {(d)

T ∗ (s)Q󸀠 (T(s)u) = Q󸀠 (u), T ∗ (s)BT(s) = B, ∗

BT 󸀠 (0) = −(T 󸀠 (0)) B, ∗

B[DT 󸀠 (0)] = D((T 󸀠 (0)) ).

284 | 4 Existence and stability of solitary waves The evolution equation which we shall study is du = JE 󸀠 (u(t)), dt

u(t) ∈ X.

(4.1.12)

Note that E and Q are formally conserved under the flow of (4.1.12). Namely, dE(u) du = ⟨E 󸀠 (u), ⟩ = ⟨E 󸀠 (u), JE 󸀠 (u)⟩ = 0 dt dt and du dQ(u) = ⟨Q󸀠 (u), ⟩ = ⟨Bu, JE 󸀠 (u)⟩ = −⟨JBu, E 󸀠 (u)⟩ dt dt = −⟨T 󸀠 (0)u, E 󸀠 (u)⟩ = 0.

Definition 4.1.1. By a solution of (4.1.12) in a time interval I , we mean a function u ∈ C(I ; X) such that d ⟨u(t), ψ⟩ = ⟨E 󸀠 (u(t)), −Jψ⟩, dt u ∈ D󸀠 (I ), ∀ψ ∈ D(J) ⊂ X ∗ .

(4.1.13)

Assumption 4.1.1 (Existence of solutions). For each u0 ∈ X there exists t0 > 0 depending only on μ, where ‖u0 ‖ ≤ μ, and there exists a solution u of equation (4.1.12) in the interval I = [0, t0 ) such that (1) u(0) = u0

and

(2) E(u(t)) = E(u0 ),

Q(u(t)) = Q(u0 )

for t ∈ I .

We remark that if u(t) is a solution of (4.1.12), so is T(s)u(t) for all s ∈ ℝ. Indeed, by (4.1.5) and (4.1.8), ⟨E 󸀠 (T(s)u(t)), Jψ⟩ = ⟨E 󸀠 (u(t)), T(−s)Jψ⟩ = ⟨E 󸀠 (u(t)), JT ∗ (s)ψ⟩ = −⟨JE 󸀠 (u(t)), T ∗ (s)ψ⟩ = − =−

d ⟨T(s)u(t), ψ⟩, dt

d ⟨u(t), T ∗ (s)ψ⟩ dt

∀ψ ∈ D(J).

Definition 4.1.2. By a bound state we mean a solution of the evolution equation of the special form u(t) = T(ωt)ϕ,

ω ∈ ℝ, ϕ ∈ X.

(4.1.14)

If ϕ ∈ D(T 󸀠 (0)) satisfies the “stationary” equation E 󸀠 (ϕ) = ωQ󸀠 (ϕ),

(4.1.15)

4.1 Orbital stability | 285

then T(ωt)ϕ is a bound state. Indeed, d T(ωt)ϕ = ωT 󸀠 (0)T(ωt)ϕ = ωJBT(ωt)ϕ dt = ωJT ∗ (−ωt)Q󸀠 (ϕ) = JT ∗ (−ωt)E 󸀠 (ϕ) = JE 󸀠 (T(ωt)ϕ). Assumption 4.1.2 (Existence of bound state). There exist real ω1 < ω2 and a mapping (a) ψ → ϕω : (ω1 , ω2 ) → X, ϕω ∈ C 1 , ω ∈ (ω1 , ω2 ), { { { { 󸀠 󸀠 { {(b) E (ϕω ) = ωQ (ϕω ), { { (c) ϕω ∈ D(T 󸀠 (0)3 ) ∩ D(JIT 󸀠 (0)2 ), { { { { 󸀠 {(d) T (0)ϕω ≠ 0. Definition 4.1.3. We define the scalar d(ω) = E(ϕω ) − ωQ(ϕω )

(4.1.16)

Hω = E 󸀠󸀠 (ϕω ) − ωQ󸀠󸀠 (ϕω ).

(4.1.17)

and the operator Hω : X → X ∗

Observe that Hω is self-adjoint in the sense that Hω∗ = Hω . This means that I −1 Hω is a bounded self-adjoint operator on X in the standard sense, since (I −1 Hu, v) = ⟨Hu, v⟩ = ⟨Hv.u⟩ = (I −1 Hu, v). The spectrum of Hω consists of the real numbers λ such that Hω −λI is not invertible. We claim that λ = 0 belongs to the spectrum of Hω . Indeed, from (4.1.5), (4.1.12) and (4.1.16), we have E 󸀠 (T(s)ϕω ) − ωQ󸀠 (T(s)ϕω ) = T ∗ (−s)[E 󸀠 (ϕω ) − ωQ󸀠 (ϕω )] = 0.

(4.1.18a)

Differentiating with respect to s at s = 0, we deduce that Hω (T 󸀠 (0)ϕω ) = 0.

(4.1.18)

Thus T 󸀠 (0)ϕω is an eigenvector with eigenvalue 0. Definition 4.1.4. The ϕω -orbit {T(ωt)ϕω , t ∈ ℝ} is called stable if for all ε > 0 there exists δ > 0 with the following property. If ‖u0 − ϕω ‖ < δ and u(t) is a solution of (4.1.12) in some interval [0, t0 ) with u(0) = u0 , then u(t) can be continued to a solution in 0 ≤ t < ∞ and 󵄩 󵄩 sup inf 󵄩󵄩󵄩u(t) − T(s)ϕω 󵄩󵄩󵄩 < ε.

0 0 then Hω has at least some negative spectrum. Indeed, differentiating (4.1.16), we have Hω ϕ󸀠ω = Q󸀠 (ϕω ),

where ϕ󸀠ω = dϕ󸀠ω /dω.

(4.1.19)

Differentiating (4.1.15) twice, we get d󸀠 (ω) = −Q(ϕω )

(4.1.20)

d󸀠󸀠 (ω) = −⟨Q󸀠 (ϕω ), ϕ󸀠ω ⟩ = −⟨Hω ϕ󸀠ω , ϕ󸀠ω ⟩.

(4.1.21)

and

Thus ⟨Hω ϕ󸀠ω , ϕ󸀠ω ⟩ < 0 if d󸀠󸀠 (ω) > 0. Now we consider the stability of this orbit. Often, when the parameter ω is unimportant, we will drop subscript ω. Thus we will write ϕ for ϕω , H for Hω , and so on. A tubular neighborhood, or simple tube, around the orbit {T(s)ϕ | s ∈ ℝ} is defined by 󵄩 󵄩 Uε = {u ∈ X : inf 󵄩󵄩󵄩u − T(s)ϕ󵄩󵄩󵄩 < ε}. s∈ℝ Lemma 4.1.1. Under Assumptions 4.1.2 and 4.1.3, either (i) T(s)ϕ = ϕ for some s > 0 or (ii) T(sn )ϕ → ϕ implies sn → 0. Proof. Consider the set of critical points of L = E − ωQ given by 0 = L󸀠 (u) − L󸀠 (ϕ) = H(u − ϕ) + O(‖u − ϕ‖2 )

4.1 Orbital stability |

287

where H = L󸀠󸀠 (ϕ). Therefore the set of critical points is locally isomorphic to the nullspace of H. Now ϕ is a critical point, and so is T(s)ϕ for every s by (4.1.18a). So there is a neighborhood N of ϕ and a number δ > 0 such that 󵄨 󵄨 {u ∈ N 󵄨󵄨󵄨 L󸀠 (u) = 0} = {T(r)ϕ 󵄨󵄨󵄨 |r| < δ}. Now suppose (ii) is false. Then there is a sequence |sn | ≥ δ with T(sn )ϕ ∈ N. Fix n. We have just proved that there exists |rn | < δ with T(sn )ϕ = T(rn )ϕ. So T(sn − rn )ϕ = 0, which means (i) is true. Lemma 4.1.2. Given Assumptions 4.1.2 and 4.1.3, there exist ε > 0 and a C 2 map s : Uε → ℝ (ℝ/period, if the orbit is periodic) such that, for all u ∈ Uε and all for r ∈ ℝ, 󵄩 󵄩 󵄩 󵄩 (i) 󵄩󵄩󵄩T(s(u))u − ϕ󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩T(r)u − ϕ󵄩󵄩󵄩, (ii) (T(s(u))u, T 󸀠 (0)ϕ) = 0, (iii) s(T(r)u) = s(u) − r, modulo the period if the orbit is periodic, (iv) s󸀠 (u) = IT(−s(u))T 󸀠 (0)ϕ/(T 󸀠 (0)2 ϕ, T(s(u))u), (v) s󸀠 maps Uε into D(J) and Js󸀠 is a C 1 function from Uε into X. Proof. The idea is to define s(u) as the minimum of ρ(s) = ‖T(s)u − ϕ‖2 for u close to the orbit of ϕ. We calculate ρ󸀠 (s) = 2(T(s)u − ϕ, T 󸀠 (0)T(s)u) = 2(T(s)u, T 󸀠 (0)ϕ)

ρ󸀠󸀠 (s) = −2(T 󸀠 (0)T(s)u, T 󸀠 (0)ϕ) = 2(T(s)u, T 󸀠 (0)2 ϕ). At u = ϕ and s = 0, ρ󸀠 (0) = 0 and ρ󸀠󸀠 (0) = 2‖T 󸀠 (0)ϕ‖2 > 0. By the implicit function theorem, there are an open ball V around ρ, an interval I around s = 0, and a C 2 map s : V → I such that the equation ρ󸀠 (s) = 0 has a unique solution s = s(u) ∈ I for all u ∈ V. Thus s(u) is the unique minimum of ρ󸀠 (s) in I for a given u ∈ V. By Lemma 4.1.1, for all δ > 0 (δ less than half the period in the periodic case) there exists η(δ) > 0 such that if ‖T(r)ϕ − ϕ‖ < η(δ), then |s| < δ (for s lies within δ of some multiple of the period in the periodic case). We choose δ less than half the period in the periodic case, and we choose I = (−δ, δ) and V = {v : ‖v − ϕ‖ < η(δ)/3}. If u ∈ V, r ∈ ℝ, and ‖T(r)u − ϕ‖ < ‖T(s(u))u − ϕ‖ < η(δ), then 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩T(r)ϕ − ϕ󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩T(r)u − ϕ󵄩󵄩󵄩 + 󵄩󵄩󵄩T(r)(u − ϕ)󵄩󵄩󵄩 < 2‖u − ϕ‖ < η(δ). Therefore r = s(u), plus a multiple of the period in the periodic case. This proves (i) and (ii) for u ∈ V. To show (iii) within V, note that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩T(s(u) − r)ϕ − ϕ󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩T(r)u − ϕ󵄩󵄩󵄩 + 󵄩󵄩󵄩T(s(u))u − ϕ󵄩󵄩󵄩 + ‖u − ϕ‖.

288 | 4 Existence and stability of solitary waves So if T(r)u ∈ V and u ∈ V, we have s(u) − r ∈ I (modulo the period). By uniqueness, s(T(r)u) = s(u) − r (modulo the period). To show (iv), we differentiate (ii) with respect to u ∈ X to obtain (T(s(u))w, T 󸀠 (0)ϕ) + ⟨s󸀠 (u), w⟩(T 󸀠󸀠 (0)T(s(u))u, T 󸀠 (0)ϕ) = 0. Since T 󸀠 (0) is skew with respect to the inner product ⟨s󸀠 (u), u⟩ =

(T(−s(u))T 󸀠 (0)ϕ, u) (T 󸀠 (0)2 ϕ, T(s(u))u)

for all u ∈ X. This implies (iv). If this formula is differentiated once more with respect to u, and if we make use of the assumption that ϕ ∈ D(T 󸀠 (0)3 ) ∩ D(JIT 󸀠 (0)2 ), then (v) follows. Finally, we extend the definition of s(u) to u ∈ Uε , where ε = η(δ)/3, as follows. If ‖u − T(s0 )ϕ‖ < ε for some s0 ∈ ℝ, we define def

s(u) = S(T(−s0 )u) − s0 . This definition is independent of the choice of s0 for the following reason. If ‖u − T(s0 )ϕ‖ < ε and ‖u − T(s1 )ϕ‖ < ε, then T(−s0 )u and T(−s1 )u belong to V. Since (iii) has already been proved within V, we have s(T(s0 − s1 )T(−s0 )u) = s(T(−s0 )u) − (s0 − s1 ) plus a multiple of the period if the orbit is periodic, where r = s0 − s1 . Thus s(T(−s1 )u) − s1 = s(T(−s0 )u) − s0

(in ℝ period).

Therefore s(u) is defined for all u ∈ Uε and satisfies properties (i)–(v). Recall that T 󸀠 (0)ϕ generates the kernel of H. Denote by χ = χω a negative eigenvector of H: 2 Hω χω = −λω Iχω ,

‖χω ‖ = 1.

(4.1.22)

Denote by P = Pω the positive subspace of H. Thus there exists δ = δω > 0 such that ⟨Hp, p⟩ ≥ δ‖p‖2

for p ∈ P.

Theorem 4.1.4. Let d󸀠󸀠 (ω) > 0. If ⟨Q󸀠 (ϕ), y⟩ = ⟨T 󸀠 (0)ϕ, y⟩ = 0 and y ≠ 0, then ⟨Hy, y⟩ ≥ 0.

(4.1.23)

4.1 Orbital stability |

289

Proof. By (4.1.21) we have ⟨Hϕ󸀠 , ϕ󸀠 ⟩ < 0. Make a spectral decomposition ϕ󸀠 = a0 + b0 T 󸀠 (0)ϕ+p0 , where p0 ∈ P. Then −a20 λ2 +⟨Hp0 , p0 ⟩ < 0. Now let y ∈ X with ⟨Q󸀠 (ϕ), y⟩ = 0 and (T 󸀠 (0)ϕ, y) = 0. Decompose y = aχ + p with p ∈ P. By (4.1.19) we have 0 = ⟨Hϕ󸀠 , y⟩ = −a0 aλ2 + ⟨Hp0 , p⟩. Therefore ⟨Hy, y⟩ = −a2 λ2 + ⟨Hp, p⟩ ≥ −a2 λ2 + > −a2 λ2

(a0 aλ2 )2 = 0. a20 λ2

⟨Hp, p0 ⟩2 −a2 λ2 + ⟨Hp0 , p0 ⟩

Corollary 4.1.1. If ⟨Q󸀠 (ϕ), y⟩ = 0 then ⟨Hy, y⟩ ≥ C‖Πy‖2 for some C > 0 where Π is the orthogonal projection onto [T 󸀠 (0)ϕ]⊥ . Theorem 4.1.5. If d󸀠󸀠 (ω) > 0, there exist C > 0 and ε > 0 such that 󵄩 󵄩2 E(u) − E(ϕ) ≥ C 󵄩󵄩󵄩T(s(u))u − ϕ󵄩󵄩󵄩 for u ∈ Uε , Q(u) = Q(ϕ). Proof. Let q = I −1 Q󸀠 (ϕ) and decompose T(s(u)) − ϕ = aq + y, where (y, q) = 0 and a is a scalar. Then Q(ϕ) = Q(u) = Q(T(s(u))u) 󵄩 󵄩2 = Q(ϕ) + ⟨Q󸀠 (ϕ), T(s(u))u − ϕ⟩ + O(󵄩󵄩󵄩T(s(u))u − ϕ󵄩󵄩󵄩 ) 󵄩 󵄩2 = Q(ϕ) + a‖q‖2 + O(󵄩󵄩󵄩T(s(u))u − ϕ󵄩󵄩󵄩 ). Hence a = O(‖T(s(u))u − ϕ‖2 ). Let L(u) = E(u) − ωQ(u). Another Taylor expansion gives 1 L(u) = L(T(s(u))u) = L(ϕ) + ⟨L󸀠 (ϕ), v⟩ + ⟨L󸀠󸀠 (ϕ)v, v⟩ + o(‖v‖2 ), 2

290 | 4 Existence and stability of solitary waves where v = T(s(u))u − ϕ = aq + y. Since Q(u) = Q(ϕ), L󸀠 (ϕ) = 0, and L󸀠 (ϕ) = H, this can be written as 1 E(u) − E(ϕ) = ⟨Hv, v⟩ + o(‖v‖2 ) 2 1 = ⟨Hy, y⟩ + O(a2 ) + O(a‖v‖) + o(‖v‖2 ) 2 1 = ⟨Hy, y⟩ + o(‖v‖2 ). 2 Now 0 = (q, y) = ⟨Q󸀠 (ϕ), y⟩ and (y, T 󸀠 (0)ϕ) = (T(s(u))u − ϕ − aq, T 󸀠 (0)ϕ) = 0. Therefore by Corollary 4.1.1, 1 E(u) − E(ϕ) ≥ C‖y‖2 + o(‖v‖2 ). 2 Finally ‖y‖ = ‖v − aq‖ ≥ ‖v‖ − |a|‖q‖ ≥ ‖v‖ + O(‖v‖2 ). Therefore for ‖v‖ small E(u) − E(ϕ) ≥

1 C‖v‖2 , 4

as we wanted to prove. Theorem 4.1.6. Given Assumption 4.1.1, 4.1.2 and 4.1.3 and d󸀠󸀠 (ω) > 0, the ϕ-orbit is stable. Proof. If it is unstable, there exists a sequence of initial data un (0) and δ > 0 such that 󵄩 󵄩 inf 󵄩󵄩un (0) − T(s)ϕ󵄩󵄩󵄩 → 0,

s∈ℝ 󵄩

󵄩 󵄩 but sup inf 󵄩󵄩󵄩un (t) − T(s)ϕ󵄩󵄩󵄩 ≥ δ, t>0 s∈ℝ

where un (t) is a solution with initial datum un (0). By continuity in t, we can pick the first time tn so that 󵄩 󵄩 inf 󵄩󵄩un (tn ) − T(s)ϕ󵄩󵄩󵄩 = δ,

s∈ℝ 󵄩

(4.1.24)

the solution un existing at least in the time interval [0, tn ]. By Assumption 4.1.1, E(un (tn )) = E(un (0)) → E(ϕ),

Q(un (tn )) = Q(un (0)) → Q(ϕ). Choose a sequence {vn } so that Q(vn ) = Q(ϕ) and ‖vn − un (tn )‖ → 0. By continuity of E, E(vn ) → E(ϕ). Choosing δ sufficiently small, we may apply Theorem 4.1.5 to obtain 󵄩 󵄩2 󵄩 󵄩2 0 ← E(vn ) − E(ϕ) ≥ c󵄩󵄩󵄩T(s(vn ))vn − ϕ󵄩󵄩󵄩 = c󵄩󵄩󵄩vn − T(−s(vn ))ϕ󵄩󵄩󵄩 . Hence ‖un (tn ) − T(−s(vn ))ϕ‖ → 0, which contradicts (4.1.24).

4.1 Orbital stability |

291

Now we consider the instability. Theorem 4.1.7. If d󸀠󸀠 (ω) < 0, then (1) E(u) is not locally minimized at u = ϕ with the constraint Q(u) = Q(ϕ), (2) there exists y ∈ D(T 󸀠 (0)2 ), such that ⟨Hy, y⟩ < 0 and ⟨Q󸀠 (ϕ), y⟩ = 0. Proof. We use the notations (4.1.22) and (4.1.23) as before. Consider ϕΩ for Ω near ω. Consider the function q(s, Ω) = Q(ϕΩ + sχω ). Then 𝜕q (0, ω) = ⟨Q󸀠 (ϕω ), ϕ󸀠ω ⟩ = −d󸀠󸀠 (ω) < 0. 𝜕Ω By the implicit function theorem, there exists a C 1 function Q(s) such that Q(0) = ω and Q(ϕΩ(s) + sχω ) = Q(ϕω ).

(4.1.25)

Now expand LΩ (u) = E(u) − ΩQ(u) near u = ϕΩ to get LΩ (ϕΩ + sχω ) = LΩ (ϕΩ ) + s⟨L󸀠 (ϕΩ ), χΩ ⟩ 1 + s2 ⟨HΩ (ϕΩ )χω , χω ⟩ + o(s2 ). 2 This may be written, for Ω = Ω(s), as E(ϕΩ(s) + sχΩ ) − ΩQ(ϕω ) 1 = d(Ω(s)) + s2 ⟨HΩ(s) χω , χω ⟩ + o(s2 ). 2

(4.1.26)

But d󸀠󸀠 (ω) < 0, so that d(Ω) < d(ω) + (Ω − ω)d󸀠 (ω) = E(ϕΩ ) − ΩQ(ϕΩ ) for Ω near ω, by (4.1.16) and (4.1.20). Furthermore ⟨HΩ χω , χω ⟩ ≤ ⟨Hω χω , χω ⟩ < 0 for Ω near ω by continuity with respect to Ω. Altogether from (4.1.26) we have the Taylor expansion E(ϕΩ(s) + sχω ) < E(ϕω ) +

1 2 s ⟨Hω χω , χω ⟩ + o(s2 ). 4

Let z=(

󵄨󵄨 d 󵄨 )(ϕΩ(s) + sχω )󵄨󵄨󵄨 . 󵄨󵄨s=0 ds

292 | 4 Existence and stability of solitary waves By (4.1.25), ⟨Q󸀠 (ϕω ), z⟩ = 0. Furthermore E(ϕΩ(s) + sχω ) = E(ϕω ) vanishes to second order at s = 0, so that ⟨Hω z, z⟩ =

1 d2 󵄨󵄨󵄨󵄨 󵄨 E(ϕΩ(s) + sχω ) < ⟨Hω χω , χω ⟩ < 0. 2 ds2 󵄨󵄨󵄨s=0

This proves (1). It also proves (2) except that z might not belong to D(T 󸀠 (0)) = D. Now D is dense in X and I − T 󸀠 (0)2 is a positive operator on X since 󵄩 󵄩2 ((I − T 󸀠 (0)2 )v, v) = ‖v‖2 + 󵄩󵄩󵄩T 󸀠 (0)v󵄩󵄩󵄩 ≥ ‖v‖2 for v ∈ D(T 󸀠 (0)2 ). Therefore ξ = [I − T 󸀠 (0)2 ] I −1 Q󸀠 (ϕω ) −1

belongs to D(T 󸀠 (0)2 ) and ⟨Q󸀠 (ϕω ), ξ ⟩ = (I −1 Q󸀠 (ϕω ), [I − T 󸀠 (0)2 ] I −1 Q󸀠 (ϕω )) > 0. −1

Give z ∈ X with ⟨Q󸀠 (ϕω ), z⟩ = 0 and ⟨Hω z, z⟩ < 0, we pick x ∈ D(T 󸀠 (0)2 ) so that ‖x − z‖ < ε. Then we let y=x−

⟨Q󸀠 (ϕω ), x⟩ ξ. ⟨Q󸀠 (ϕω ), q⟩

Then ⟨Q󸀠 (ϕω ), y⟩ = 0 and ‖x−z‖ = O(ε). If ε is small enough. ⟨Hy, y⟩ < ε. This completes the proof. Lemma 4.1.3. For ε sufficiently small there exists a C 1 functional A : Uω → ℝ such that (i) A(T(s)u) = A(u); (ii) [Range of A󸀠 (ϕ)] ⊂ D(J); (iii) JA󸀠 (ϕ) = −y with y given in Theorem 4.1.7; (iv) ⟨Q󸀠 (u), JA󸀠 (u)⟩ = 0 for all u ∈ Uε and s ∈ ℝ; (v) JA󸀠 ∈ C 1 : Uω → X. Proof. With y given in Theorem 4.1.7, let y ∈ D(J) such that Jy = y. With s(u) given in Lemma 4.1.1, we define A(u) = −⟨Y, T(s(u))u⟩.

(4.1.27)

Invariance (i) is clearly satisfied. The derivative of (4.1.27) is A󸀠 (u) = −T ∗ (s(u))y − ⟨y, T(s(u))T 󸀠 (0)u⟩s󸀠 (u).

(4.1.28)

By (4.1.7) the first term belongs to D(J). By Assumption 4.1.2, IT 󸀠 (0)ϕ ∈ D(J). By (4.1.8) IT 󸀠 (−s)T 󸀠 (0)ϕ ∈ D(J). By Lemma 4.1.1 (iii), s󸀠 (u) ∈ D(J) and therefore A󸀠 (u) ∈ D(J). Putting u = ϕ, A󸀠 (ϕ) = −y − ⟨y, T 󸀠 (0)ϕ⟩s󸀠 (ϕ).

4.1 Orbital stability |

293

The last term vanishes because ⟨y, T 󸀠 (0)ϕ⟩ = ⟨y, JBϕ⟩ = −⟨Bϕ, y⟩ = −⟨Q󸀠 (ϕ), yϕ⟩ = 0 by (4.1.9) and (4.1.21). Therefore A󸀠 (ϕ) = −y. Next, from (i) we have for u ∈ D(T 󸀠 (0))∩Uε , 0=

d 󵄨󵄨󵄨󵄨 󸀠 󸀠 󵄨 A(T(s)u) = ⟨A (u), T (0)u⟩ ds 󵄨󵄨󵄨s=0

= ⟨A󸀠 (u), JQ󸀠 (u)⟩ = −⟨Q󸀠 (u), JA(u)⟩.

By a passage to a limit, the same is true for all u ∈ Uε . Finally, in order to prove (v), we shall apply J to (4.1.28) and differentiate once more. To justify this procedure, recall that T 󸀠 (0)J = −JT 󸀠 (0)∗ with the same domain and that Jy = y ∈ D(T 󸀠 (0)2 ). Thus y ∈ D(JT 󸀠 (0)∗2 ). Now the derivative of J applied to (4.1.28) has several terms. The first term −JT ∗ (s(u))y is differentiable because y ∈ D(JT 󸀠 (0)∗ ). The factor ⟨y, T(s(u))T 󸀠 (0)u⟩ is differentiable because y ∈ D(T 󸀠 (0)∗2 ). The last factor Js󸀠 (u) is differentiable by Lemma 4.1.2. Definition 4.1.5. Solve the differential equation du = −JA󸀠 (u) dλ

(4.1.29)

with the initial condition u(0) = v ∈ Uε . Call the solution u = R(λ, v). It exists in some interval |λ| < λ0 (v) with value in Uε . It satisfies T(s)R(λ, v) = R(λ, T(s)v)

d Q(R(λ, v)) = ⟨Q󸀠 (u), −JA󸀠 (u)⟩ dλ dR(λ, ϕ) 󵄨󵄨󵄨󵄨 = −JA󸀠 (ϕ) = y. 󵄨 dλ 󵄨󵄨󵄨λ=0

(4.1.30) (4.1.31) (4.1.32)

Lemma 4.1.4. There exists a C 1 functional Λ : {v ∈ Uε : Q(v) = Q(ϕ)} → ℝ such that E(R(Λ(v), v)) > E(ϕ)

(4.1.33)

for all v ∈ Uε with Q(v) = Q(ϕ) and v ∈ ̸ {T(s)ϕ : s ∈ ℝ}. Proof. Letting L = E − ωQ and M(u) = T(s(u)u), we have L(M(u)) = L(u). So Taylor’s expansion gives 1 󵄩 󵄩2 L(u) = L(ϕ) + ⟨H[M(u) − ϕ], M(u) − ϕ⟩ + o(󵄩󵄩󵄩M(u) − ϕ󵄩󵄩󵄩 ). 2

(4.1.34)

294 | 4 Existence and stability of solitary waves Recall that M(u) − ϕ is orthogonal to T 󸀠 (0)ϕ. We define λ = Λ(v) as the unique solution of the equation f (λ, v) = (M(R(λ, v)) − ϕ, χ) = 0,

(4.1.35)

where χ is the negative eigenvector of H. Indeed, f (0, ϕ) = (M(ϕ) − ϕ, χ) = 0 and 𝜕R 𝜕f (0, ϕ) = (M 󸀠 (ϕ) (0, ϕ), χ) = (M 󸀠 (ϕ)y, χ) 𝜕λ 𝜕λ = (y, χ) + (T 󸀠 (0)ϕ, χ)⟨s󸀠 (ϕ), y⟩ = (y, χ) ≠ 0 because ⟨Hy, y⟩ < 0. By the implicit function theorem, λ = Λ(v) exists in a neighborhood of v = ϕ. Since f (λ, T(r)v) = (M(T(r)R(λ, v)) − ϕ, χ) = f (λ, v), the function Λ extends to all v in the tube Uε for some ε > 0. Thus M(u) − ϕ is orthogonal to both T 󸀠 (0)ϕ and χ. By (4.1.23), c󵄩 󵄩2 󵄩 󵄩2 L(u) ≥ L(ϕ) + 󵄩󵄩󵄩M(u) − ϕ󵄩󵄩󵄩 + o(󵄩󵄩󵄩M(u) − ϕ󵄩󵄩󵄩 ). 2

Hence L(u) ≥ L(ϕ). Since Q(u) = Q(v) = Q(ϕ), we have E(u) ≥ E(ϕ). Equality occurs only if M(u) = ϕ. That is, u is in the ϕ-orbit and so is v. Lemma 4.1.5. For v ∈ Uε with Q(v) = Q(ϕ) and v ∈ ̸ {T(s)ϕ | s ∈ ℝ} we have E(ϕ) < E(v) + Λ(v)P(v),

(4.1.36)

P(v) = ⟨E 󸀠 (v), −JA󸀠 (v)⟩.

(4.1.37)

where we define

Proof. We note that d 󵄨󵄨󵄨󵄨 du 󵄨󵄨󵄨󵄨 󵄨󵄨 E(R(λ, v)) = ⟨E(v), 󵄨 ⟩ = P(v) dλ 󵄨󵄨λ=0 dλ 󵄨󵄨󵄨λ=0 and 󵄨󵄨 d2 u d2 du du 󵄨󵄨 󸀠 E(R(λ, v)) , ⟩ + ⟨L (ϕ), ⟩ = ⟨H 󵄨 󵄨 󵄨󵄨λ=0,v=ϕ dλ dλ dλ2 dλ2 = ⟨Hy, y⟩ < 0. So the second derivative is negative for v in a neighborhood of ϕ and the Taylor expansion in λ gives E(R(λ, v)) ≤ E(v) + λP(v) for all small λ. Combining this inequality with (4.1.33), E(ϕ) < E(R(Λ(v), v)) ≤ E(v) + Λ(v)P(v).

4.1 Orbital stability |

295

Lemma 4.1.6. There is a C 2 curve ψ : (−δ, δ) → Uε such that ψ(0) = ϕ, ψ󸀠 (0) = y, Q(ψ(s)) = Q(ϕ), P(ψ(s)) changes sign at s = 0, and E(ψ(s)) has a strict local maximum at s = 0. Proof. Since y is a vector tangent to the smooth manifold {v | Q(v) = Q(ϕ)}, pick a curve through ϕ tangent to y lying in this manifold. We have to show that P changes sign along this curve. Now d 󵄨󵄨󵄨󵄨 d 󵄨󵄨󵄨󵄨 󸀠 󵄨󵄨 E(ψ(s)) = 󵄨 L(ψ(s)) = ⟨L (ϕ), y⟩ = 0 ds 󵄨󵄨s=0 ds 󵄨󵄨󵄨s=0 by (4.1.15) and d2 󵄨󵄨󵄨󵄨 󸀠󸀠 󸀠 󸀠󸀠 󵄨 E(ψ(s)) = ⟨L (ϕ)y, y⟩ + ⟨L (ϕ), ψ (0)⟩ = ⟨Hy, y⟩ < 0. ds2 󵄨󵄨󵄨s=0 So E(ψ(s)) has a strict local maximum at s = 0. By (4.1.36), 0 < E(ϕ) − E(ψ(s)) ≤ Λ(ψ(s))P(ψ(s))

(4.1.38)

for small s. The inequality is strict because ψ󸀠 (0) = y ≠ T 󸀠 (0)ϕ. So it suffices to show that Λ(ψ(s)) changes sign at s = 0. Differentiating the defining equation (4.1.35) for Λ(ψ(s)), we get M 󸀠 (R(Λ(ψ(s)), ψ(s)){

𝜕R dΛ 𝜕R dψ + }, χ) = 0. 𝜕λ ds 𝜕v ds

If s = 0, then ψ(0) = ϕ, Λ(ψ(0)) = Λ(ϕ) = 0, 𝜕R(0, ϕ) 𝜕R(0, v) = I, ψ󸀠 (0) = y, = y. 𝜕v 𝜕λ Hence (M 󸀠 (ϕ){y

dΛ 󵄨󵄨󵄨󵄨 󵄨 + y}, χ) = 0. ds 󵄨󵄨󵄨s=0

However, (M 󸀠 (ϕ)y, χ) = (y + ⟨s󸀠 (ϕ), χ⟩T 󸀠 (0)ϕ, χ) = (y, χ) = 0, whence dΛ(ψ(s)) 󵄨󵄨󵄨󵄨 = −1 ≠ 0. 󵄨󵄨 󵄨󵄨s=0 ds

(4.1.39)

296 | 4 Existence and stability of solitary waves Lemma 4.1.7. Let X and W be real Banach spaces with W densely embedded in X ∗ . Let u ∈ C(I , X) ∩ C 1 (I ; W ∗ ), where I is an open interval of ℝ. Let A ∈ C 1 (X, ℝ) with A󸀠 ∈ C(X, W). Then A ⋅ u ∈ C 1 (I ) and dA(u(t)) du = ⟨ (t), A󸀠 (u(t))⟩. dt dt

(4.1.40)

Proof. Since W ⊂ X ∗ , it follows that X ⊂ X ∗∗ ⊂ W ∗ . The last pairing is between W ∗ and W. First we cut off and mollify in the time variable. Let ρ ∈ Cc∞ (ℝ) be a positive function with integral 1. Let ζn ∈ Cc∞ (I ) with ζn → 1 in I . Let un (t) = ∫ nρ(n(t − τ))ζn (τ)u(τ)dτ.

(4.1.41)

Then un ∈ C 1 (I , X) → u in C(I , X) and u󸀠n → u󸀠 in C(I , W ∗ ). Now du d A(un (t)) = ⟨ n (t), A󸀠 (un (t))⟩, dt dt

(4.1.42)

the pairing being between X and X ∗ . Since A󸀠 ∈ C(X, W), the pairing may also be regarded between W ∗ and W. As n → ∞, A󸀠 ⋅ un → A ∘ u in C(I , W) and A ⋅ un → A ⋅ u in C(I ). Therefore a passage to the limit yields (4.1.40). Theorem 4.1.8. If d󸀠󸀠 (ω) < 0, the ϕ-orbit is unstable. Proof. Recall that J : D(J) ⊂ X ∗ → X is a closed linear operator. Let W = D(J) with the graph norm ‖v‖2W = ‖v‖2 + ‖Jv‖2 . Then W is a Hilbert space, and J : W → X and J ∗ : X ∗ → W ∗ are continuous. By definition (4.1.13), a solution of the evolution equation is a function u ∈ C(I , X) such that d ⟨u(t), ψ⟩ = ⟨E 󸀠 (u(t)), −Jψ⟩ = ⟨J ∗ E 󸀠 (u(t)), ψ⟩ dt for all ψ ∈ W. Therefore u ∈ C 1 (I , W ∗ ) and du = −J ∗ E 󸀠 (u(t)) for t ∈ I . dt

(4.1.43)

Now fix ω and ϕ = ϕω . Fix the tube width ε so small that Lemma 4.1.5 is applicable. Let u0 = ψ(s) be given in Lemma 4.1.6 so that u0 is arbitrarily near ϕ, Q(u0 ) = Q(ϕ), E(u0 ) < E(ϕ), and P(u0 ) > 0. (P(u0 ) < 0 would also do.) According to Assumption 4.1.1, there is an interval [0, t0 ) and a solution u(t) of (4.1.43) which satisfies u(0) = u0 and Q(u(t)) = Q(u0 ) = Q(ϕ),

E(u(t)) = E(u0 ) < E(ϕ).

(4.1.44)

Since t0 depends only on μ where ‖u0 ‖ ≤ μ, either the solution u(t) can be continued to a solution for all time 0 ≤ t < ∞ which satisfies (4.1.43) or else it can be continued

4.1 Orbital stability |

297

until it blows up at a finite time T : u(t) → ∞ as t → T. In the latter case we surely have instability. In the former case we argue as follows. In any interval [0, t1 ) in which u(t) ∈ Uε , we may apply Lemma 4.1.5 and (4.1.44) to obtain 0 < E(ϕ) − E(u0 ) = E(ϕ) − E(u(t)) < Λ(u(t))P(u(t)). Therefore P(u(t)) > 0. Taking ε small if necessary, we may assume Λ(u(t)) < 1. Therefore P(u(t)) > E(ϕ) − E(u0 ) = ε0 > 0.

(4.1.45)

By Lemma 4.1.7, A ⋅ u is differentiable and d du A(u(t)) = ⟨ , A󸀠 (u)⟩ = ⟨−J ∗ E 󸀠 (u), A󸀠 (u)⟩ dt dt = ⟨E 󸀠 (u), −JA󸀠 (u)⟩ = P(u) > ε0

in the interval [0, t1 ). But 󵄨󵄨 󵄨 󵄨󵄨A(v)󵄨󵄨󵄨 ≤ ‖y‖X ∗ (‖ϕ‖ + ε) for v ∈ Uε . Therefore t1 ≤

2‖y‖X ∗ (‖ϕ‖ + ε) < ∞. E(ϕ) − E(u0 )

So the solution must exist from the tube and the ϕ-orbit is unstable. This completes the proof of Theorem 4.1.3. Most of the following is taken from the proof of Theorem 4.1.2, that is, the extension to the case when d󸀠󸀠 (ω) = 0. After that we prove Theorem 4.1.1 and then extend our results to an arbitrary Banach space X. Corollary 4.1.2. Given Assumptions 4.1.1, 4.1.2 and 4.1.3, if d(⋅) is not convex in a neighborhood of ω, then the ϕω -orbit is unstable. Proof. Let S = {ω | ω1 < ω < ω2 , the ϕω -orbit is stable}. The curve ω → ϕω is continuous. Therefore from the definition of stability, the set S is open. Let R = {ω | ω1 < ω < ω2 , d(⋅) is convex in a neighborhood of ω}. Obviously R is open. If d(⋅) is not convex in a neighborhood of ω, then ω ∈ ̸ R. Hence there is a sequence ωn → ω such that d󸀠󸀠 (ωn ) < 0. By Theorem 4.1.3, already proved, ωn ∈ ̸ S. Therefore ω ∈ ̸ S. Theorem 4.1.9. Suppose d󸀠󸀠 (ω) = 0 for some ω. Then ⟨Hy, y⟩ > 0 for all y ∈ X such that y ≠ 0 and y is orthogonal in X to each of there vectors I −1 Q󸀠 (ϕ), ϕ󸀠 , and T 󸀠 (0)ϕ.

298 | 4 Existence and stability of solitary waves Proof. By (4.1.9) and Assumption 4.1.2, JBϕ = T 󸀠 (0)ϕ ≠ 0. So by (4.1.21), Hϕ󸀠 = Q󸀠 (ϕ) = Bϕ ≠ 0 where ϕ󸀠 = dϕ/dω. By (4.1.19), ⟨Hϕ󸀠 , ϕ󸀠 ⟩ = d󸀠󸀠 (ω) = 0. It follows that ϕ󸀠 must have nontrivial components in both χ and P. We spectrally decompose ϕ󸀠 = a0 χ + b0 T 󸀠 (0)ϕ + p0 ,

p0 ∈ P.

(4.1.46)

Then a0 ≠ 0, p0 ≠ 0 and Q󸀠 (ϕ) = Hϕ󸀠 = a0 Hχ + Hp0 .

(4.1.47)

The proof of Theorem 4.1.4 is repeated exactly, except that the inequality is not strict. So if y is orthogonal to both I −1 Q󸀠 (ϕ) and T 󸀠 (0)ϕ, then ⟨Hy, y⟩ ≥ 0. Suppose now that ⟨Hy, y⟩ = 0. From the proof of Theorem 4.1.4, we would then have Schwartz’s equality ⟨Hp, p0 ⟩2 = ⟨Hp, p⟩⟨Hp0 , p0 ⟩, where y has the spectral decomposition y = aχ + p, p ∈ P. Therefore p and p0 are linearly dependent, so that y = aχ + cp0 ,

(4.1.48)

for some scalars a, c. We want to show that a = c = 0 is y is orthogonal to ϕ󸀠 as well. Thus 0 = (I −1 Q󸀠 (ϕ), y) = ⟨aHχ + Hp0 , aχ + cp0 ⟩ and 0 = (ϕ󸀠 , y) = (a0 χ + p0 , aχ + cp0 ). Hence a and c satisfy the linear system with matrix a0 ⟨Hχ, χ⟩ a0

(

⟨Hp0 , p0 ⟩ ). ‖p0 ‖2

This matrix is nonsingular since ⟨Hχ, χ⟩ and ⟨Hp0 , p0 ⟩ have opposite signs (strictly). Hence a = c = 0. Corollary 4.1.3. Suppose d󸀠󸀠 (ω0 ) = 0. If ω is sufficiently close to ω0 and if y ≠ 0 is orthogonal to I −1 Q󸀠 (ϕ), ϕ󸀠ω0 and T 󸀠 (0)ϕω , then ⟨Hy, y⟩ > 0. Proof. This is obvious by continuity in the variable ω. Theorem 4.1.10. Suppose d󸀠󸀠 (ω0 ) = 0 where ω is fixed. Let d󸀠󸀠 ≥ 0 in an open interval containing ω. Then there exists ε > 0 such that E(u) > E(ϕ) for all u ∈ Uε with Q(u) = Q(ϕ), and u ≠ T(s)ϕ for s ∈ ℝ.

4.1 Orbital stability |

299

Proof. Recall that we have abbreviated ϕ = ϕω with ω fixed. Given u ∈ Uω we define y = T(s)u − ϕΩ − aI −1 Q󸀠 (ϕΩ ).

(4.1.49)

We claim that the three parameters, s, Ω and a, can be chosen depending on u so that y is orthogonal to the three vectors T 󸀠 (0)ϕΩ , ϕ󸀠ω , and I −1 Q󸀠 (ϕΩ ). This can be accomplished in some neighborhood of ϕω by the implicit function theorem provided a certain 3 × 3 determinant does not vanish. First note that when s = 0, Ω = ω, and a = 0, we have y = u − ϕω . Now the three orthogonality conditions 0 = (y, T 󸀠 (0)ϕΩ ) = (y, ϕ󸀠ω ) = (y, I −1 Q󸀠 (ϕΩ ))

(4.1.50)

with y given by (4.1.49) are three scalar equations for s, Ω and a. The Jacobian, evaluated at s = 0, Ω = ω, a = 0, and u = ϕω = ϕ, is ‖T 󸀠 (0)ϕ‖2 (T 󸀠 (0)ϕ, ϕ󸀠 ) 󸀠 󸀠 [⟨Q (ϕ), T (0)ϕ⟩ = 0 [ [

0 −‖ϕ󸀠 ‖2 −⟨Q󸀠 (ϕ), ϕ⟩ = 0

−⟨Q󸀠 (ϕ), T 󸀠 (0)ϕ⟩ = 0 ] −⟨Q󸀠 (ϕ), ϕ⟩ = 0 ] . −‖I −1 Q󸀠 (ϕ)‖2 ]

This is a triangular matrix with nonzero diagonal entries and so it is nonsingular. This proves the claim. Furthermore |s| + |Ω − ω| + |a| = O(‖u − ϕω ‖) = O(‖y‖)

(4.1.51)

as u → ϕω . Denoting v = T(s)u, a Taylor expansion yields Q(u) = Q(v) = Q(ϕΩ ) + ⟨Q󸀠 (ϕΩ ), v − ϕΩ ⟩ + O(‖v − ϕΩ ‖2 ) = Q(ϕω ) + ⟨Q󸀠 (ϕω ), ϕ󸀠ω ⟩(Ω − ω) + O((Ω − ω)2 )

+ ⟨Q󸀠 (ϕΩ ), y + aI −1 Q󸀠 (ϕΩ )⟩ + O(‖v − ϕΩ ‖2 ) 󵄩 󵄩2 = Q(ϕω ) + O((Ω − ω)2 ) + a󵄩󵄩󵄩I −1 Q󸀠 (ϕΩ )󵄩󵄩󵄩 + O(‖v − ϕω ‖2 ).

Since Q(u) = Q(ϕω ) by assumption and since Q󸀠 (ϕω ) ≠ 0, we deduce that |a| = O((Ω − ω)2 + ‖v − ϕΩ ‖2 ) = O(‖y‖2 ).

(4.1.52)

Now we expand E(v) − ΩQ(v) around v = ϕΩ , noting that E 󸀠 (ϕΩ ) − ΩQ󸀠 (ϕΩ ) = 0 by (4.1.16) and E 󸀠󸀠 (ϕΩ ) − ΩQ󸀠󸀠 (ϕΩ ) = HΩ by definition. Thus E(u) − ΩQ(u) = E(v) − ΩQ(v) 1 = d(Ω) + ⟨HΩ (v − ϕΩ ), v − ϕΩ ⟩ + o(‖v − ϕΩ ‖2 ) 2 We substitute v − ϕΩ = y + I −1 Q󸀠 (ϕΩ ) to get 1 d(Ω) + ⟨HΩ y, y⟩ + O(a‖y‖) + O(a2 ) + o(‖v − ϕΩ ‖2 ). 2

300 | 4 Existence and stability of solitary waves By (4.1.52) all the error terms may be written as o(‖y‖2 ). By (4.1.50), the conditions of Corollary 4.1.3 are satisfied with ω0 replaced by ω and ω by Ω. Therefore we deduce E(u) − ΩQ(u) ≥ d(Ω) + δ‖y‖2 + O(‖y‖2 ). This is strictly greater than d(Ω) provided y ≠ 0. By assumption u ≠ T(r)ϕω for all r ∈ ℝ. Taking a small enough tube Uε , we have u ≠ T(r)ϕΩ for all r ∈ ℝ, so that y ≠ 0. Therefore E(u) − ΩQ(u) > d(Ω) ≥ d(ω) + d󸀠 (ω)(Ω − ω) = E(ϕω ) − ωQ(ϕω ) − Q(ϕω )(Ω − ω) = E(ϕω ) − ΩQ(ϕω )

because d is convex. Since Q(u) = Q(ϕω ), we conclude that E(u) > E(ϕω ). Proof of Theorem 4.1.2. The cases when d is not convex and d󸀠󸀠 (ω) > 0 were treated in Corollary 4.1.2 and Theorem 4.1.5. Let d be convex near ω and let d󸀠󸀠 (ω) = 0. Mimicking the proof of Theorem 4.1.5, we have vn ∈ Uε , Q(vn ) = Q(ϕ). By Theorem 4.1.10 we must have 󵄩 󵄩 inf 󵄩󵄩vn − T(s)ϕ󵄩󵄩󵄩 → 0.

s∈ℝ 󵄩

Therefore infs∈ℝ ‖un (tn ) − T(s)ϕ‖ → 0, which contradicts (4.1.24). This completes the proof. Proof of Theorem 4.1.1. By the assumption on H, it is obvious that ⟨Hy, y⟩ > 0 for any nonzero vector y orthogonal to the kernel T 󸀠 (0)ϕ. By a simple Taylor expansion we have exactly the conclusion of Theorem 4.1.5. Finally, the stability is proved exactly as in Theorem 4.1.6. Extension to Banach spaces. Let X be any real Banach space. Let J and E be as before. Let T be a one-parameter strongly continuous group of isometries of X onto X. Assume (4.1.4), (4.1.8), and (4.1.9), except that B is merely symmetric, ⟨Bu, v⟩ = ⟨Bv, u⟩ for u, v ∈ X. Define Q(u) by (4.1.10). Assumption 4.1.1 and 4.1.2 are unchanged with an exception noted below. Without the inner product we have to rephrase Assumption 4.1.3 as follows. For each ω ∈ (ω1 , ω2 ), let Hω = E 󸀠󸀠 (ϕω ) − ωQ󸀠󸀠 (ϕω ). We assume (i) There exists χ ∈ X such that ⟨Hχ, χ⟩ < 0. (ii) There exists a closed subspace P ⊂ X such that ⟨Hp, p⟩ ≥ δ‖p‖2

for p ∈ P.

(iii) For all u ∈ X, there exist unique constants a, b, and a unique p ∈ P such that u = aχ + bT 󸀠 (0)ϕ + p.

4.1 Orbital stability | 301

Then we define Πp (u) = p, Π0 (u) = b, and ΠN (u) = a. These operators are continuous. We replace Assumption 4.1.2 (c) by (C1 ) The functional u → Π0 (T 󸀠 (s)u) belongs to D(J); and (C2 ) The functional u → Π0 (T 󸀠 (s)2 u) belongs to X ∗ for all s ∈ ℝ. Theorem 4.1.11. If X is a Banach space and the above assumptions hold, then Theorem 4.1.2 and 4.1.3 are valid. Example. Traveling waves of nonlinear wave equations. Traveling wave solutions of the nonlinear wave equation utt − uxx + f (u) = 0

(4.1.53)

are generated by the invariance of the equation under space translation. Equation (4.1.53) can be written in the form (4.1.12) as du = JE 󸀠 (u), dt where u u = [ ], v

J=[

0 −1

1 ], 0

1 1 E(u) = ∫( v2 + u2x + F(u))dx, 2 2 F󸀠 = f ,

(4.1.54)

F(0) = 0.

If we define the space X = H 1 (ℝ) × L2 (ℝ), then X ∗ = H −1 (ℝ) × L2 (ℝ) and we have the isomorphism I : X → X ∗ , where I=[

−Δ + 1 0

0 ]. 1

The above initial value problem is well-defined on the space X; T(s) is a well-defined unitary group on X with 𝜕 T 󸀠 (0) = [ x 0

0 ], −𝜕x

D(T 󸀠 (0)) = H 2 (ℝ) × H 1 (ℝ) ⊂ X

and J −1 T(s) = T ∗ (s)J −1 . The invariance generates the conserved quantity (momentum) 1 Q(u) = ⟨Bu, u⟩ = ∫ ux vdx, 2

(4.1.55)

302 | 4 Existence and stability of solitary waves where 0 B=[ −𝜕x

𝜕x ]. 0

Now for traveling waves to exist we should find nonzero solutions of the equation E 󸀠 (Φ) − ωQ󸀠 (Φ) = 0. ϕ

The components of Φ = [ ψ ] will satisfy the equations −(1 − ω2 )ϕxx + f (ϕ) = 0,

(4.1.56)

ψ = ωϕx .

(4.1.57)

Suppose that f satisfies the following conditions: (i) f 󸀠 (0) > 0. (ii) ∃η such that F(η) < 0. (Consequently from (i), F(η) < 0 has at least one zero u0 .) (iii) If u0 is the zero of F with smallest nonzero absolute value, then f 󸀠 (u0 ) ≠ 0. Lemma 4.1.8. If f satisfies (i)–(iii), then the equation −pxx − f (p) = 0 has a unique solution that satisfies (1) p(x) > 0, p(x) = p(−x), p(0) = u0 . (2) p(x) decays exponentially like e−c|x| with c > 0. Denote ϕω (x) = p(x/√1 − ω2 ), ω ∈ (−1, 1). Then ϕω satisfies (4.1.56) and we have a nontrivial traveling wave. The linearized operator is Lω = −(1 − ω2 )𝜕x2 + f 󸀠 (ϕω ).

(4.1.58)

The kernel of Lω is spanned by 𝜕x ϕω and has a simple zero at x = 0, Lω has exactly 2 one strictly negative eigenvalue −αω , with an eigenfunction χω , 2 Lω χω = −αω χω .

(4.1.59)

In order to verify Assumption 4.1.3 of the theorem, we shall compute the spectrum of the operator L Hω = E 󸀠󸀠 (ϕω ) − ωQ󸀠󸀠 (ϕω ) = [ 0 −ω𝜕x

ω𝜕x ]. 1

Lemma 4.1.9. The spectrum of the operator Hω is as follows: (1) There is a negative simple eigenvalue.

(4.1.60)

4.1 Orbital stability | 303

(2) The kernel is spanned by T 󸀠 (0)ϕω . (3) The positive spectrum of Hω is bounded away from zero. ψ

Proof. Let Ψ = [ ψ1 ] be an eigenfunction of Hω with eigenvalue λ. Writing the compo2 nents of the equation Hω Ψ = λΨ, we have −𝜕x2 ψ1 + f 󸀠 (ϕω )ψ1 + ω𝜕x ψ2 = λψ1 ,

−ω𝜕x ψ1 + ψ2 = λψ2 .

For λ ≠ 1 we can rewrite the above equations as −(1 − ω2 )𝜕x2 ψ1 + f (ϕω )ψ1 = ψ2 =

ω 𝜕ψ. λ−1 x 1

λ(1 − ω2 ) − λ2 ψ1 , 1−λ

(4.1.61) (4.1.62)

If λ < 0 then by (4.1.59) we have λ(1 − ω2 ) − λ2 2 = −αω , 1−λ or 2 2 λ2 − (1 − ω2 − αω )λ − αω = 0,

which has exactly one negative root. Therefore Hω has exactly one negative eigenvalue λ− (ω). Next we substitute λ = 0 in (4.1.61) and we get the kernel of Hω spanned by T (0)ϕω . By Weyl’s theorem on the hypotheses of the theorem, the stability of the traveling waves is determined by the sign of d󸀠󸀠 (ω). But 󸀠

d󸀠 (ω) = −Q(ϕω ) = −ω ∫ |𝜕x ϕω |2 dx. Therefore d󸀠󸀠 (ω) = (√1 − ω2 ) ∫ |𝜕x p|2 dx < 0. 󸀠󸀠

Hence all traveling waves are unstable.

304 | 4 Existence and stability of solitary waves Remark. The well-known kink solution ϕω (x−ωt) travels monotonically from one zero of f to another. Therefore, 𝜕x ϕω does not vanish and is the lowest eigenfunction of the operator Lω with eigenvalue zero. Hence Lω ≥ 0. Now Hω is again given by formula (4.1.60). By (4.1.61), Hω cannot have any negative eigenvalues. By Theorem 4.1.1, the kinks are always stable. In order to obtain the conditions of the stability and instability of the solitary wave solution, we seek the conditions of the self-adjoint operator H. Let u(t) = T(etω )ϕ, where ω ∈ g and ϕ ∈ X, then T(etω )ϕ is a bound state if only if E 󸀠 (ϕ) = Q󸀠ω (ϕ). Assume that H is positive definite, JH is similar to √HJ √H. But the spectrum of H is imaginary. Because H is the second order derivative of E(ϕ) − Qω (ϕ), we hope it is stable when E(ϕ) − Qω (ϕ) arrives at the minimum. But Qω keeps invariant, so it is sufficient for us to show that the energy E can arrive at a minimum when the constant charges are concentrated. We have the following theorem: Theorem 4.1.12. If E(v) is restricted to the set 󵄨 {v ∈ X 󵄨󵄨󵄨 Qσ (v) = Qσ (ϕω ), ∀σ ∈ gω }, when ϕω arrives at a minimum, then ϕω is stable. Here gω is a Lie group. In general, the minimal problem for the constrained, linearized Hamiltonian H has some negative spectrum. But if there is not too much negative spectrum, the solitary wave is still stable. We give the following decision theorems: Definition 4.1.6. Define d(σ) : gω → ℝ by d(σ) = E(ϕσ ) − Qσ (ϕσ ),

(4.1.63)

and let d󸀠󸀠 (σ) denote its Hessian matrix. Theorem 4.1.13. If the number of negative eigenvalues of H, denoted neg(H), is equal to the number of positive eigenvalues of d󸀠󸀠 (σ), denoted pos(d󸀠󸀠 ), then the solitary wave is stable. Theorem 4.1.14. If neg(H) − pos(d󸀠󸀠 ) is odd, then the solitary wave is unstable. As an application of the above theorems, we give some examples. Example 1. Nonlinear Schrödinger equation iut − Δu − |u|p−1 u = 0,

x ∈ ℝn .

We know that 1 1 |u|p+1 )dx, E(u) = ∫( |∇u|2 − 2 p+1

(4.1.64)

4.1 Orbital stability | 305

Q(u) =

1 ∫ |u|2 dx, 2

J = i, G = S1 : u(x) → exp(iθ)u(x), θ is a real constant. Now we apply Theorems 4.1.13 and 4.1.14. Choose space X = H 1 (ℝn , ℂ) = H 1 (ℝn , ℝ) + H 1 (ℝn , ℝ), that is, we split the space into two parts, real and imaginary. The linearized Hamiltonian H is a 2 × 2 diagonal matrix, the diagonal elements are denoted by R and S, respectively, and given by R = −Δ + ω − p|ϕ|p−1 ,

S = −Δ + ω − |ϕ|p−1 .

The kernel space of S is one-dimensional and spanned by ϕ. Because ϕ(x) is positive, 𝜕ϕ the minimal eigenvalue of S is positive, S ≥ 0. The kernel space of R is spanned by 𝜕x , k k = 1, 2, . . . , n; they are not positive. The minimal eigenvalue of R is negative and ϕ is the minimizer of inf

1 ∫ |∇ϕ|2 dx, n

with the constraint 1 1 1 1 |ϕ|p+1 ]dx = 0. ∫[− ωϕ2 + ( − )|∇ϕ|2 − 2 2 n p+1 So we can get neg(H) = neg(R) + neg(S) = 1 + 0 = 1. Moreover, d(ω) = E(ϕ) − ωQ(ϕ), { { { { { { {d󸀠 (ω) = −⟨E(ϕ) − ωQ(ϕ), dϕ ⟩ − Q(ϕ) = −Q(ϕ), dω { { { { { { {d󸀠󸀠 (ω) = −⟨Q󸀠 (ϕ), dϕ ⟩ = −⟨H( dϕ ), dϕ ⟩ dω dω dω { yield 1 d󸀠 (ω) = − ∫ |ϕ|2 dx, 2 where ϕ satisfies −ωϕ − Δϕ − |ϕ|p−1 ϕ = 0. Set 1

ϕ(x) = (−ω) p−1 Ψ(√−ωx), where Ψ satisfies Ψ − ΔΨ − |Ψ|p−1 Ψ = 0.

(4.1.65)

306 | 4 Existence and stability of solitary waves Hence 1 d󸀠 (ω) = −(−ω)b ∫ |Ψ|2 dx, 2 󸀠󸀠 b−1 1 d (ω) = b(−ω) ∫ |Ψ|2 dx, 2 where b =

2 p−1

− n2 . If 1 < p < 1 + 41 , then b > 0, d󸀠󸀠 (ω) > 0 and neg(H) = 1 =

1 pos(d󸀠󸀠 ), so by Theorem 4.1.13, the solitary wave is stable. If 1 + 41 < p < 1 + n−2 , then 󸀠󸀠 neg(H) − pos(d ) = 1 − 0 = 1 is odd, so by Theorem 4.1.14, the solitary wave is instable.

Example 2. The coupled wave equation is utt − uxx + u − |u|2 u = 0,

(4.1.66)

where x ∈ ℝ, u(x, t) ∈ ℝ3 . Consider the solitary wave as follows: u = ets ϕ(x + ω0 t),

(4.1.67)

where ω0 is a real constant, S is a 3 × 3 skew-symmetric matrix. Write Sy = ω ∧ y, here ω ∈ ℝ3 . Substituting (4.1.67) into (4.1.66), ϕ satisfies the equation −𝜕x2 ϕ + ϕ − |ϕ|2 ϕ + (ω0 𝜕x + S2 )ϕ = 0. Let ϕ(x) = η(x) exp(αxS)ν, where ν is a unit vector orthogonal to ω. Then η(x) satisfies −(1 − ω20 )𝜕x2 η − |ω|2 (1 − ω20 ) η + η − η3 = 0. −1

Let ω20 − |ω|2 < 1. Then there is a unique positive solution η(x), and when |x| → ∞, the solution decays exponentially to 0. Equation (4.1.66) can be written in Hamiltonian form. The energy E is 1 1 1 1 E(u) = ∫[ u2t + |∇u|2 + |u|2 − |u|4 ]dx. 2 2 2 4 Also J is an skew-symmetric matrix, and the Lie group G = SO(3) × ℝ acts as u(x) → Ru(x + a), for R ∈ SO(3), a ∈ ℝ, that is, rotation acts on u and scaling acts on x. The charges are ∫ Su ⋅ vdx and ∫ 𝜕x u ⋅ vdx, where S is an arbitrary skew-symmetric matrix, u = [u, v]T ∈ X. The linearized Hamiltonian is −𝜕2 + 1 + 3ϕ2 H = [ xx 2 −ω0 𝜕x − S

ω0 𝜕x + S2 ]. 1

(4.1.68)

4.1 Orbital stability | 307

Similar to the Schrödinger equation, we can prove that H has exactly one negative eigenvalue. Also d(ω) takes the form 1 1 󵄨2 󵄨 ∫[|𝜕x ϕ|2 + |ϕ|2 − |ϕ|4 + 󵄨󵄨󵄨(ω0 𝜕x + S)ϕ󵄨󵄨󵄨 ]dx 2 2 1 1 −1 = ∫[(1 − ω20 )|𝜕x η|2 + (1 − |ω|2 (1 − ω20 ) )η2 − η4 ]dx. 2 4

d(ω0 , ω1 , ω2 , ω3 ) =

We must compute the Hessian of d(ω0 , |ω|), where 3

d(ω0 , |ω|) = C(1 − ω20 ) (1 − ω20 − |ω|2 ) 2 . −1

(4.1.69)

If ω0 or |ω| is small, then d󸀠󸀠 (ω0 , |ω|) has two negative eigenvalues. Hence neg(H) − pos(d󸀠󸀠 ) = 1 − 0 = 1. By Theorem 4.1.14, the solitary wave is unstable. 1 On the other hand, if |ω| is near (1 − ω20 ) 2 , then d󸀠󸀠 (ω0 , |ω|) has exactly one positive eigenvalue and one negative eigenvalue. Hence neg(H) − pos(d󸀠󸀠 ) = 1 − 1 = 0. By Theorem 4.1.13, the solitary wave solution is stable. Example 3. The generalized KdV equation is ut + uxx + (up )x = 0,

x ∈ ℝ.

(4.1.70)

Consider the real valued solution of (4.1.70) which vanishes when |ω| → ∞. For p > 1, obviously, (4.1.70) is translation invariant for x. Solitary waves are the form u(x, t) = ϕ(x − ct). Because ϕ satisfies an ordinary differential equation, it is easy to show that solitary wave solutions exist for all c > 0, p > 1. Equation (4.1.70) becomes the classical KdV for p = 2, it has solitary wave solutions, and the solitary wave solutions are stable. The solitary wave solutions of (4.1.70) are unstable if p is sufficiently large. The stability for p < 5 can be proved. 𝜕 Let X be a real H 1 (ℝ) space, J = 𝜕x , 1 1 p+1 E(u) = ∫[ u2x − u ]dx, 2 p+1 1 Q(u) = ∫ u2 dx. 2 The group G = ℝ is a scaling group with respect to x, and J is not a surjection, so we cannot use the above theorems, yet we can apply the modified instability theorem to be proved. In fact, we need another invariant of the KdV equation I(u) = ∫ udx.

308 | 4 Existence and stability of solitary waves 2 Suppose that p > 5, the linearized Hamiltonian is H = −𝜕xx + c − pϕp−1 , and its kernel space is spanned by 𝜕x ϕ. We can prove that there exists y ∈ X such that

⟨Hy, y⟩ < 0,

⟨y, ϕ⟩ = 0.

x

Let Y = J −1 y, Y(x) = ∫−∞ y(z)dz, ∞

A(t) = ∫ Y(x − β(t))u(x, t)dx. −∞ 𝜕ϕ

Chose β(t) such that u(x + β(t), t) is orthogonal to 𝜕x . When x → −∞, Y(x) → 0, and ∞ when x → −∞, y(x) → γ = ∫−∞ y(z)dz. Hence A(t) is approximately equal to ∞

∫ u(x, t)dx. β(t)

Using the estimate 󵄨󵄨 ∞ 󵄨󵄨 2 󵄨󵄨 󵄨 󵄨󵄨 ∫ u(x, t)dx󵄨󵄨󵄨 ≤ c1 (1 + t 3 ), 󵄨 󵄨 󵄨󵄨 −∞ 0 and δ > 0, such that if 󵄩󵄩 󵄩 󵄩󵄩u(0) − ϕ󵄩󵄩󵄩 < δ, we have 󵄩 󵄩 inf 󵄩󵄩u(t) − T(s)ϕc 󵄩󵄩󵄩 > ε1 .

s∈ℝ 󵄩

Hence it is unstable.

4.2 The nonlinear derivative Schrödinger equation Consider the following nonlinear quintic derivative Schrödinger equation: ut = iuxx + ig(|u|2 )u + [(s0 + s2 |u|2 )u]x ,

x ∈ ℝ,

(4.2.1)

4.2 The nonlinear derivative Schrödinger equation

| 309

with g(|u|2 ) = c3 |u|2 + c5 |u|4

(4.2.2)

and s0 , s2 , c3 , c5 being real constants. It is obvious that if u(x, t) is a solution of (4.2.1), then u(x − s0 t, t) must be a solution of the following equation: ut = iuxx + ig(|u|2 )u + s2 (|u|2 u)x ,

x ∈ ℝ.

(4.2.3)

Furthermore if e−iωt eiψ(x−vt) a(x − vt) is a solitary wave of (4.2.3), then e−iωt eiψ(x−(v−s0 )t) a(x − (v − s0 )t) is a solitary wave of (4.2.1). Consider the following equation: ut = iuxx + i(c3 |u|2 + c5 |u|4 )u + s2 (|u|2 u),

u ∈ ℝ,

(4.2.4)

with s2 , c3 , c5 being real numbers. The solitary waves of (4.2.4) are assumed in the following form: e−iωt eiψ(x−vt) a(x − vt)

(4.2.5)

with ω, v real numbers, and ψ(⋅), a(⋅) real functions. It was show in [94] that there exist solitary waves in the form of (4.2.5) with a2 (ξ ) = (d3 + d5 cosh d6 ξ )−1 , ξ ∈ ℝ, v 3 ψ󸀠 (ξ ) = + s2 a2 (ξ ), 2 4 d3 = −(d2 /2d0 ), d52 = (d22 − 4d0 d4 )/4d02 ,

(4.2.8)

d62 = 4d0 ,

(4.2.9)

d0 = −ω −

1 d4 = − (c5 + 3s22 /16). 3

2

v , 4

1 1 d2 = − c3 − s2 v, 2 4

(4.2.6) (4.2.7)

(4.2.10)

We only consider the nonnegative solution of (4.2.4) Assumption 4.2.1. The following conditions are assumed to hold: (a) d0 > 0, d2 < 0, (b)

either d4 ≤ 0 or d4 > 0 and d22 − 4d0 d4 > 0.

(4.2.11)

By (4.2.6)–(4.2.10) and Assumption 4.2.1, we have the following existence results: Theorem 4.2.1. For any fixed real constants c3 , c5 , s2 , if ω, v satisfy Assumption 4.2.1, then there exist solitary wave solutions e−iωt eiψ(x−vt) a(x − vt) of (4.2.1), with a(ξ ), ψ(ξ ) in the form of (4.2.6)–(4.2.7).

310 | 4 Existence and stability of solitary waves Let ̂ a(x) = eiψ(x) a(x).

(4.2.12)

̂ satisfies By (4.2.4)–(4.2.12), it is easy to see that a(x) − â xx − g(a2 )â + is2 (a2 a)̂ x − ωâ + ivâ x = 0

(4.2.13)

and a(x) satisfies 2

a󸀠󸀠 + g(a2 ) + s2 a2 ψ󸀠 − (ψ󸀠 ) + ω + vψ󸀠 = 0,

(4.2.14)

with g(a2 ) = c3 a2 + c5 a4 . Now we consider the following initial value problem: ut = iuxx + i(c3 |u|2 + c5 |u|4 )u + s2 (|u|2 u)x ,

u(0, x) = u0 (x),

x ∈ ℝ,

x ∈ ℝ.

(4.2.15) (4.2.16)

The function space in which we shall work is the complex space X = H 1 (ℝ), with real inner product ̄ (u, v) = Re ∫(ux + v̄x + uv)dx.

(4.2.17)

ℝ

The dual space of X is X = H (ℝ), there is a natural isomorphism I : X → X ∗ defined by ∗

−1

⟨Iu, v⟩ = (u, v),

(4.2.18)

where ⟨⋅, ⋅⟩ denotes the pairing between X and X ∗ , ̄ ⟨f , u⟩ = Re ∫ f udx.

(4.2.19)

ℝ

By (4.2.17)–(4.2.19), it is obvious that I=−

𝜕2 + 1. 𝜕x2

(4.2.20)

Let T1 , T2 be one-parameter groups of unitary operators on X defined by T1 (s1 )ϕ(⋅) = e−s1 i ϕ(⋅),

T2 (s2 )ϕ(⋅) = ϕ(⋅ − s2 ),

ϕ(⋅) ∈ X ϕ(⋅) ∈ X

s1 ∈ ℝ,

s2 ∈ ℝ.

(4.2.21) (4.2.22)

Obviously, T1󸀠 (0) = −i, T2󸀠 (0) = −𝜕/𝜕x. By Theorem 4.2.1, we have obtain the existence of solitary wave solutions T1 (ωt) × T2 (vt)â ω,v of (4.2.15), with â ω,v defined by (4.2.6)–(4.2.10) and (4.2.12). In this and in the following, we shall consider the orbital stability of solitary waves T1 (ωt)T2 (vt)â ω,v . Note that equation (4.2.15) has phase and translation symmetries. We define the orbital stability as follows.

4.2 The nonlinear derivative Schrödinger equation

| 311

Definition 4.2.1. The solitary wave T1 (ωt)T2 (vt)â ω,v is orbitally stable if for all ε > 0 there exists δ > 0 with the following property. If ‖u0 − â ω,v ‖X < δ and u(t) is a solution of (4.2.15) in some interval [0, t0 ) with u(0) = u0 , then u(t) can be continued to a solution in 0 ≤ t < ∞ and 󵄩 󵄩 sup inf inf 󵄩󵄩󵄩u(t) − T1 (ωt)T2 (vt)â ω,v 󵄩󵄩󵄩X < ε. s1 ∈ℝ s2 ∈ℝ

0 0.

(4.2.46)

Thus we have the following spectral properties of L̄ 11 . Proposition 4.2.1. Operator L̄ 11 has exactly one negative simple eigenvalue and has its kernel spanned by a󸀠 (x) while the rest of its spectrum is positive and bounded away from zero. It follows from Proposition 4.2.1 that for any real function z1 ∈ H 1 (ℝ) satisfying ⟨z1 , a󸀠 ⟩ = ⟨z1 , χ11 ⟩,

(4.2.47)

there exists a positive number δ̄ 1 > 0 such that ⟨L̄ 11 z1 , z1 ⟩ = δ̄ 1 ‖z1 ‖2L2 .

(4.2.48)

Furthermore, we can prove the following lemma: Lemma 4.2.1. For any real function z1 ∈ H 1 (ℝ) satisfying (4.2.47), there exists a positive number δ1 > 0 such that ⟨L̄ 11 z1 , z1 ⟩ ≥ δ1 ‖z1 ‖2H 1 ,

(4.2.49)

where δ1 is independent of z1 . It follows from (4.2.6) and (4.2.41) that 0 is the first eigenvalue of L22 , with an eigenfunction a(x). Note that L22 =

𝜕2 + d0 + M2 (x), 𝜕x2

(4.2.50)

316 | 4 Existence and stability of solitary waves and (4.2.6), (4.2.8)–(4.2.10) imply M2 (x) → 0,

as |x| → ∞,

(4.2.51)

thus σess (L22 ) = [d0 , ∞),

d0 > 0.

(4.2.52)

We have the following spectral properties of L22 : Proposition 4.2.2. Operator L22 has its kernel spanned by a(x) while the rest of its spectrum is positive and bounded away from zero. By Proposition 4.2.2 and (4.2.50)–(4.2.51), similarly we have Lemma 4.2.2. For any real function z2 ∈ H 1 (ℝ) satisfying ⟨z2 , a⟩ = 0,

(4.2.53)

there exists a positive number δ2 > 0 such that ⟨L22 z2 , z2 ⟩ ≥ δ2 ‖z2 ‖2H 1 ,

(4.2.54)

where δ2 is independent of z2 . To prove Theorem 4.2.3, we first verify that Assumption 4.2.2 holds and n(Hω,v ) = 1.

(4.2.55)

For any ϕ(x) ∈ X, let ϕ(x) = eiψ(x) (z1 (x) + iz2 (x)) and

z2 (x) = a(x)z3 (x),

(4.2.56)

with z1 , z2 , z3 real functions, z1 , z2 ∈ H 1 (ℝ). Denote M̄ 2 = d0 + M2 (x), and notice that ⟨L22 z2 , z2 ⟩ = ⟨−(z2 )󸀠󸀠 , z2 ⟩ + ⟨M̄ 2 (x)z2 , z2 ⟩ = ⟨L22 a, az32 ⟩ − ⟨(a2 z3󸀠 ) , z3 ⟩ 󸀠

= ⟨a2 z3󸀠 , z3󸀠 ⟩ = ⟨az3󸀠 , az3󸀠 ⟩.

(4.2.57)

It follows from (4.2.38) and (4.2.56)–(4.2.57) that ⟨Hω,v ϕ, ϕ⟩ = ⟨L̄ 11 z1 , z1 ⟩ + ⟨az3󸀠 , az3 ⟩

9 + ⟨ s22 a4 z1 , z1 ⟩ − 3⟨s2 a3 z3󸀠 , z1 ⟩ 4 2

3 = ⟨L̄ 11 z1 , z1 ⟩ + ∫( s2 a2 z1 − az3󸀠 ) dx. 2 ℝ

(4.2.58)

4.2 The nonlinear derivative Schrödinger equation

| 317

Choose χ− = (χ11 + iχ12 )eiψ(x)

(4.2.59)

and x

3 χ12 = aχ13 = a( s2 ∫ a(s)χ11 (s)ds + k1 ), 2

(4.2.60)

−∞

with k1 any real number. Then (4.2.56) and (4.2.58)–(4.2.60) imply 2 ⟨Hω,v χ− , χ− ⟩ = ⟨L̄ 11 χ11 , χ11 ⟩ = −λ11 < 0.

(4.2.61)

⟨χ12 , a⟩ = 0.

(4.2.62)

N = {kχ− | k ∈ ℝ},

(4.2.63)

Fix k1 such that

Denote

then (4.2.61) and (4.2.63) imply (4.2.34). Let χ1 = (a󸀠 (x) + i(k2 + χ2 = ia(x)eiψ(x)

3 s a2 (x))a(x))eiψ(x) , 4 2

(4.2.64) (4.2.65)

and choose k2 such that ⟨(k2 +

3 s a2 )a, a⟩ = 0, 4 2

(4.2.66)

then Z can be rewritten as Z = {k1 χ1 + k2 χ2 | k1 , k2 ∈ ℝ}.

(4.2.67)

Choose subspace P as 󵄨 P = {p ∈ X 󵄨󵄨󵄨 p = (p1 + ip2 )eiψ(x) , ⟨p1 , χ11 ⟩ = 0, ⟨p1 , a󸀠 ⟩ = 0, ⟨p2 , a⟩ = 0}.

(4.2.68)

For this subspace P, we shall prove that (4.2.33) holds. Indeed, for any ϕ(x) ∈ X, ϕ(x) can be uniquely represented by ϕ(x) = a1 χ− + b1 χ1 + b2 χ2 + p,

(4.2.69)

318 | 4 Existence and stability of solitary waves where p ∈ P, a1 , b1 , b2 are real numbers. Let ϕ(x) = (z1 + iz2 )eiψ(x) and choose a1 = ⟨z1 , χ11 ⟩,

⟨z1 , a󸀠 ⟩ , ‖a󸀠 ‖2L

b1 =

b2 =

2

⟨z2 , a⟩ . ‖a‖2L

(4.2.70)

2

Then (4.2.40), (4.2.42), (4.2.62)–(4.2.66) imply that (4.2.69) holds with p ∈ P. For subspace P, it remains to prove that (4.2.35) holds. And indeed, Lemma 4.2.3. For any p ∈ P defined by (4.2.68), there exists a constant δ > 0 such that ⟨Hω,v p, p⟩ ≥ δ‖p‖2H1 ,

(4.2.71)

where δ is independent of p. Proof. For any p = (p1 + ip2 )eiψ(x) ∈ P, let p2 = ap. Then by (4.2.58), we have 2

3 ⟨Hω,v p, p⟩ = ⟨L̄ 11 p1 , p1 ⟩ + ∫( s2 a2 p1 − ap󸀠3 ) dx. 2

(4.2.72)

ℝ

Equation (4.2.68) and Lemmas 4.2.1–4.2.2 assure that ⟨L̄ 11 p1 , p1 ⟩ ≥ δ1 ‖p1 ‖2H 1 ,

⟨L22 p2 , p2 ⟩ ≥

(4.2.73)

δ2 ‖p2 ‖2H 1 .

(4.2.74)

(1) If 󵄨 󵄨2 C0 = 3s3 󵄨󵄨󵄨a(0)󵄨󵄨󵄨 ,

󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩ap3 󵄩󵄩L2 ≥ C0 ‖p1 ‖L2 ,

(4.2.75)

then 2

2

‖ap󸀠3 ‖L2 ‖ap󸀠3 ‖L2 3 3 − ∫( s2 a2 p1 ) dx ≥ ∫( s2 a2 z1 − az3󸀠 ) dx ≥ 2 2 2 4

ℝ

ℝ

1 = ⟨L22 p2 , p2 ⟩, 4

(4.2.76)

thus (4.2.72)–(4.2.75) imply ⟨Hω,v p, p⟩ ≥ δ1 ‖p1 ‖2H 1 +

δ2 ‖p ‖2 1 ≥ δ‖p‖2H 1 . 4 2 H

(2) If 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩ap3 󵄩󵄩L2 < C0 ‖p1 ‖L2 ,

󵄨 󵄨2 C0 = 3s3 󵄨󵄨󵄨a(0)󵄨󵄨󵄨 ,

then (4.2.72)–(4.2.74), (4.2.77) and (4.2.57) imply ⟨Hω,v p, p⟩ ≥ δ1 ‖p1 ‖2H 1 ≥ ≥ This completes the proof of Lemma 4.2.3.

δ 󵄩 δ1 󵄩2 ‖p1 ‖2H 1 + 12 󵄩󵄩󵄩ap󸀠3 󵄩󵄩󵄩L2 2 2C0

δ1 δδ ‖p1 ‖2H 1 + 1 22 ‖p2 ‖2H 1 . 2 2C0

(4.2.77)

4.2 The nonlinear derivative Schrödinger equation

| 319

It follows from (4.2.63) and (4.2.67)–(4.2.71) that Assumption 4.2.2 holds and n(Hω,v ) = 1. Finally, we shall prove that for any c3 , c5 , s2 , ω, v satisfying Assumption 4.2.1, p(d󸀠󸀠 ) = 1

(4.2.78)

which will complete the proof of Theorem 4.2.3. Note that ̂ d(ω, v) = E(a)̂ − ωQ1 (a)̂ − vQ2 (a), ̂ dω = −Q1 (a),

̂ dv = −Q2 (a),

̂ dω,ω = −⟨Q󸀠1 (a),

𝜕â 𝜕â ⟩ = −⟨a,̂ ⟩, 𝜕ω 𝜕ω

𝜕â , 𝜕v 𝜕â 𝜕â ̂ ⟩ = ⟨iâ x + s2 a2 a,̂ ⟩, = −⟨Q󸀠2 (a), 𝜕v 𝜕v

dvω = dωv = − dvv (

𝜕 − 21 𝜕ω ⟨a, a⟩

𝜕 − 21 𝜕v ⟨a, a⟩

𝜕 ⟨a, a⟩ − 21 𝜕v

â x = eiψ(x) a(x), ⟨iâ x + s2 a2 a,̂

) ̂ ⟨iâ x + s2 a2 a,̂ 𝜕𝜕va ⟩

𝜕ψ 𝜕â 𝜕a ⟩ = Re ∫(−ψ󸀠 a + ia󸀠 + s2 a3 )( − i )dx 𝜕v 𝜕v 𝜕v ℝ

= ∫[(−ψ󸀠 a + s2 a3 ) ℝ

𝜕ψ 𝜕a + a󸀠 a ]dx 𝜕v 𝜕v

v 1 𝜕a 1 2 𝜕2 ψ = ∫[( + s2 a3 )a − a ]dx 2 4 𝜕v 2 𝜕v𝜕x ℝ

=−

v 𝜕 1 1 𝜕 ⟨a, a⟩ − ⟨a, a⟩ − s2 ∫ a4 (x)dx, 4 𝜕v 4 8 𝜕v

v 𝜕 𝜕 1 𝜕 ⟨a, a⟩ ⟨a, a⟩ + ⟨a, a⟩ ⟨a, a⟩ det(d ) = 8 𝜕ω 𝜕v 8 𝜕ω

ℝ

󸀠󸀠

+

2

s2 𝜕 𝜕 1 𝜕 ⟨a, a⟩ ∫ a4 (x)dx − ( ⟨a, a⟩) . 16 𝜕ω 𝜕v 4 𝜕v ℝ

and a2 (x) = (d3 + d5 cosh d6 x)−1 , d3 = −(d2 /2d0 ), 𝜕d0 = −1, 𝜕ω

d52 = (d22 − 4d0 d4 )/4d02 ,

𝜕d0 1 =− v 𝜕v 2

d62 = 4d0

(4.2.79)

320 | 4 Existence and stability of solitary waves 𝜕d2 1 = − s2 , 𝜕v 4

𝜕d2 = 0, 𝜕ω

𝜕d4 𝜕d4 = = 0. 𝜕ω 𝜕v

Case (A), d4 < 0. In this case, we have ⟨a, a⟩ = ∫(d3 + d5 cosh d6 x)−1 dx ℝ

=

d3 2 1 π ( − arctan ) d6 √d2 − d2 2 2 − d2 √d 5 5 3 3

−d2 1 π ( − arctan ) > 0, √−d4 2 2√−d0 d4 d2 𝜕 ⟨a, a⟩ = 2 < 0, 𝜕ω (d2 − 4d0 d4 )√d0 =

−√d0 vd 𝜕 (s2 − 2 ), ⟨a, a⟩ = 2 𝜕v d0 2(d2 − 4d0 d4 )√d0 ∫ a4 (x)dx = ∫(d3 + d5 cosh d6 x)−1 dx ℝ +∞

ℝ

= ∫ −∞ +∞

= ∫ 0

[2d3

e2d0 x

4e2d0 x dx + d5 e2d0 x + d5 ]2

4y dy d6 [2d3 y + d5 e2d0 x + d5 y + d5 ]2

−√d0 d2 −d2 π = − ( − arctan ), d4 2d4 √−d4 2 2√−d0 d4

s2 −d2 v π 𝜕 ( − arctan ) − ∫ a4 (x)dx = 𝜕v 2√−d0 d4 4d4 √d0 8(−d4 )3/2 2 ℝ

+

√d0 d2 vd (s2 − 2 ). d0 4(d22 − 4d0 d4 )d4

By (4.2.79), we have det(d󸀠󸀠 ) =

1 𝜕 1 𝜕 𝜕 𝜕 ⟨a, a⟩ ⟨a, a⟩ + ⟨a, a⟩(v ⟨a, a⟩ − 2 ⟨a, a⟩) 8 𝜕ω 8 𝜕v 𝜕ω 𝜕v s 𝜕 𝜕 + 2 ⟨a, a⟩ ∫ a4 (x)dx 16 𝜕ω 𝜕v ℝ

=

d 𝜕 1 𝜕 1 𝜕 ⟨a, a⟩ ⟨a, a⟩ + (s 0 ⟨a, a⟩) 8 𝜕ω 8 𝜕v 2 d2 𝜕ω s 𝜕 𝜕 + 2 ⟨a, a⟩ ∫ a4 (x)dx 16 𝜕ω 𝜕v ℝ

(4.2.80)

4.2 The nonlinear derivative Schrödinger equation

=

| 321

s22 𝜕 1 ⟨a, a⟩ ⟨a, a⟩ + [ 8 𝜕ω 64(d22 − 4d0 d4 )d4 × (1 +

= I1 + I2 .

d2 −d2 π ( − arctan ))] 2 2√−d0 d4 2√−d0 d4

Equations (4.2.11) and (4.2.80) imply I1 < 0. Let y = −d2 /(2√−d0 d4 ), then y > 0 and I2 =

64(d22

s22 Y1 (y), − 4d0 d4 )d4

Y1 (y) = 1 − y(

π − arctan y). 2

Note that Y1 (0) = 1, Y1 (+∞) = lim [1 − y( y→+∞

π π/2 − arctan y − arctan y)] = 1 − lim y→+∞ 2 1/y

−1/(1 + y2 ) = 0, y→+∞ −1/y2 y π Y1󸀠 (y) = − + arctan y + , 2 1 + y2 π Y1󸀠 (0) = − , Y1󸀠 (+∞) = 0, 2 2 2y2 2 Y1󸀠󸀠 (y) = − = >0 2 1+y (1 + y2 )2 (1 + y2 )2 = 1 − lim

(4.2.81) (4.2.82)

for all y ∈ ℝ,

(4.2.83)

and then (4.2.82) and (4.2.83) imply Y1󸀠 (y) < 0 for all y ∈ ℝ. Thus by (4.2.81) we have Y1 (y) > 0

for all y ∈ ℝ,

which assures I2 < 0 (since d4 < 0). Finally, we have det(d󸀠󸀠 ) < 0, which implies that d󸀠󸀠 has exactly one negative and one positive eigenvalue, thus (4.2.78) holds for Case (A). Case (B), d4 = 0. In this case, we have a2 (x) = −

2d0 (1 + cosh d6 )−1 , d2

⟨a, a⟩ = −

−2√d0 2d0 , ∫(1 + cosh d6 )−1 dx = d2 d2 ℝ

𝜕 1 ⟨a, a⟩ = < 0, 𝜕ω d2 √d0

322 | 4 Existence and stability of solitary waves −2√d0 vd 𝜕 ⟨a, a⟩ = (s2 − 2 ), 𝜕v d0 d22 ∫ a4 (x)dx = ℝ

4d02 ∫(1 + cosh d6 )−1 dx d22 ℝ

+∞

=

16d02 y 4 d0 √d0 dy = , ∫ 3 d22 (1 + y)4 d22 d6 0

𝜕 ∫ a4 (x)dx = 𝜕v ℝ

det(d󸀠󸀠 ) =

d03/2 (s2 d23

−

3/2 vd2 1 d ) − s2 03 . d0 3 d2

1 𝜕 1 𝜕 𝜕 𝜕 ⟨a, a⟩ ⟨a, a⟩ + ⟨a, a⟩(v ⟨a, a⟩ − 2 ⟨a, a⟩) 8 𝜕ω 8 𝜕v 𝜕ω 𝜕v s 𝜕 𝜕 ⟨a, a⟩ ∫ a4 (x)dx + 2 16 𝜕ω 𝜕v ℝ

=

s2 d sd sd vd vd 1 − 2 04 (s2 − 2 ) + 2 04 (s2 − 2 ) − 2 04 2 d0 d0 48 d2 4d2 16d2 16d2

=−

s2 d 1 − 2 04 < 0, 2 4d2 48 d2

thus d󸀠󸀠 has one negative and one positive eigenvalue, which assures that (4.2.78) holds for Case (B). Case (C), d4 > 0, d22 − 4d0 d4 > 0. In this case, we have d52 − d32 = −d4 /d0 < 0, ⟨a, a⟩ =

d3 > d5 ,

d2 < 0,

d0 > 0,

−d /2d + √d4 /d0 1 ln( 2 0 ), 2√d4 −d2 /2d0 − √d4 /d0

d 1 𝜕 ⟨a, a⟩ = 2 2 , 𝜕ω √d0 d2 − 4d0 d4

−√d0 vd 𝜕 ⟨a, a⟩ = 2 (s2 − 2 ), 𝜕v d0 2d2 − 4d0 d4 ∫ a4 (x)dx = − ℝ

√d0 2√d4 d0 − d2 d2 − ln( ), −d4 4d4 √d4 −2√d4 d0 − d2

2√d4 d0 − d2 s2 𝜕 v + ln( ) ∫ a4 (x)dx = 3/2 𝜕v 16(d ) √d −2√d4 d0 − d2 4d4 0 4 ℝ

+

√d0 d2 vd (s2 − 2 ). d0 4(d22 − 4d0 d4 )d4

(4.2.84)

4.2 The nonlinear derivative Schrödinger equation

| 323

By (4.2.79), we have det(d󸀠󸀠 ) =

𝜕 1 𝜕 𝜕 𝜕 1 ⟨a, a⟩ ⟨a, a⟩ + ⟨a, a⟩(v ⟨a, a⟩ − 2 ⟨a, a⟩) 8 𝜕ω 8 𝜕v 𝜕ω 𝜕v s 𝜕 𝜕 + 2 ⟨a, a⟩ ∫ a4 (x)dx 16 𝜕ω 𝜕v ℝ

=

d 𝜕 s 𝜕 𝜕 𝜕 1 ⟨a, a⟩ ⟨a, a⟩ + 2 ⟨a, a⟩(2 0 ⟨a, a⟩ + ∫ a4 (x)dx) 8 𝜕ω 16 𝜕ω d2 𝜕v 𝜕v ℝ

s 𝜕 𝜕 1 ⟨a, a⟩ = ⟨a, a⟩ ⟨a, a⟩ + 2 8 𝜕ω 16 𝜕ω d √d d2 √d0 vd vd × [− 2 0 0 (s2 − 2 ) + (s2 − 2 ) d0 d0 (d2 − 4d0 d4 )d2 4(d22 − 4d0 d4 )d4

2√d4 d0 − d2 s2 v + ln( )] 3/2 −2√d4 d0 − d2 4d4 √d0 16(d4 ) s2 d2 1 𝜕 = ⟨a, a⟩ ⟨a, a⟩ + 8 𝜕ω 16√d0 (d22 − 4d0 d4 ) +

×( =

2√d4 d0 − d2 s2 √d0 s2 + ln( )) 4d2 d4 16(d4 )3/2 −2√d4 d0 − d2

s22 𝜕 1 ⟨a, a⟩ ⟨a, a⟩ + 8 𝜕ω 64d4 (d22 − 4d0 d4 ) × (1 +

= I1 + I2 .

2√d4 d0 − d2 d2 ln( )) 4√d0 d4 −2√d4 d0 − d2

Equation (4.2.84) now implies I1 < 0. Let y = −d2 /(2√d0 d4 ), then in Case (C), y > 1 and s22 Y2 (y), − 4d0 d4 )

I2 =

64d4 (d22

with Y2 (y) = 1 −

y y+1 ln( ). 2 y−1

Note that y2 (1) = −∞, lim Y2 (y) = 1 + lim

y→+∞

y→+∞

1/(1 + y) − 1/(y − 1) 2/y2

−y2 = 0, y→+∞ y 2 − 1

= 1 + lim

(4.2.85)

324 | 4 Existence and stability of solitary waves and y+1 y 1 , Y2󸀠 (y) = − ln( )+ 2 2 y−1 y +1

(4.2.86)

2 − 1)2

(4.2.87)

Y2󸀠 (+∞) = 0, Y2󸀠󸀠 (y) = −

(y2

for all 1 < y < +∞.

Then (4.2.86) and (4.2.87) imply Y2󸀠 (y) > 0

for all 1 < y < +∞,

(4.2.88)

thus (4.2.85) and (4.2.88) assure that Y2 (y) < 0

for all 1 < y < +∞,

which implies I2 < 0 (since d4 > 0, d22 − 4d0 d4 > 0). Finally, in Case (C), we have det(d󸀠󸀠 ) < 0, which implies that (4.2.78) holds. This completes the proof of (4.2.78) under Assumption 4.2.1.

4.3 Long wave–short wave resonance equations Consider the long wave–short wave resonance equations arising in the study of surface waves with both gravity and capillary model present and also in plasma physics: iεt + εxx = nε + α|ε|2 ε,

(4.3.1)

nt = (|ε| )x ,

(4.3.2)

2

x ∈ ℝ,

where α ∈ ℝ, n is a real function, and ε a complex function; n is the amplitude of long wave, while ε is the envelope of the short wave. Let ε(t, x) = eiωt eiq(x−vt) φω,c (x − vt), { n(t, x) = nω,c (x − vt)

(4.3.3)

be the solitary wave of (4.3.1) and (4.3.2), where ω, q, c are real numbers, φω,c and nω,c are real functions. Putting (4.3.3) into (4.3.1) and (4.3.2), we obtain 󸀠 2 3 φ󸀠󸀠 ω,c + i(2q − c)φω,c + (ω + qc − q − nω,c )φω,c − αφω,c = 0,

−cn󸀠ω,c

=

2φω,c φ󸀠ω,c .

(4.3.4) (4.3.5)

Suppose nω,c , φω,c → 0, as x → ∞. Then by (4.3.5) we have 1 nω,c = − φ2ω,c . c

(4.3.6)

4.3 Long wave–short wave resonance equations | 325

Equation (4.3.4) implies 2q = c,

(4.3.7)

1 3 2 φ󸀠󸀠 ω,c + (ω + qc − q )φω,c + ( − α)φω,c = 0. c

(4.3.8)

Let φω,c − c1 sech c2 x satisfy (4.3.8) with constants c1 , c2 to be determined later. Then we have ω + qc − q2 = −c22 ,

1 2c22 = ( − α)c12 . c

(4.3.9)

It follows from (4.3.7) and (4.3.9) that −(4ω + c2 )c , 2(1 − αc)

(4.3.10)

√−4ω − c2 { −(4ω + c2 )c { √ { φ (x) = sech( x), { ω,c { 2(1 − αc) 2 { { { { 2 −(4ω + c2 )c 2 √−4ω − c { { x), { nω,c (x) = 2(1 − αc) sech ( { 2 { { { { c { q= . { 2

(4.3.11)

q=

c , 2

c2 = √−ω −

c2 , 4

c1 = √

thus

Finally, we have Theorem 4.3.1. For any real constants ω, v, α satisfying c0 = 4ω + c2 < 0,

c>0

and 1 − αc > 0,

(4.3.12)

there exist solitary waves of (4.3.1) and (4.3.2) in the form of (4.3.3), with φω,c , nω,c , q, c, ω, and α satisfying (4.3.11). Rewrite (4.3.1) and (4.3.2) splitting real and imaginary parts, with ε = u + iv, ut = −vxx + nv + α(u2 + v2 )v, { { { vt = uxx − nu − α(u2 + v2 )u, x ∈ ℝ. { { { 2 2 {nt = (u + v )x ,

(4.3.13)

Let u = (u, v, n). The function space in which we shall work is X = H 1 (ℝ) × H 1 (ℝ) × L (ℝ), with inner product 2

(f, g) = ∫(f1 g1 + f1x g1x + f2 g2 + f2x g2x + f3 g3 )dx, ℝ

for f, g ∈ X.

(4.3.14)

326 | 4 Existence and stability of solitary waves The dual space of X is X ∗ = H −1 (ℝ)×H −1 (ℝ)×L2 (ℝ), and there is a natural isomorphism I : X → X ∗ defined by ⟨If, g⟩ = (f, g),

(4.3.15)

where ⟨⋅, ⋅⟩ denotes the pairing between X and X ∗ , 3

⟨f, g⟩ = ∫(∑ fi gi ).

(4.3.16)

i=1

By (4.3.14)–(4.3.16), it is obvious that I=(

1−

𝜕2 𝜕x2

0 0

0

0

𝜕2 𝜕x2

1− 0

0) . 1

Let T1 , T2 be one-parameter groups of unitary operators on X defined by T1 (s1 )u(⋅) = u(⋅ − s1 ), cos s2 T2 (s2 )u(⋅) = (− sin s2 0

(4.3.17) sin s2 cos s2 0

0 u(⋅) 0) (v(⋅)) 1 n(⋅)

for u(⋅) ∈ X,

s2 ∈ ℝ.

(4.3.18)

Obviously, T1󸀠 (0) = (

T2󸀠 (0)

𝜕 − 𝜕x

0 = (−1 0

𝜕 − 𝜕x

1 0 0

𝜕 − 𝜕x

),

0 0) . 0

It follows from Theorem 4.3.1 and (4.3.1)–(4.3.2) that there exist solitary waves T1 (ct)T2 (ωt) Φω,c (x) of (4.3.1) with Φω,c (x) defined by c c Φω,c (x) = (φω,c (x) cos( x), φω,c (x) sin( x), nω,c (x)). 2 2 We shall consider the orbital stability of solitary waves T1 (ct)T2 (ωt)Φω,c (x). Define 1 1 α 2 E(u) = ∫[ (u2x + vx2 ) + n(u2 + v2 ) + (u2 + v2 ) ]dx. 2 2 4 ℝ

(4.3.19)

4.3 Long wave–short wave resonance equations | 327

It is easy to verify that E(u) is invariant under T1 and T2 , and formally conserved under the flow of (4.3.1) and (4.3.2). Indeed, E(T1 (s1 )T2 (s2 )u) = E(u)

for any s1 , s2 ∈ ℝ,

(4.3.20)

and for any t ∈ ℝ, u(t) is a flow of (4.3.1) and (4.3.2) since E(u(t)) = E(u(0)).

(4.3.21)

Note that equations (4.3.1) and (4.3.2) can be written as the following Hamiltonian system: du = JE 󸀠 (u) dt

(4.3.22)

with a skew-symmetric linear operator J defined by 0 J = (−1 0

1 0

0 0 ),

𝜕2 2 𝜕x 2

0

(4.3.23)

and E 󸀠 (u) is the Fréchet derivative of E(u). We define 0 𝜕 B1 = (− 𝜕x 0

𝜕 𝜕x

0 0

0 0 ), − 21

1

),

such that T1󸀠 (0) = JB1 , 1 B2 = (

0

and also T2󸀠 (0) = JB2 , 1 Q1 (u) = ⟨B1 u, u⟩ 2 1 1 = − ∫ n2 dx + ∫(vx u − vux )dx, 4 2 ℝ

(4.3.24)

ℝ

1 1 Q2 (u) = ⟨B2 u, u⟩ = ∫(u2 + v2 )dx. 2 2

(4.3.25)

ℝ

By (4.3.19) and (4.3.25), we can prove that Q1 (T1 (s1 )T2 (s2 )u) = Q1 (u),

Q2 (T1 (s1 )T2 (s2 )u) = Q2 (u),

(4.3.26)

328 | 4 Existence and stability of solitary waves for any s1 , s2 ∈ ℝ. Moreover, Q1 (u(t)) = Q1 (u(0)),

Q2 (u(t)) = Q2 (u(0)).

(4.3.27)

Furthermore, E 󸀠 (Φω,c ) − cQ󸀠1 (Φω,c ) − ωQ󸀠2 (Φω,c ) = 0,

(4.3.28)

where E 󸀠 , Q󸀠1 , Q󸀠2 are the Fréchet derivatives of E, Q1 and Q2 , with −uxx + nu + α(u2 + v2 )u E (u) = ( −vxx + nv + α(u2 + v2 )v ) , 1 2 (u + v2 ) 2 󸀠

vx Q󸀠1 (u) = ( −ux ) , − 21 n

u Q󸀠2 (u) = ( v ) . 0

Define an operator from X to X ∗ by 󸀠󸀠 Hω,c = E 󸀠󸀠 (Φω,c ) − cQ󸀠󸀠 1 (Φω,c ) − ωQ2 (Φω,c ).

(4.3.29)

Since ω, c are fixed we write φ for φω,c , and we have L + αφ2 cos( c2 x)

𝜕 αφ2 sin( c2 x) − c 𝜕x

φ cos( c2 x)

φ cos( c2 x)

φ sin( c2 x)

c 2

𝜕 Hω,c = (αφ2 sin( c2 x) + c 𝜕x

L − αφ2 cos( c2 x)

φ sin( c2 x) ) ,

(4.3.30)

where L = −𝜕2 /𝜕x2 − ω + n + 2αφ2 . ∗ Observe that Hω,c is self-adjoint in the sense that Hω,c = Hω,c , and that I −1/2 Hω,c I −1/2 2 2 2 is a self-adjoint operator on L (ℝ) × L (ℝ) × L (ℝ), where I −1/2

(1 − =(

𝜕2 −1/2 ) 𝜕x2

(1 −

𝜕2 −1/2 ) 𝜕x2

). 1

The “spectrum” of Hω,c consists of real numbers λ such that Hω,c − λI is not invertible. We claim that λ = 0 belongs to the spectrum of Hω,c . By (4.3.20), (4.3.26), (4.3.28), and (4.3.29), it is easy to prove that Hω,c T1󸀠 (0)Φω,c (x) = 0,

Hω,c T2󸀠 (0)Φω,c (x)

= 0.

(4.3.31) (4.3.32)

Let Z = {k1 T1󸀠 (0)Φω,c (x) + k2 T2󸀠 (0)Φω,c (x) | k1 , k2 ∈ ℝ}. By (4.3.31) and (4.3.32), Z is contained in the kernel of Hω,c .

(4.3.33)

4.3 Long wave–short wave resonance equations | 329

Assumption 4.3.1 (Spectral decomposition of Hω,c ). The space X is decomposed as a direct sum X = N + Z + P,

(4.3.34)

where Z is the kernel of Hω,c , N is a finite-dimensional subspace such that ⟨Hω,c u, u⟩ < 0,

0 ≠ u ∈ N

(4.3.35)

and P is a closed subspace such that ⟨Hω,c u, u⟩ ≥ ‖u‖2X ,

u ∈ P,

(4.3.36)

with some constant δ > 0 independent of u. We define d(ω, c) : ℝ × ℝ → ℝ by d(ω, c) = E(Φω,c ) − cQ󸀠1 (Φω,c ) − ωQ󸀠2 (Φω,c )

(4.3.37)

and define d󸀠󸀠 (ω, c) to be the Hessian of function d. It is a symmetric bilinear form. By (4.3.24) we know that J is not onto. We give the following modified abstract stability results for problem (4.3.1)–(4.3.2): Theorem 4.3.2. Assume that there exist solitary wave solutions T1 (ct)T2 (ct)Φω,c (x) of (4.3.13) and that Assumption 4.3.1 holds. Let n(Hω,c ) be the number of negative eigenvalues of Hω,c . Assume d(ω, c) is nondegenerate at (ω, c) and let p(d󸀠󸀠 ) be the number of positive eigenvalues of its Hessian at (ω, c). Furthermore, if p(d󸀠󸀠 ) = n(Hω,c ), then the solitary wave T1 (ct)T2 (ct)Φω,c (x) is orbitally stable. Theorem 4.3.3. Under the condition of Theorem 4.3.1, the solitary waves T1 (ct)T2 (ct) × Φω,c (x) of (4.3.13) are orbitally stable if ω 0. It follows from (4.3.3), (4.3.7), (4.3.8), and (4.3.28) that L1 φx = 0,

L2 φ = 0.

(4.3.49) (4.3.50)

By (4.3.11) and (4.3.49), we see that φx has a simple zero at x = 0, and then Sturm– Liouville theorem implies that 0 is the second eigenvalue of L1 , and L1 has exactly one strictly negative eigenvalue −σ−2 , with an eigenfunction χ1 , L1 χ1 = −σ−2 χ1 ,

⟨χ1 , χ1 ⟩ = 1.

(4.3.51)

Also, by (4.3.11), equation (4.3.50) implies that 0 is the first simple eigenvalue of L2 .

4.3 Long wave–short wave resonance equations | 331

By virtue of (4.3.44)–(4.3.50), we have the following lemmas: Lemma 4.3.1. For any real functions y1 ∈ H 1 (ℝ) satisfying ⟨y1 , χ1 ⟩ = ⟨y1 , φx ⟩ = 0,

(4.3.52)

there exists a positive number δ1 > 0 such that ⟨L1 y1 , y1 ⟩ ≥ δ1 ‖y1 ‖2H 1 .

(4.3.53)

Lemma 4.3.2. For any real functions y2 ∈ H 1 (ℝ) satisfying ⟨y2 , φ⟩ = 0,

(4.3.54)

there exists a positive number δ2 > 0 such that ⟨L2 y2 , y2 ⟩ ≥ δ2 ‖y2 ‖2H 1 .

(4.3.55)

For any Ψ ∈ X, from (4.3.39), we can simply denote Ψ by Ψ = (y1 , y2 , y3 ).

(4.3.56)

Choose 2 z2− = − φχ1 , c

(4.3.57)

⟨Hω,c Ψ− , Ψ− ⟩ = −σ−2 ⟨χ1 , χ1 ⟩ < 0.

(4.3.58)

y1− = χ1 ,

y2− = 0,

Ψ = (y1− , y2− , z2− ), then

Also, note that the following two vectors are spanning the kernel of Hω,c : 2 Ψ0,1 = (φx , 0, − φφx ), c Ψ0,2 = (0, φ, 0).

(4.3.59) (4.3.60)

Let Z = {k1 Ψ0,1 + k2 Ψ0,2 | k1 , k2 ∈ ℝ}, P = {p ∈ X | p = (p1 , p2 , p3 ), −

⟨p1 , χ1 ⟩ = ⟨p1 , φx ⟩ = ⟨p2 , φ⟩ = 0},

N = {kΨ | k ∈ ℝ}. Obviously, (4.3.35) holds.

(4.3.61) (4.3.62) (4.3.63)

332 | 4 Existence and stability of solitary waves For any u ∈ X, u = (y1 , y2 , y3 ), choose a = ⟨y1 , χ1 ⟩, b1 = ⟨φx , y1 ⟩/⟨φx , φx ⟩, b2 = ⟨φ, y2 ⟩/⟨φ, φ⟩, then u can be uniquely represented by u = aΨ− + b1 Ψ0,1 + b2 Ψ0,2 + p

(4.3.64)

with p ∈ P, which implies (4.3.34). For subspace P, it remains to prove (4.3.36). Lemma 4.3.3. For any p ∈ P, defined by (4.3.62), there exists a constant δ > 0 such that ⟨Hω,c p, p⟩ ≥ δ‖p‖X ,

(4.3.65)

with δ independent of p. Proof. For any p ∈ P, by (4.3.62) and Lemmas 4.3.1 and 4.3.2, we have ⟨Hω,c p, p⟩ ≥ δ1 ‖p1 ‖2H 1 + δ2 ‖p2 ‖2H 1 +

2

2 c ∫(p3 + φp1 ) dx. 2 c

(4.3.66)

ℝ

(1) If ‖p3 ‖2L2 ≥

8M ‖p ‖2 2 , c2 1 L

M = |φ|2∞ ,

(4.3.67)

then 2

c c c 2M c ‖p1 ‖2L2 ≥ ‖p3 ‖2L2 . ∫(p3 + φ1 p1 ) dx ≥ ‖p3 ‖2L2 − 2 2 2 c 4

(4.3.68)

ℝ

(2) If ‖p3 ‖2L2 ≤

8M ‖p ‖2 2 c2 1 L

(4.3.69)

then δ1 ‖p1 ‖2H 1 ≥

δ1 δ c2 ‖p1 ‖2H 1 + 1 ‖p3 ‖2L2 . 2 16M

(4.3.70)

Thus, for any p ∈ P, it follows from (4.3.66)–(4.3.70) that ⟨Hω,c p, p⟩ ≥ δ3 ‖p3 ‖2L2 +

δ1 ‖p ‖2 1 + δ2 ‖p2 ‖2H 1 , 2 1 H

(4.3.71)

with δ3 = min{c/4, δ1 c2 /(16M)} > 0. Finally, with (4.3.71) we have ⟨Hω,c p, p⟩ ≥ δ‖p‖2X , where δ > 0 is independent of p.

(4.3.72)

4.3 Long wave–short wave resonance equations | 333

Thus, under the condition of (4.3.12), Assumption 4.3.1 holds, and n(Hω,c ) = 1. In the following, we shall verify that p(d󸀠󸀠 ) = 1 under the conditions of Theorem 4.3.3. Note that (4.3.28) and (4.3.37) imply dω (ω, c) = −Q2 (Φω,c ), dc (ω, c) = −Q1 (Φω,c ), 1 −Q2 (Φω,c ) = − ∫(u2 + v2 )dx 2 ℝ

=

√−4ω − c2 −c(ω + c2 /4) x)dx ∫ sech2 ( 1 − αc 2 ℝ

c √ =− −4ω − c2 < 0, 1 − αc 1 1 −Q1 (Φω,c ) = ∫(n2 )dx − ∫(vx u − ux v)dx 4 2 ℝ

= dωω (ω, c) =

ℝ

4(ω − c2 /4)3/2 c2 √−4ω − c2 , − 2 2(1 − αc) 3(1 − αc) 2c

(1 − αc)√−4ω − c2

dcω (ω, c) = dωc (ω, c) = − dcc (ω, c) =

> 0,

√−4ω − c2 c2 , + (1 − αc)2 (1 − αc)√−4ω − c2

(−4ω − c2 )3/2 c3 3c √ 2+ −4ω − c . − 2(1 − αc) 3(1 − αc)3 2(1 − αc)√−4ω − c2

Denote y = −4ω − c2 > 0, then det(d󸀠󸀠 ) = dωω dcc − dωc dcω =

1 2αcy2 y2 2c2 y [ − 3c2 y + c4 − − c4 + ] 2 2 2 1 − αc (1 − αc) y 3(1 − αc) (1 − αc)

=

−1 2 [(1 − αc)y + (1 − 3αc)(1 − αc)c2 ] 4 3 (1 − αc)

=

−4(1 − 32 αc) (1 − αc)4

[−ω +

α(10 − 9αc)c3 12(1 − 32 αc)

] < 0.

Thus, d󸀠󸀠 has exactly one positive and one negative eigenvalue, whence, p(d󸀠󸀠 ) = 1. This completes the proof of Theorem 4.3.3.

334 | 4 Existence and stability of solitary waves

4.4 The generalized Kadomtsev–Petviashvili equations Recently, Bounard, Sout, Liu and others [18, 19, 76] have studied the existence and stability and instability of solitary waves of generalized KP equations, and got the same results of Wang, Ablowitz and Segur (see [97]) obtained earlier using a different method. We will consider the following generalized Kadomtsev–Petviashvili equations: ut + f (u)ux + uxxx + εvy = 0, { { { u = u(x, y, t), (x, y) ∈ ℝ2 , t ≥ 0, { { { {vx = uy ,

(4.4.1)

ut + f (u)ux + uxxx + εavy + bwz = 0, { { { u = u(x, y, z, t), (x, y, z) ∈ ℝ3 , t ≥ 0, { { { {vx = uy , wx = uz .

(4.4.2)

and

The constants ε, a, b measure the transverse dispersion effects and are normalized to ±1. The “usual” Kadomtsev–Petviashvili equations correspond to f (u) = u. We will consider therein power nonlinearities. We recall that (4.4.1) with f (u) = u is integrable by the inverse scattering method and is classically called KPI (ε = −1) or KPII (ε = +1). We firstly consider the existence and nonexistence of solitary waves. We shall denote for d = 2, 3, X = {φ ∈ H 1 (ℝd ), 𝜕x2 φ ∈ L2 (ℝd )}. Let Y be the closure of 𝜕x (C0∞ (ℝd )) for the norm 󵄩 󵄩2 1/2 ‖𝜕x φ‖Y = (‖∇φ‖2L2 + 󵄩󵄩󵄩𝜕x2 φ󵄩󵄩󵄩L2 ) ,

where H m (ℝd ) denotes the space of functions of the form 𝜕x φ with φ ∈ C0∞ (ℝd ) (i. e., ∞ the space of functions ψ in C0∞ (ℝd ) such that ∫−∞ ψ(x, x 󸀠 )dx = 0, for every x 󸀠 ∈ ℝd−1 ). Definition 4.4.1. A solitary wave of (4.4.1) (resp. (4.4.2)) is a solution of the type u(x − ct, y) (resp. u(x − ct, y, z)) where u ∈ Y and c > 0. We are thus looking for “localized” solutions to the systems −cux + f (u)ux + uxxx + εvy = 0,

{

vx = uy ,

(4.4.3)

and −cux + f (u)ux + uxxx + εavy + bwz = 0, { { { vx = uy , { { { {wx = uz .

We may from now on assume that c = 1, and f (u) = up , p = 1, 2, 3, . . . .

(4.4.4)

4.4 The generalized Kadomtsev–Petviashvili equations | 335

In order to prove the existence of solitary waves, our strategy is to consider the minimization problem Iλ = inf{‖u‖2Y , u ∈ Y, with ∫ up+2 dxdx 󸀠 = λ},

(4.4.5)

ℝd

where x󸀠 = y, if d = 2, x󸀠 = (y, z) if d = 3 and λ > 0. We shall use the concentration– compactness principle of Lions [75]. Theorem 4.4.1. (i) Assume that d = 2. Equation (4.4.1) does not admit any nontrivial 2 2 2 ∞ 2 solitary wave satisfying u = 𝜕x φ ∈ Y, u ∈ H 1 (ℝ2 ) ∩ L∞ loc (ℝ ), 𝜕x u and 𝜕y φ ∈ Lloc (ℝ ) if ε = −1

and

p>4

(4.4.6)

or ε = 1 and p is arbitrary.

(4.4.7)

(ii) Assume that d = 3. Equation (4.4.2) does not admit any nontrivial solitary wave 3 1 3 2(p+1) satisfying u = 𝜕x φ ∈ Y, 𝜕x2 u, 𝜕y2 φ and 𝜕z2 φ ∈ L∞ (ℝ3 ) ∩ loc (ℝ ), u ∈ H (ℝ ) ∩ L 3 L∞ loc (ℝ ) if ab = −1

(resp. a = b = 1) and p is arbitrary

(4.4.8)

or a = b = −1

p ≥ 2.

(4.4.9)

Proof. It is based on Pohojaev type identities. The regularity assumptions of Theorem 4.4.1 are needed to justify them by the following standard truncation argument. 2 Let χ0 ∈ C0∞ (ℝ), 0 ≤ χ0 ≤ 1, χ0 (t) = 1 if 0 ≤ |t| ≤ 1, χ0 (t) = 0, |t| ≥ 2. We set χj = χ0 ( |⋅|j2 ), j = 1, 2, . . . . To begin with, we treat the 2-dimensional case. We multiply (4.4.3)1 by xχj u and we integrate over ℝ2 to get (note that the third integral has to be interpreted as a H 1 –H −1 duality) − ∫ xχj 𝜕x (

u2 1 )dxdy + ∫ xχj 𝜕x (up+2 )dxdy 2 p+2

+ ∫ xχj uuxxx dxdy + ε ∫ xχj vy udxdy = 0, and after several integrations by parts we obtain 1 3 1 ∫ χj u2 dxdy − ∫ χj 𝜕x up+2 dxdy + ∫ χj u2x dxdy 2 p+2 2

(4.4.10)

336 | 4 Existence and stability of solitary waves

+ − − − −

ε 1 r2 ∫ χj v2 dxdy + 2 ∫ xχ0󸀠 ( 2 )u2 dxdy 2 j j

2 3 r2 r2 ∫ x2 χ0󸀠 ( 2 )u2 dxdy − 2 ∫ χ0󸀠 ( 2 )u2 dxdy j(p + 2) j j j 6 6 r2 r2 ∫ χ0󸀠󸀠 ( 2 )u2 dxdy − 4 ∫ xχ0󸀠󸀠 ( 2 )u2 dxdy 4 j j j j

4 r2 3 r2 ∫ x3 χ0󸀠󸀠󸀠 ( 2 )u2 dxdy + 2 ∫ xχ0󸀠 ( 2 )u2x dxdy 6 j j j j

r2 1 r2 2ε ∫ xyχ0󸀠 ( 2 )uvdxdy + 2 ∫ xχ0󸀠 ( 2 )v2 dxdy = 0, 2 j j j j

(4.4.11)

where r 2 = x2 +y2 . By Lebesgue dominated convergence theorem, we infer from (4.4.12) that 1 up+2 3 2 ε 2 − u − v ]dxdy = 0. ∫[− u2 + 2 p+2 2 x 2

(4.4.12)

We multiply (4.4.3)1 by yv and integrate. After several integrations by parts and using (4.4.3)2 we obtain finally u2 ε 1 1 up+2 + x + v2 ]dxdy = 0. ∫[ u2 − 2 (p + 1)(p + 2) 2 2

(4.4.13)

We multiply (4.4.3)1 by u and integrate. After several integrations by parts we obtain ∫[−u2 +

up+2 − u2x + εv2 ]dxdy = 0. p+1

(4.4.14)

Subtracting (4.4.12) from (4.4.13), we get ∫[u2 −

up+2 + 2u2x + εv2 ]dxdy = 0. p+1

(4.4.15)

Adding (4.4.14) and (4.4.15) yields ∫[u2x + 2εv2 ]dxdy = 0. The identity (4.4.15) for ε = −1, namely ∫ u2x dxdy = ∫ 2v2 dxdy, gives, when inserted into (4.4.12) and (4.4.14), 1 up+2 5 2 − v ]dxdy = 0, ∫[− u2 + 2 p+2 2

(4.4.16)

4.4 The generalized Kadomtsev–Petviashvili equations | 337

∫[−u2 +

up+2 − 3v2 ]dxdy = 0. p+2

Eliminating v2 leads to ∫[u2 +

p−4 up+2 ]dxdy = 0. 2(p + 1)(p + 2)

(4.4.17)

On the other hand, adding (4.4.12) and (4.4.13) yields ∫ u2x dxdy =

p ∫ up+2 dxdy. (p + 1)(p + 2)

And (4.4.6) follows from the equality reported in (4.4.17). Let us now consider the case d = 3. Again we give a formal proof, which can be justified by the aforementioned truncation process. We multiply successively (4.4.4)1 by xu, yv and zw, and integrate to get 1 up+2 3 2 a 2 b 2 − u − v − w ]dxdydz = 0, ∫[− u2 + 2 p+2 2 x 2 2

1 up+2 1 a b + u2 + v2 − w2 ]dxdydz = 0, ∫[ u2 − 2 (p + 1)(p + 2) 2 x 2 2

up+2 1 a b 1 + u2 − v2 + w2 ]dxdydz = 0. ∫[ u2 − 2 (p + 1)(p + 2) 2 x 2 2

(4.4.18) (4.4.19) (4.4.20)

Integrating (4.4.4)1 once with respect to x, and taking the duality product of the resulting equation with u ∈ Y as in dimension 2, one obtains ∫[−u2 +

up+2 − u2x + av2 + bw2 ]dxdydz = 0. p+1

(4.4.21)

Subtracting (4.4.20) from (4.4.19) yields ∫[av2 − bw2 ]dxdydz = 0,

(4.4.22)

which rules out the case (4.4.8) when ab = −1. Now adding (4.4.19) and (4.4.20) implies up+2 1 1 + u2x ]dxdydz = 0. ∫[ u2 − 2 (p + 1)(p + 2) 2 Multiplying (4.4.23) by p + 1 and adding (4.4.18), we obtain ∫[pu2 + (p − 2)u2x − av2 − bw2 ]dxdydz = 0, which rules out (4.4.9). On the other hand, from (4.4.19) and (4.4.22) we infer 1 1 ∫ up+2 dxdydz = ∫[u2 + u2x ]dxdydz. (p + 1)(p + 2) 2

(4.4.23)

338 | 4 Existence and stability of solitary waves This identity plugged in (4.4.21) yields p p ∫[ u2 + u2x + av2 + bw2 ]dxdydz = 0, 2 2 which proves (4.4.8) for a = b = 1. Now we prove the existence of solitary waves solutions of equations (4.4.1) and (4.4.2). Theorem 4.4.2. Let d = 2, ε = 1 and p be such that 1 ≤ p < 4. Then equation (4.4.3) possesses a solution (u, v) with u ∈ Y, u ≠ 0. Proof. First, observe that λ = 0 for any Iλ > 0. This follows from the imbedding theorem for anisotropic Sobolev spaces ([5] p. 323) which gives ‖u‖Lq ≤ C‖u‖Y

for any u ∈ Y and 2 ≤ q < 6.

Hence | ∫ up+2 dxdy| ≤ C‖u‖p+2 Y for any u ∈ Y and 2/(p+2)

λ Iλ ≥ ( ) C

> 0,

∀λ > 0.

Now, let λ > 0 and let un be a minimizing sequence for (4.4.5). Then there is a sequence of functions φn , φn ∈ Lqloc (ℝ2 ), q > 0, un = 𝜕x φn . Let vn = 𝜕y φn = D−1 x uny , we apply the 2 2 concentration–compactness lemma of [75] to ρn = |un | + |vn | + |𝜕x un |2 . Note that lim ∫ ρn dxdy = lim ‖un ‖2Y = Iλ > 0.

n→+∞

n→+∞

(i) Assume first that “vanishing” occurs, i. e., that for any R > 0, lim

n→+∞

sup

(x,y)∈ℝ2

∫

(|un |2 + |vn |2 + |𝜕x un |2 )dxdydz = 0,

(4.4.24)

(x,y)+BR

where BR is the ball of radius R centered at 0. Let q be such that 2 < q < 6, then from the Sobolev inequalities in anisotropic Sobolev spaces (see [5]), there is a positive constant C independent of (x, y) ∈ ℝ2 such that if φx ∈ Y, then q/2

∫

|φx |q ≤ C( ∫ (|φx |2 + |φy |2 + |φxx |2 ))

(x,y)+B1

(x,y)+B1

≤ C( sup

(x,y)∈ℝ2

×

∫ (|φx |2 + |φy |2 + |φxx |2 )) (x,y)+B1

∫ (|φx |2 + |φy |2 + |φxx |2 ). (x,y)+B1

(q−1)/2

4.4 The generalized Kadomtsev–Petviashvili equations | 339

Now, covering ℝ2 by balls of radius 1 in such a way that each point of ℝ2 is contained in at most 3 balls, we have (q−2)/2

∫ |φx |q ≤ 3C( sup

(x,y)∈ℝ2

ℝ2

∫ (|φx |2 + |φy |2 + |φxx |2 ))

‖φx ‖2Y

(x,y)+B1

for any φ such that φx ∈ Y. From this, we conclude that under assumption (4.4.24), un → 0 in Lq for any q such that 2 < q < 6, which contradicts the constraint in Iλ . (ii) Assume now that a “dichotomy” occurs, i. e., that lim Q(t) = α ∈ (0, Iλ ), where for t ≥ 0, { { { { t→+∞ { Q(t) = lim sup ∫ ρn dxdy. { { { t→+∞ (x,y)∈ℝ2 (x0 ,y0 )+B1 {

(4.4.25)

2

Note that for λ > 0, Iλ = λ p+2 I1 . Assumption (4.4.25) will then give a contradiction, provided that it leads to the splitting of un into two sequences of u1n and u2n with disjoint supports. Lemma 4.4.1. Let q be such that 2 ≤ q ≤ +∞, then there exists a positive constant C such that for all f ∈ L1loc (ℝ2 ) with ∇f ∈ L2loc (ℝ2 ) for all R > 0 and for all x0 ∈ ℝ2 , 1/q

(

∫

R≤|x−x0 |≤2R

󵄨󵄨 󵄨q 󵄨󵄨f (x) − mR (f )󵄨󵄨󵄨 dx)

1/2

≤ CR2/q (

|∇f |2 dx) ,

∫

R≤|x−x0 |≤2R

where mR (f ) =

1 vol(Ωx0 ,R )

f (x)dx,

∫

x = (x1 , x 2 ) ∈ ℝ2

R≤|x−x0 |≤2R

and Ωx0 ,R = {x ∈ ℝ2 , R < |x − x0 | < 2R}. Proof. Applying Poincaré inequality for zero mean-valued H 1 functions on the bounded open set Ωx0 ,R , and then using Sobolev imbedding theorem, we obtain the existence of a positive constant C(x0 , R) such that 1/q

(

∫

R≤|x−x0 |≤2R

󵄨󵄨 󵄨q 󵄨󵄨f (x) − mR (f )󵄨󵄨󵄨 dx)

≤ C(x0 , R)(

1/2

∫

R≤|x−x0 |≤2R

|∇f |2 dx) .

340 | 4 Existence and stability of solitary waves Then the translation invariance of Lebesgue measure and the scale change f → f ( R⋅ ) show that C(x0 , R) = CR2/q where C is independent of x0 and R. We are now able, using Lemma 4.4.1, to prove the following Lemma 4.4.2. Lemma 4.4.2. Assume that (4.4.25) holds, and fix ε > 0. Then we can find δ(ε) (as ε → 0, δ(ε) → 0) such that there are u1n and u2n in Y satisfying for n > n0 : 󵄩󵄩 1 󵄩 2 󵄩󵄩un + un − un 󵄩󵄩󵄩Y ≤ δ(ε), 󵄨󵄨󵄩󵄩 1 󵄩󵄩 󵄨 󵄨󵄨󵄩󵄩un 󵄩󵄩Y − α󵄨󵄨󵄨 ≤ δ(ε), 󵄨󵄨󵄨󵄩󵄩󵄩u1 󵄩󵄩󵄩 − (Iλ − α)󵄨󵄨󵄨 ≤ δ(ε), 󵄨󵄩 n 󵄩Y 󵄨 󵄨󵄨 󵄨 󵄨󵄨 1 p+2 2 p+2 p+2 󵄨󵄨 󵄨󵄨 ∫ [(un ) + (un ) − (un ) ]󵄨󵄨󵄨 ≤ δ(ε), 󵄨󵄨 󵄨󵄨 ℝ2

and lim dist(supp u1n , supp u2n ) = +∞.

n→+∞

Proof. Assume that (4.4.25) holds, and fix ε > 0. Then we can find ε > 0, Rn > 0 with Rn → +∞ and xn ∈ ℝ2 such that α≥

∫ (|un |2 + |v|2 + |𝜕x un |2 ) ≥ α − ε, xn +BR0

Qn (2Rn ) ≤ α + ε, for n ≥ n0 , where Qn (t) =

sup

(x0 ,y0

)∈ℝ2

∫

(|un |2 + |vn |2 + |𝜕x un |2 ),

(x0 ,y0 )+Bt

it follows that (|un |2 + |vn |2 + |𝜕x un |2 ) ≤ 2ε.

∫ R0 ≤|x− xn |≤2Rn

Let ξ and η ∈ C0∞ (ℝ2 ), 0 ≤ ξ ≤ 1, 0 ≤ η ≤ 1, ξ ≡ 1 on B1 , supp ξ ⊂ ℝ2 \ B1 . We set ξn = ξ (

⋅ − xn ), R1

ηn = η(

⋅ − xn ), Rn

and consider u1n = 𝜕x (ξn (φn − an )),

u2n = 𝜕x (ηn (φn − an )),

4.4 The generalized Kadomtsev–Petviashvili equations | 341

where (an ) and (bn ) are sequences of real numbers which will be chosen later. Lastly we set 1 u1n = D−1 x (un )y = 𝜕y (ξn (φn − an ))

and 2 u2n = D−1 x (un )y = 𝜕y (ηn (φn − bn )).

Then we have, for example, 󵄩󵄩 1 󵄩 󵄩 󵄩 󵄩 󵄩 2 󵄩󵄩un + un − un 󵄩󵄩󵄩L2 ≤ 󵄩󵄩󵄩(𝜕x ξn )(φn − an )󵄩󵄩󵄩L2 + 󵄩󵄩󵄩(𝜕x ηn )(φn − an )󵄩󵄩󵄩L2 + √2ε and 󵄩󵄩 󵄩 󵄩󵄩(𝜕x ξn )(φn − an )󵄩󵄩󵄩L2 = (

1/2

|𝜕x ξn |2 |φn − an |2 dx)

∫

R1 ≤|x−x0 |≤2R1

≤ ‖𝜕x ξn ‖Lp (

1/q

|φn − an |q dx)

∫

,

R1 ≤|x−x0 |≤2R1

where

1 p

+

1 q

=

1 . 2

1 ∫ vol(Ωx0 ,R1 ) R1 ≤|x−x0 |≤2R1

Now choosing an =

Lemma 4.4.1, we get

󵄩󵄩 󵄩 󵄩󵄩(𝜕x ξn )(φn − an )󵄩󵄩󵄩L2 2

≤ CR1p

+ q2 −1

(

1/2

∫

(|un |2 + |vn |2 )dx)

1 ∫ vol(Ωx0 ,Rn ) Rn ≤|x−x0 |≤2Rn

φn (x)dx leads to the bound

󵄩󵄩 󵄩 󵄩󵄩(𝜕x ηn )(φn − an )󵄩󵄩󵄩L2 2

≤ C 󸀠 √ε.

R1 ≤|x−x0 |≤2R1

In the same way, choosing an =

≤ CR1p

φn (x)dx, and applying

+ q2 −1

(

1/2

∫

(|un |2 + |vn |2 )dx)

≤ C √ε.

Rn ≤|x−x0 |≤2Rn

This implies the desired estimate on ‖u1n + u2n − un ‖L2 ; the bound on ‖vn1 + vn2 − vn ‖L2 is obtained in the same way. All the other terms of Lemma 4.4.2 are bounded in a similar way; the last bound follows from the first one, the fact that supp u1n ∩ supp u2n = 0 and the injection of Y into Lp+2 (ℝ2 ).

342 | 4 Existence and stability of solitary waves We now continue the proof of Theorem 4.4.2. Taking subsequences if necessary, we may assume that lim ∫ (u1n )

n→∞

p+2

= λ1 (ε),

ℝ2

lim ∫ (u2n )

n→∞

p+2

= λ2 (ε),

ℝ2

with |λ1 (ε) + λ2 (ε) − λ| ≤ δ(ε). Assume first that limε→0 λ1 (ε) = 0. Then choosing ε sufficiently small, we have for n large enough limn→∞ ∫ℝ2 (u2n )p+2 dxdy > 0. Hence by considering 1/(p+2)

(

λ2 (ε) ) ∫ℝ2 (u2n )p+2

u2n ,

we get 󵄩 󵄩 Iλ2 (ε) ≤ lim inf 󵄩󵄩󵄩u2n 󵄩󵄩󵄩Y ≤ Iλ − α + δ(ε) n→∞

but this contradiction since limε→0 λ2 (ε) = λ. Thus, we may assume that limε→0 |λ1 (ε)| > 0 and limε→0 |λ1 (ε)| > 0. In the same way as before we then obtain 󵄩 󵄩 󵄩 󵄩 I|λ1 (ε)| + I|λ2 (ε)| ≤ lim inf 󵄩󵄩󵄩u1n 󵄩󵄩󵄩Y + lim inf 󵄩󵄩󵄩u2n 󵄩󵄩󵄩Y n→∞

n→∞

≤ Iλ + δ(ε). We reach a contradiction by letting ε → 0 and by using the fact that Tμ = μ2/(p+2) I1 for any positive μ. This ends to rule out the “dichotomy” case. (iii) The only remaining possibility is the following: There is a sequence {xn }, xn ∈ ℝ2 such that for all ε > 0, there exists a finite R > 0 and n0 > 0, such that ∫ (|un |2 + |vn |2 + |𝜕x un |2 )dxdy ≥ Iλ − ε,

n ≥ n0 .

xn +BR

Note that this implies, for n large enough, ∫ |un |2 ≥ ∫ |un |2 − 2ε. xn +BR

ℝ2

Since un is bounded in Y, we may assume that un (⋅ − xn ) converges weakly in Y to some u ∈ Y. We then have 󵄨 󵄨 ∫ |un |2 ≤ lim inf ∫ |u2n | ≤ lim inf ∫ 󵄨󵄨󵄨u2n 󵄨󵄨󵄨 + 2ε. n→∞ n→∞

ℝ2

ℝ2

xn +BR

Lemma 4.4.3. Let un be a bounded sequence in Y, and let R > 0. Then there is a subsequence {unk } which converges strongly to u in L2 (BR ).

4.4 The generalized Kadomtsev–Petviashvili equations | 343

We first end the proof of Theorem 4.4.2, after what we prove Lemma 4.4.3. By Lemma 4.4.3, we may assume that un (⋅−xn ) converges to u strongly in L2loc . But then, the inequality preceding Lemma 4.4.3 shows that, in fact, un (⋅−xn ) converges to u strongly in L2 (ℝ2 ), and by interpolation, using the imbedding Y ⊂ L6 (ℝ2 ), un (⋅ − xn ) also converges to u strongly in Lp+2 so that ∫ up+2 = λ. Since ‖u‖Y ≤ lim infn→∞ ‖u‖Y = Iλ , this shows that u is a solution of Iλ . Proof. Let un be a bounded sequence in Y, with un = 𝜕x φn , φn ∈ L2loc (ℝ2 ), and let 2 vn = 𝜕y φn ∈ L2 (ℝ2 ). Multiplying φn by a function ψ ∈ C∞ (ℝ2 ) with 0 ≤ ψ ≤ 1, ψ ≡ 1 on BR and supp ψ ⊂ B2R we may assume that supp φn ⊂ B2R . Now since un is bounded in Y, we may assume that un ⇀ u = 𝜕x φ weakly in Y, and replacing if necessary φn by φn − φ, we may also assume that φ = 0. Then we have ∫ |un |2 = ∫ |û n |2 = B2R

ℝ2

|û n |2

∫ {|ξ1 |≤R1 ,|ξ2 |≤R1 }

+ ∫ |û n |2 +

∫

|û n |2 ,

{|ξ1 |≤R1 ,|ξ2 |≥R21 }

|ξ1 |≥R1

where f ̂(ξ1 , ξ2 ) is the Fourier transform of f (x, y). The third term satisfies |û n |2 =

∫ {|ξ1 |≤R1 ,|ξ2 |≥R21 }

∫ {|ξ1 |≤R1 ,|ξ2 |≥R21 }

|ξ1 |2 1 |v̂ |2 ≤ 2 ‖vn ‖2L2 . |ξ2 |2 n R1

The second term is bounded in the following way: ∫ |û n |2 ≤ |ξ1 |≥R1

1 ‖𝜕x vn ‖2L2 . R21

Fix ε > 0. Then choosing R1 sufficiently large leads to ∫ |û n |2 + |ξ1 |≥R1

∫ {|ξ1 |≤R1 ,|ξ2 |≥R21 }

ε |û n |2 ≤ . 2

We then use Lebesgue dominated convergence theorem for the first term, having noted that since un tends to 0 weakly in L2 (ℝ2 ), ̂ 1 , ξ2 ) = lim ∫ e−ixξ1 −iyξ2 un (x, y)dxdy = 0 lim u(ξ

n→+∞

n→+∞

B2R

for (ξ1 , ξ2 ) ∈ ℝ2 , and that 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨u(ξ )󵄨󵄨 ≤ |un |L1 (B2R ) .

344 | 4 Existence and stability of solitary waves We now turn to the 3-dimensional case. Theorem 4.4.3. Let d = 3, a = b = −1 and p = 1, then equation (4.4.4) possesses a solution (u, v, w) with u ∈ Y, y ≠ 0. Proof. Again, we prove the existence of a minimum for Iλ , by using the concentration– compactness principle. First, we also have Iλ > 0 for any λ > 0. We have ‖u‖Lq ≤ C‖u‖y

for any q, 2 ≤ q
0 and let un be a minimizing sequence for Iλ . Then there exists φn ∈ L6 (ℝ3 ), with 𝜕x φn = un ; let vn = 𝜕y φn , and wn = 𝜕z φn . We apply the concentration– compactness lemma to ρn = |φn |6 . Since φn is bounded in L6 (ℝ3 ) by Sobolev’s inequality, there exists a subsequence still denoted by ρn such that limn→∞ ∫ ρn dxdydz = β ≥ 0. Applying the next lemma with r = 6 shows that β > 0. Lemma 4.4.4. Let φ ∈ L6 (ℝ3 ) with 𝜕x φ ∈ Y. Then φ ∈ L10 (ℝ3 ) and there is a constant C > 0 such that 󵄩 󵄩1/5 󵄩 󵄩2/5 󵄩 󵄩2/5 ‖φ‖L10 ≤ C 󵄩󵄩󵄩𝜕x2 φ󵄩󵄩󵄩L2 󵄩󵄩󵄩𝜕y2 φ󵄩󵄩󵄩L2 󵄩󵄩󵄩𝜕z2 φ󵄩󵄩󵄩L2 . For any r with 6 ≤ r ≤ 10, there exist αj with 0 < αj < 1 for j = 0, 1, 2, 3 and a constant C > 0 such that if φ ∈ L6 (ℝ3 ) and 𝜕x φ ∈ Y α

α

α

α

α

‖𝜕x φ‖L3 ≤ C‖φ‖Lr0 ‖𝜕x φ‖L21 ‖𝜕x2 φ‖L22 ‖𝜕y φ‖L23 ‖𝜕z φ‖L23 . Proof. The first inequality and the case r = 6 in the second inequality follow directly from the generalized Sobolev inequality. If 6 ≤ r ≤ 10, then we cannot take directly q = 3 in the generalized Sobolev in4r < q < 10 . And then the generalized equality, rather we first consider q such that 3 < 2+r 3

Sobolev inequality applies with μ1 = we have

3−(r/q) , 5−(r/2)

μ

μ

μ2 = μ3 = 2μ1 − 1 and μ0 = r( q1 − μ

μ

μ1 ), 2

i. e.,

μ

‖𝜕x φ‖Lq ≤ C‖φ‖Lr0 ‖𝜕x φ‖L21 ‖𝜕x2 φ‖L22 ‖𝜕y φ‖L23 ‖𝜕z φ‖L23 . We obtain the desired inequality by interpolation, writing 1 θ 1−θ = + , 3 q 2

with θ ∈ (0, 1).

(i) We first show that “vanishing” cannot occur. If it were to occur, then we could prove that φn tends to zero in Lr (ℝ2 ). But then, using Lemma 4.4.4, we obtain a contradiction to the constraint of Iλ . (ii) Next, assume that “dichotomy” occurs, i. e., that lim Q(t) = α ∈ (0, β),

t→+∞

where for t ≥ 0,

4.4 The generalized Kadomtsev–Petviashvili equations | 345

Q(t) = lim

sup

t→+∞ (x ,y ,z )∈ℝ3 0 0 0

∫

|φn |6 dxdydz.

(x0 ,y0 ,z0 )+Bt

We define R0 , Rn , ξn , ηn in the same way as in the proof of Lemma 4.4.2, with |un |2 + |vn |2 + |𝜕x un |2 replaced by |φn |6 . We then set φ1n = ξn φn , φ2n = ηn φn , and (u1n , vn1 , wn1 ) = ∇φ1n , (u2n , vn2 , wn2 ) = ∇φ2n . By doing so, we have for n sufficiently large, 󵄨 󵄨󵄨󵄩󵄩 1 󵄩󵄩 1/6 󵄨󵄨󵄩󵄩φn 󵄩󵄩L6 − α󵄨󵄨󵄨 ≤ Cε , {󵄨󵄩 2 󵄩 󵄨󵄨󵄩󵄩φ 󵄩󵄩 6 − (β − α)󵄨󵄨󵄨 ≤ Cε1/6 . 󵄨󵄩 n 󵄩L 󵄨

(4.4.26)

Now, using the fact that |φn |6 dx ≤ 2ε,

∫

x = (x, y, z) ∈ ℝ3 ,

R0 ≤|xn −x0 |≤2Rn

it is not difficult, although quite technical, to show that 󵄨󵄨󵄩󵄩 1 󵄩󵄩2 󵄩󵄩 2 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 󵄨󵄨 󵄨󵄨󵄩󵄩un 󵄩󵄩Y + 󵄩󵄩un 󵄩󵄩Y − 󵄩󵄩un 󵄩󵄩Y 󵄨󵄨 ≤ δ(ε) → 0, ε → 0.

(4.4.27)

Consider, for example, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ [ξ 2 󵄨󵄨󵄨∇φn 󵄨󵄨󵄨2 − 󵄨󵄨󵄨∇(ξn φn )󵄨󵄨󵄨2 ]󵄨󵄨󵄨 n󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 3 󵄨󵄨 ℝ ≤

(|∇ξn |2 φ2n + 2|ξn ∇ξn ||φn ∇φn |)

∫ R0 ≤|xn −x0 |≤2Rn

≤ C‖∇ξn ‖2L3 ε1/3 + ‖∇φn ‖L2 ε1/6 ‖ξn ∇ξn ‖L3 and conclude the proof of this case by using the fact that ∇φn is bounded in L2 and that ∇ξn is bounded in L3 , independently of n. Using this and the second inequality in Lemma 4.4.4, we also have 󵄩󵄩 1 󵄩 2 󵄩󵄩un + un − un 󵄩󵄩󵄩L3 ≤ δ(ε) → 0,

ε → 0.

(4.4.28)

Finally, (4.4.26), (4.4.27) and (4.4.28) lead to a contradiction to the subadditivity condition implied by the relation Iλ = λ2/3 I1 , exactly as in the 2-dimensional case. (iii) The only remaining possibility is that ∃xn ∈ ℝ3 , ∀ε > 0, ∃R < +∞ such that for n sufficiently large ∫ |φn |6 ≥ β − ε.

(4.4.29)

xn +BR

Now, it is easily checked that Lemma 4.4.3 is also true in dimension 3. Hence the sequence (un ) is relatively compact in L6loc (ℝ3 ) by Sobolev inequality. This, together with

346 | 4 Existence and stability of solitary waves (4.4.29), shows that, modulo a subsequence, φn (⋅ − xn ) → φ strongly in L6 (ℝ3 ) and un (⋅ − xn ) → u = 𝜕x φ ∈ Y weakly in Y. Lastly, by using Lemma 4.4.4 with r = 6, un → u strongly in L3 (ℝ3 ) and u is a solution of Iλ . For the regularity property of the solitary waves, we have the following theorem: Theorem 4.4.4. Any solitary wave solution of (4.4.1) (resp. (4.4.1)) belongs to H ∞ (ℝd ) provided ε = −1 and p = 1, 2, 3 (resp. a = b = −1 and p = 1). Moreover, v = D−1 x uy (resp. −1 −1 ∞ d v = Dx uy ), and v = Dy uz belong to H (ℝ ). For the 2D and 3D versions of a fifth order KdV equation {

ut + up ux + uxxx + δuxxxxx − vy = 0, vx = uy ,

(4.4.30)

and u + up ux + uxxx + δuxxxxx − vy − wz = 0, { { t v = uy , { { x {wx = uz ,

(4.4.31)

we have the following result: Theorem 4.4.5. Equation (4.4.30) has no nontrivial solitary wave, with δ = 1, p ≥ 4, or δ = 1, 1 ≤ p < 4 and c is sufficiently large. But for δ = −1 and p arbitrary, it admits a nontrivial solitary wave belonging to H ∞ (ℝ2 ). Equation (4.4.31) has no nontrivial solitary wave, with p ≥ 3,

δ = −1,

p ≥ 2,

δ = 1,

or

or p = 1, δ = 1, and c being sufficiently large. But for δ = −1 and p = 1, 2, it admits a nontrivial solitary wave belonging to H ∞ (ℝ2 ). In order to study the stability and instability of solitary waves for KP equations, we first consider the stability of solitary wave for the 2-dimensional KP equation (ut + (up+1 )x + uxxx )x = uxx .

(4.4.32)

󵄨 2 2 V(Ω) = {u 󵄨󵄨󵄨 u ∈ L2 (ℝ2 ), ux ∈ L2 (ℝ2 ), D−1 x uy ∈ L (ℝ )}

(4.4.33)

Define a function space

4.4 The generalized Kadomtsev–Petviashvili equations | 347

with the norm 1 2

2

̃ 2 )dxdy) , |u|V = (∫ (u + |∇u| ℝ2

̃ = (ux , D−1 uy )T . We call u(x, y, t) = φω (x − ωt, y) the solitary wave of equation where ∇u x (4.4.32), if it satisfies p+1 ωφ + D−2 . x φyy − φxx = φ

(4.4.34)

When p = 1, it is the pulsed solitary wave v(x, y) = 8

ω − x2 /3 + y2 /(3ω) . ω + x2 /3 + y2 /(3ω)2

It is easy to know that equation (4.4.34) is the following functional of Euler–Lagrange equation: ω 1 ̃ 2 1 Lω (u) = ∫( u2 + |∇u| − up+2 )dxdy. 2 2 (p + 1)(p + 2)

(4.4.35)

So, if ∃φ ∈ V(ℝ2 ) is such that Lω (φ) = M(ω),

K(φ) = 1,

then φ is the solution of the following equation m+1 ωφ − φxx + D−2 , x φyy = λφ

where λ is the Lagrange multiplier. Let 󵄨 M(ω) = inf {Iω (u) 󵄨󵄨󵄨 k(u) = 1}, u∈V

̃ 2 )dxdy, Iω (u) = ∫ (ωu2 + |∇u| ℝ2

k(u) = ∫ up+2 dxdy. ℝ2

Theorem 4.4.6. Assume that 0 < p < 43 , p = p1 /p2 , where p1 is an arbitrary even number, p1 is an arbitrary odd number, ω > 0. Then p+2

Sω = {φ ∈ V(ℝ2 ), k(φ) = Iω (φ) = (M(ω)) is orbitally stable, where V is defined by equation (4.4.33).

p}

348 | 4 Existence and stability of solitary waves Consider the following 3-dimensional KP equations: u + up ux + uxxx − vy − wz = 0, { { t v = uy , { { x 3 {wx = uz , (x, y, z) ∈ ℝ .

(4.4.36)

We have Theorem 4.4.7. Assume that 0 ≤ p < 43 . Then there is a solitary wave u(x, y, z, t) = φc0 (x − ct, y, z), c0 > 0 of the 3-dimensional KP equation (4.4.36), and it is orbitally unstable with respect to the norm ‖ ⋅ ‖Y , where 󵄩 󵄩2 1 ‖u‖Y = ‖𝜕x v‖Y = (‖∇v‖22 + 󵄩󵄩󵄩𝜕x2 v󵄩󵄩󵄩2 ) 2 ,

v ∈ C0∞ (ℝ3 ).

4.5 The generalized Davey–Stewartson system Consider the N-dimensional Davey–Stewartson system iu + Δu = a|u|α u + b1 uux1 , { t −Δv = b2 (|u|2 )x ,

(4.5.1)

1

where Δ is the usual Laplacian operator in ℝn , a ∈ ℝ and α, bl , b2 are positive. The system (4.5.1) may be reduced to a single equation in u by applying the Fourier transform. Indeed, let E1 be the (nonlocal) linear operator defined by F (E1 (ψ))(ξ ) = σ1 (ξ )F (ψ)(ξ ),

where σ1 (ξ ) = ξ12 /|ξ |2 , ξ ∈ ℝN and F denotes the Fourier transform: N

F (ψ)(ξ ) = (

1 2 ) ∫ e−iξx ψ(x)dx. 2π

Then we have formally vx1 − b2 E1 (|u|2 ), and (4.5.1) may be written as the following nonlinear Schrodinger equation: iut + Δu = a|u|α u + b1 b2 E1 (|u|2 )u.

(4.5.2)

Consider periodic solutions of the form u(x, t) = eiωt φ(x), { v(x) = Φ(x),

(4.5.3)

where ω > 0, φ, Φ ∈ H 1 (ℝN ), φ, Φ ≠ 0, N = 2 and 3. If u is a solution of (4.5.2) satisfying (4.5.3), then we can see that φ must solve the following problem: φ ∈ H 1 (ℝN ), φ ≠ 0, { −Δφ + ωφ = bE1 (|φ|2 )φ − a|φ|α φ,

(4.5.4)

4.5 The generalized Davey–Stewartson system

| 349

where b = b1 b2 > 0, φ is a solution of (4.5.4) if and only if φ is a critical point of the Lagrangian S defined by S(φ) =

b a ω 1 ∫ |∇φ|2 − ∫ |φ|2 E1 (|φ|2 ) + ∫ |φ|α+2 + ∫ |φ|2 . 2 4 a+2 2

(4.5.5)

Let L2 be a real Hilbert space when equipped with the scalar product ̄ (u, v)2 = ∫ Re(u(x)v(x))dx. Also let W m,p and H s be the usual Sobolev spaces on ℝN . We assume that b and ω are positive constants and we denote Sob(N) = {

2N/(N − 2), +∞

if N ≥ 3, if N = 2.

We define the function sets 1󵄨 X = {ψ ∈ H 󵄨󵄨󵄨 ψ ≠ 0, ψ solves (4.5.4)},

󵄨 G = {φ ∈ X 󵄨󵄨󵄨 S(φ) ≤ S(ψ), ∀ψ ∈ X },

(G being the set of ground-states), and we introduce the set of admissible parameters ∗

ℛω,b = {(α, a) | 0 < α < Sob(N) − 2 and a < aα },

where +∞ { { { a∗α = {b { { α 2−α α α−2 α−2 {b 2 ω 2 ( 2 )( 2 ) 2

if α < 2, if α = 2, if α > 2.

We consider the following functionals on N 1 : T(φ) = |∇φ|22 ,

a ω b B (|φ|2 ) − |φ|α+2 − |φ|2 , 4 1 α + 2 α+2 2 2 1 S(φ) = T(φ) − V(φ), 2 ω E(φ) = S(φ) − |φ|22 . 2 V(φ) =

Proposition 4.5.1. If φ ∈ H 1 is a solution of (4.5.4), then (i) T(φ) + ω|φ|22 = bB1 (|φ|2 ) − a|φ|α+2 α+2 ; bN 2 (ii) (N − 2)T(φ) + Nω|φ|2 = 2 B1 (|φ|2 ) − 2Na |φ|α+2 α+2 ; α+2 N a(α−2) α+2 2 (iii) 4T(φ) + 2(α+2) |φ|α+2 = ω|φ|2 + 2 ∑j=2 |𝜕j φ|2 + Θ1 (|φ|2 );

350 | 4 Existence and stability of solitary waves where Θ1 is the functional defined by Θ1 (ψ) = ∫

2ξ12 (ξ22 + ⋅ ⋅ ⋅ + ξN2 ) 󵄨󵄨 󵄨2 󵄨󵄨F {ψ}󵄨󵄨󵄨 dξ . |ξ |4

Proof. If φ is a solution of (4.5.4), then (i), (ii) and (iii) are easily obtained by differentiating S along the curves in H 1 defined, for λ > 0, respectively by λ → λφ, λ → φ( λ⋅ ), 1

and λ → φλ , where φλ = λ 4 φ(Λ λ x), Λ λ = diag(1, . . . , λ, . . . , 1). Corollary 4.5.1. If φ is a solution of (4.5.4), then (i) S(φ) = N1 T(φ); (ii) (N − 2)T(φ) = 2NV(φ); 2−α (iii) E(φ) = (2−α) bB1 (|φ|2 ) + Nα−4 T(φ) = 2(α+2) a|φ|α+2 α+2 + 4α 2Nα (iv) b(B1 (|φ|2 ) + Θ1 (|φ|2 )) =

2aα |φ|α+2 α+2 α+2

+ 4|𝜕1 φ|22 .

N−2 T(φ); 2N

We prove the existence of ground-states for the problem (4.5.2) in 2D and 3D, i. e., of positive solutions of (4.5.4) that minimize the Lagrangian S. Theorem 4.5.1. Let N = 2 and (α, a) ∈ ℛω,b . Then the following holds: (i) X and G contain a real-valued positive function; (ii) φ ∈ G if and only if φ solves the minimization problem {

φ ∈ Σ0 , T(φ) = min{T(ψ) | ψ ∈ Σ0 },

(4.5.6)

where Σ0 = {ψ ∈ H 1 | ψ ≠ 0, V(ψ) = 0}. Theorem 4.5.2. Let N = 3 and (α, a) ∈ ℛω,b . Then the following holds: (i) X and G contain a real-valued positive function; (ii) there exists a constant μ0 > 0 such that φ ∈ G if and only if φ solves the minimization problem φ ∈ Σμ0 , { T(φ) = min{T(ψ) | ψ ∈ Σμ0 },

(4.5.7)

where Σμ0 = {ψ ∈ H 1 | ψ ≠ 0, V(ψ) = μ0 }. To prove these theorems we need the following lemmas: Lemma 4.5.1. Let 0 < q < Sob(N) − 2. Then there exists a constant C > 0 such that, for all ψ ∈ H 1 , q

2 2 2 2 |ψ|q+2 q+2 ≤ C(sup ∫ (|∇ψ| + |ψ| ) ‖ψ‖H 1 ). y

B1 (y)

4.5 The generalized Davey–Stewartson system |

351

Proof. Let us cover ℝN by a sequence of unit cubes {Cj } such that Cj ∩ Ck = 0 if j ≠ k. Then we obtain ∞

q+2 |ψ|q+2 q+2 = ∑ ∫ |ψ| j=1 C

j

and ∞

‖ψ‖2H 1 = ∑ ∫(|∇ψ|2 + |ψ|2 ). j=1 C

j

Therefore, it follows from Sobolev imbedding theorem that q+2

∫ |ψ|

2

2

≤ C(∫(|∇ψ| + |ψ| ))

Cj

q+2 2

Cj 2

q 2

≤ C(sup ∫(|∇ψ| + |ψ| )) ∫(|∇ψ|2 + |ψ|2 ), j∈N

2

Cj

Cj

with C depending only on N and q. Summing up the above inequality over j, we obtain |ψ|q+2 q+2

2

2

q 2

≤ C(sup ∫(|∇ψ| + |ψ| )) ∫(|∇ψ|2 + |ψ|2 ), j∈N

Cj

Cj

from which the result follows. For each μ ∈ ℝ we define 󵄨 Σμ = {ψ ∈ H 1 󵄨󵄨󵄨 ψ ≠ 0, V(ψ) = μ}, 󵄨󵄨 1 󵄨 j(μ) = inf{ T(ψ) 󵄨󵄨󵄨 ψ ∈ Σμ }. 󵄨󵄨 2

(4.5.8) (4.5.9)

Then we have Theorem 4.5.3. Let N ∈ {2, 3} and (α, a) ∈ ℛω,b . Then the following holds: (i) Σμ ≠ 0 for all μ ∈ ℝ; (ii) if N = 2 then there exists a constant I > 0 such that j(μ) = I, ∀μ ∈ ℝ; (iii) if N = 3 then there exists a constant I > 0 such that j(μ) = μ1/3 I, ∀μ > 0. Proof. Let φ ∈ H 1 and define for λ > 0, φλ (x) = φ(λ−1/N x). If ε > 0 is small enough, then V(εφ) < 0 and since V((εφ)λ ) = λφV(εφ) for all λ > 0, it follows that Σμ ≠ 0, ∀μ < 0.

352 | 4 Existence and stability of solitary waves In order to prove that Σμ ≠ 0, ∀μ ≥ 0, it suffices to show that ∃φ0 ∈ H 1

such that V(φ0 ) > 0.

(4.5.10)

Indeed, if (4.5.10) holds, then there exists τ0 < 1 such that V(τ0 φ0 ) = 0 and we conclude that Σμ ≠ 0. Moreover, since V(φ0λ ) = λV(φ0 ), ∀λ > 0, it follows that Σμ ≠ 0, ∀μ > 0. Assuming (α, a) ∈ Rω,b , we have: First, if α < 2 or if a < 0, then (4.5.10) holds by taking φ0 = τφ, with τ large enough. Second, if α = 2 and a < b, then it follows that there exists φ ∈ H 1 such that a 1 B (|φ|2 ) − |φ|44 > 0 4 1 4 and (4.5.10) holds, where B1 (|φ|2 ) = ∫ σ1 (Λ λ ξ )|F {|φ|2 }|2 dξ , with φ0 = τφ, for τ large enough. α 2−α 2−α Last, if α > 0 and a < b 2 ω 2 ( α2 )( α−2 ) 2 , we define α Gλ (s) =

b 4 a α−2 ω 1 s − λ 4 sα+2 − λ− 2 s2 . 4 α+2 2

It is easy to check that, for all λ > 0, ∃s0 > 0 | Gλ (s0 ) > 0

⇐⇒

α 2

a 0. With ψλ (x) = λ 41 ψ(Λ λ x), Λ λ = diag(1, . . . , λ, . . . , 1), there exists λ > 0 such that α−2

b aλ 4 ω 1 V(ψλ ) > |ψ|44 − ε − |ψ|α+2 − λ− 2 |ψ|22 = mes(BR )Gλ (s) − ε. 4 α + 2 α+2 2 From (4.5.11) there exists s0 such that Gλ (s0 ) > 0, and we get V(ψλ ) > 0 if we chose R large enough. We obtain (4.5.10) by denseness. Let us assume that N = 2 and let 󵄨󵄨 1 󵄨 I = j(0) = inf{ T(ψ) 󵄨󵄨󵄨 ψ ∈ Σ0 }. 󵄨󵄨 2 In order to prove that I > 0, we consider ψ ∈ ∑0 . By Bj = ∫ Ej (ψ)ψ̄ ≤ |ψ|22 , we have |a| ω 2 b 4 |ψ| ≤ |ψ| + |ψ|α+2 . 2 2 4 4 α + 2 α+2

(4.5.12)

From Gagliardo–Nirenberg–Sobolev inequality we obtain |ψ|44 ≤ C1 |∇ψ|22 |ψ|22 , { α+2 |ψ|α+2 ≤ C2 |∇ψ|α2 |ψ|22 .

(4.5.13)

4.5 The generalized Davey–Stewartson system |

353

Merging (4.5.12) and (4.5.13) we get ω ≤ C1󸀠 T(ψ) + C2󸀠 T(ψ)α/2 , 2 from which we conclude that I > 0. With ψλ (x) = ψ(λ−λ/N x) we have ψ ∈ Σμ

⇐⇒

ψλ ∈ Σλμ .

Since T(ψλ ) = T(ψ) for all λ > 0, we conclude that j(μ) must be constant on [−∞, 0] and on [0, +∞]. Let μn ↓ 0 and ε > 0, there exists ψ ∈ Σ0 such that 1 I < T(ψ) < I + ε. 2 Taking τn > 1 such that V(τn ψ) = μn , we have 1 j(μn ) − I < (τn2 − 1)T(ψ) + ε 2 and so we obtain lim sup(j(μn ) − I) ≤ 0. n→∞

(4.5.14)

On the other hand, let ψn ∈ Σμn be such that 1 T(ψn ) < j(μn ) + ε. 2 Taking τn < 1 such that τn ψn ∈ Σμn , it follows that 1 1 I < T(τn ψn ) = τn2 T(ψn ) < j(μn ) + ε, 2 2 and then we obtain lim sup(j(μn ) − I) ≥ 0. n→∞

From (4.5.14) and (4.5.15) we obtain j(μ) = I,

∀μ ∈ (0, +∞).

The same arguments with μn ↑ 0 give us j(μ) = I,

∀μ ∈ (−∞, 0).

Assume now N = 3 and let 󵄨󵄨 1 󵄨 I = j(1) = inf{ T(ψ) 󵄨󵄨󵄨 ψ ∈ Σ1 }. 󵄨󵄨 2 As before, we get from Gagliardo–Nirenberg–Sobolev inequality that I > 0.

(4.5.15)

354 | 4 Existence and stability of solitary waves With ψλ (x) = ψ(λ−1/N x), we have ψ ∈ Σμ

⇐⇒

ψλ ∈ Σλμ ,

∀λ > 0,

and the conclusion follows since T(ψλ ) = λ1/3 T(ψ). Corollary 4.5.2. Let N ∈ {2, 3}, and let us consider j(μ) defined by (4.5.8)–(4.5.9). Then we have the following subadditivity property: ∀μ > 0, { j(μ) < j(λ) + j(μ − λ),

∀λ ∈ [0, μ].

(4.5.16)

If N = 2 then (4.5.16) holds for all λ, μ ∈ ℝ. The minimization problems (4.5.6) and (4.5.7) have the following equivalent formulations: Lemma 4.5.2. Let N = 2 and (α, a) ∈ Rω,b . Then the problem (4.5.6) is equivalent to {

V(φ) = 0, φ ≠ 0, T(φ) = min{T(ψ) | V(ψ) ≥ 0}.

Let N = 3, (α, α) ∈ Rω,b and μ0 > 0. Then the problem (4.5.7) is equivalent to {

V(φ) = μ0 , φ ≠ 0, T(φ) = min{T(ψ) | V(ψ) ≥ μ0 }.

Proof. We prove only for N = 2, the other case being similar. Let I ̄ = inf{T(ψ) | V(ψ) ≥ 0},

I = inf{T(ψ) | V(ψ) = 0}.

It is evident that I ̄ ≤ I. Now, if ψ ∈ H 1 , ψ ≠ 0 is such that V(ψ) ≥ 0, we can get 0 < τ ≤ 1 for which V(τψ) = 0 and the conclusion follows from I ≤ T(τψ) = τ2 T(ψ) ≤ T(ψ). Lemma 4.5.3. Let a(x) : ℝN → ℝ be a continuous function and let us assume that a(x) → 0 as |x| → +∞. Let us assume further that there exists v ∈ H 1 such that J(v) = ∫(|∇v|2 − a|v|2 )dx < 0. Then there exists λ > 0 and a positive solution u ∈ H 1 ∩ C of the equation −△u + λu = au. In addition, if ω ∈ H 1 is nonnegative, ω ≠ 0, is such that −△ω + νω = aω for some ν ∈ ℝ, then ω = cu for some c > 0. In particular, ν = λ.

4.5 The generalized Davey–Stewartson system |

355

Proof. See Cazenave [9, p. 168]. We are now ready to prove Theorems 4.5.1 and 4.5.2. Proof of Theorem 4.5.1. Let j(μ) as defined by (4.5.8)–(4.5.9), and let us consider the minimization problem (4.5.6). We proceed in several steps. Step 1. Problem (4.5.6) has a solution. Let {ψn } be a minimizing sequence, and let us define φn (x) = ψn (√Λ n x), where Λ n = |ψn |22 . Then {ψn } is also a minimizing sequence because T(φn ) = T(ψn ), Since |φn |22 =

1 |ψn |22 Λn

V(φn ) =

1 V(ψn ) = 0. Λn

(4.5.17)

= 1, it follows that {φn } is bounded in H 1 , and we may assume

the existence of φ ∈ H 1 such that

φn → φ weakly in H 1 .

(4.5.18)

We apply the concentration–compactness principle with ρn = |∇φn |2 + |φn |2 and observe that lim ∫ ρn = lim λ̄n = λ̄ = 2I + 1 > 0.

n→+∞

n→+∞

If “vanishing” occurs, namely limn→+∞ sup(x,y)∈ℝ2 ∫(x,y)+B ρn = 0, then letting n → ∞, R we obtain from Lemma 4.5.1, B1 (|φn |2 ) → 0,

|φn |α+2 → 0,

(4.5.19)

which is impossible because |φn |22 = 1. So “vanishing” cannot occur. Next, if “dichotomy” were to occur, then for all ε > 0, we would find R0 > 0, {yn } ⊂ ℝ2 , Rn 󳨀󳨀󳨀󳨀󳨀→ +∞ and φ1n , φ2n bounded in H 1 , all depending on ε, such that n→∞

(i) supp φ1n ⊂ BRn (yn ),

(ii) supp φ2n ⊂ ℝ2 \ BR0 (yn ), 󵄩 󵄩 (iii) 󵄩󵄩󵄩φn − φ1n − φ2n 󵄩󵄩󵄩H 1 ≤ ε 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 (iv) 󵄩󵄩󵄩∇φn 󵄩󵄩󵄩2 − 󵄩󵄩󵄩∇φ1n 󵄩󵄩󵄩2 − 󵄩󵄩󵄩∇φ2n 󵄩󵄩󵄩2

(4.5.20) (4.5.21) (4.5.22) ≥ −Cε,

C > 0 independent of ε.

Since φkn ≠ 0, k = 1, 2, it follows from (4.5.23) that, for n large enough, 1 1 C 1 I + ε > T(φn ) ≥ T(φ1n ) + T(φ2n ) − ε 2 2 2 2 C 1 2 ≥ j(V(φn )) + j(V(φn )) − ε. 2

(4.5.23)

356 | 4 Existence and stability of solitary waves From Lemma 4.5.2 we obtain I + ε > 2I − C2 ε, which is impossible if ε is small enough. Thus dichotomy does not occur. Therefore, we have “concentration”, which means that there exists a sequence {yn } in ℝ2 for which we have ∀ε > 0, ∃Rε ≥ 1/ε such that ρn (x)dx ≤ ε,

∫ BRε (yn

(4.5.24)

)c

where BRε (yn )c = ℝ2 \ BRε (yn ). Let φ̃ n (⋅) = φn (⋅ − yn ). Then from (4.5.18) we have φ̃ n → φ̃ weakly in H 1 . Moreover, from (4.5.24) and Sobolev inequality, it follows that ∫ |φ|̃ p dx ≤ εp/2 ,

∀p ∈ [2, ∞),

(4.5.25)

BRc ε

where BRε = BRε (0). Denoting by a ω b |ψ|α − }, VΩ (ψ) = ∫ |ψ|2 { E1 (|ψ|2 ) − 4 α+2 2 Ω

we obtain from (4.5.25) that 󵄨󵄨 󵄨 󵄨󵄨VBRc (φ̃ n )󵄨󵄨󵄨 ≤ δ(ε), ε

(4.5.26)

with δ(ε) → 0 as ε → 0. Since the injection H 1 (BRε ) ⊂ Lq (BRε ) is compact for q ∈ [1, ∞), we have ̃ VBR (φ̃ n ) 󳨀󳨀󳨀󳨀󳨀→ VBR (φ). n→∞

ε

ε

(4.5.27)

Moreover, 0 = V(φ̃ n ) = VBR (φ̃ n ) + VBc (φ̃ n ) and, from (4.5.26), we have ε

Rε

󵄨󵄨 󵄨 󵄨󵄨VBRε (φ̃ n )󵄨󵄨󵄨 < δ(ε).

(4.5.28)

Letting n → ∞ in (4.5.28), we have by (4.5.27) 󵄨󵄨 󵄨 󵄨󵄨VBRε (φ)̃ 󵄨󵄨󵄨 < δ(ε). Letting now ε → 0, we get φ̃ ∈ Σ0 and the conclusion follows by the semicontinuity of T. Step 2. X is nonempty. Let φ be a solution of (4.5.6). Then there exists a Lagrange multiplier λ such that −Δφ = λ(bE1 (|φ|2 )φ − a|φ|α φ − ωφ).

(4.5.29)

4.5 The generalized Davey–Stewartson system |

357

In order to prove that λ > 0, let φ ∈ H 1 be such that ⟨V 󸀠 (φ); φ⟩ > 0, where ⟨⋅, ⋅⟩ denotes the H −1 –H 1 duality pairing. Since T, V ∈ C 1 (H 1 ; ℝ), we obtain t

V(φ + tϕ) = V(φ) + ∫⟨V 󸀠 (φ + sϕ), ϕ⟩ds, 0

t2 T(φ + tϕ) = T(φ) + tλ⟨V 󸀠 (φ), ϕ⟩ + |∇ϕ|22 . 2

(4.5.30)

If λ < 0, we obtain from (4.5.30) for t small enough that V(φ + tϕ) > 0 and T(φ + tϕ) < T(φ), which is in contradiction with Lemma 4.5.3. Hence λ > 0. Now, letting φλ (x) = φ(x/√λ), we conclude that φλ ∈ X . Step 3. We prove (ii). Let φ be a solution of (4.5.6) and let ψ ∈ X . From Proposition 4.5.1 we have ψ ∈ Σ0 . Therefore S(φ) ≤ S(ψ) and φ ∈ G . Conversely, assume φ ∈ G . Then S(φ) ≤ S(ψ) for all ψ ∈ S. Again Proposition 4.5.1 implies V(φ) = V(ψ) = 0, and we conclude that φ solves (4.5.6). Step 4. We prove (i). First we remark that from Steps 1 and 3 we obtain G ≠ 0. The conclusion follows from Lemma 4.5.3 (see [9]). Proof of Theorem 4.5.2. Let j(μ) be defined by (4.5.8)–(4.5.9) and consider the minimizing problem φ ∈ Σμ , μ > 0, { T(φ) = min{T(ψ) | ψ ∈ Σμ }.

(4.5.31)

We proceed in several steps: Step 1. Problem (4.5.31) has a solution. Let {φn } be a minimizing sequence for (4.5.31). Then {φn } is bounded in H 1 . Indeed, since ψn ∈ Σμ we have ω 2 b |a| |ψ|2 + μ ≤ B1 (|ψ|2 ) + |ψ|α+2 . 2 4 α + 2 α+2

(4.5.32)

From Gagliardo–Nirenberg–Sobolev inequality and Bj (ψ) = ∫ Ej (ψ)ψn̄ ≤ |ψn |22 , we have ω 2 |ψ| + μ ≤ C1 |∇ψn |32 |ψn |2 + C2 |∇ψn |3α/2 |ψn |2−α/2 . 2 2 2 2 Since {∇φn } is bounded in L2 , (4.5.33) implies that {φn } is bounded in H 1 . Hence, taking a subsequence if necessary, there exists φ ∈ H 1 such that φn → φ weakly in H 1 .

(4.5.33)

358 | 4 Existence and stability of solitary waves We apply the concentration–compactness lemma of Lions with ρn = |∇ψn |2 + |ψn |2 . We remark that, without loss of generality, we may assume that lim ∫ ρn = lim λ̃n = λ,̃

n→∞

n→∞

1

with λ̃ ≥ 2j(μ) = 2μ 3 I > 0 (see Theorem 4.5.3). If “vanishing” were to occur, then from Lemma 4.5.1 we would have B1 (|ψn |2 ) → 0,

|ψn |α+2 α+2 → 0,

but this would contradict (4.5.32). Next, if “dichotomy” were to occur, then for all ε > 0, we would find {yn } ⊂ ℝ3 , R0 > 0, limn→∞ Rn = ∞ and φ1n , φ2n ≠ 0 bounded in H 1 satisfying (4.5.20)–(4.5.23). From (4.5.22) we have 󵄩󵄩 1 󵄩󵄩 󵄩 2󵄩 󵄩󵄩φn 󵄩󵄩H 1 + 󵄩󵄩󵄩φn 󵄩󵄩󵄩H 1 ≥ ‖φn ‖H 1 − ε.

(4.5.34)

Taking subsequences if necessary, we may assume that 󵄩 󵄩 limn→∞ 󵄩󵄩󵄩φkn 󵄩󵄩󵄩H 1 = λ̃k (ε) > 0,

{

limn→∞ V(φkn ) = λk (ε),

which, from (4.5.34), imply 1 λ̃2 (ε) ≥ λ̃ > 0. 2

(4.5.35)

From (4.5.20)–(4.5.21), Ej (ψ) = Ej (ψ)̄ and Bj (ψ) = ∫ Ej (ψ)ψ̄ ≤ |ψ|22 , we have 󵄨 󵄨 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 B1 (󵄨󵄨󵄨φ1n + φ2n 󵄨󵄨󵄨2 ) = B1 (󵄨󵄨󵄨φ1n 󵄨󵄨󵄨 ) + B1 (󵄨󵄨󵄨φ2n 󵄨󵄨󵄨 ) + 2(󵄨󵄨󵄨φ2n 󵄨󵄨󵄨 ; E1 (󵄨󵄨󵄨φ1n 󵄨󵄨󵄨 ))2 , for n large enough and b 󵄨 󵄨 2 󵄨2 󵄨 1 󵄨2 󵄨 󵄨󵄨 󵄨 1 2 󵄨 1 2 󵄨 󵄨󵄨V(φn ) − V(φn ) − V(φn )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨V(φn ) − V(φn + φn )󵄨󵄨󵄨 + 󵄨󵄨󵄨(󵄨󵄨󵄨φn 󵄨󵄨󵄨 ; E1 (󵄨󵄨󵄨φn 󵄨󵄨󵄨 ))2 󵄨󵄨󵄨. 2 Since V ∈ C 1 (H 1 ; ℝ), we can write 1

V(φ) − V(ψ) = ∫⟨V 󸀠 (tφ + (1 + t)ψ), φ − ψ⟩dt, 0

(where ⟨⋅, ⋅⟩ denotes the H −1 –H 1 duality pairing) and then we obtain 󵄨󵄨 󵄩 1 2 󵄨 1 2󵄩 󵄨󵄨V(φn ) − V(φn + φn )󵄨󵄨󵄨 ≤ C(‖φn ‖H 1 )󵄩󵄩󵄩φn − φn − φn 󵄩󵄩󵄩H 1 .

(4.5.36)

4.5 The generalized Davey–Stewartson system |

359

On the other hand, setting φ̃ kn (⋅) = φkn (⋅ − yn ),

k = 1, 2,

by the properties of Ej (ψ) we have 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 (󵄨󵄨󵄨φ2n 󵄨󵄨󵄨 ; E1 (󵄨󵄨󵄨φ1n 󵄨󵄨󵄨 ))2 = ∫ 󵄨󵄨󵄨φ̃ 2n 󵄨󵄨󵄨 E1 (󵄨󵄨󵄨φ1n 󵄨󵄨󵄨 ). BRc

n

Since H 1 (BR0 ) ⊂ Lq (BR0 ) is compact, for all q ∈ [2, 6], and since {φ1n } is bounded in H 1 , we conclude that φ̃ 1n → φ1 in L4 (BR0 ) for some φ1 ∈ H01 (BR0 ). In particular, 󵄨 󵄨2 lim E (󵄨󵄨φ̃ 1 󵄨󵄨 ) n→∞ 1 󵄨 n 󵄨

󵄨 󵄨2 = E1 (󵄨󵄨󵄨φ1 󵄨󵄨󵄨 ) in L2 (BR0 ).

(4.5.37)

Since Rn → ∞, it follows from (4.5.20)–(4.5.21) and Lebesgue convergence theorem that 󵄨 󵄨2 󵄨 󵄨2 (󵄨󵄨󵄨φ2n 󵄨󵄨󵄨 ; E1 (󵄨󵄨󵄨φ1n 󵄨󵄨󵄨 ))2 → 0,

n → ∞.

(4.5.38)

Putting (4.5.36) and (4.5.37) together, we obtain for n large enough 󵄨󵄨 1 2 󵄨 󵄨󵄨V(φn ) − V(φn ) − V(φn )󵄨󵄨󵄨 ≤ δ(ε),

(4.5.39)

with δ(ε) 󳨀󳨀󳨀󳨀→ 0. Letting n → ∞ in (4.5.39) we obtain ε→0

󵄨󵄨 󵄨 󵄨󵄨μ − λ1 (ε) − λ2 (ε)󵄨󵄨󵄨 ≤ δ(ε).

(4.5.40)

We have three possibilities for λ1 (ε) (or λ2 (ε)): either λ1 (ε) 󳨀󳨀󳨀󳨀→ 0, ε→0

or

λ1 (ε) ≤ −λ1 < 0,

or λ1 (ε) ≥ λ1 > 0,

for some λ1 . Suppose the first case occurs. Since there exists a constant C1 > 0 independent of n and ε such that |∇φ1n |22 ≥ 2C1 , we get j(μ) = lim

n→∞

1 1 T(φn ) ≥ C1 + lim inf T(φ2n ) − Cε n→∞ 2 2 ≥ C1 + j(λ2 (ε)) − Cε.

Letting ε → 0 in the previous inequality, we get a contradiction to (4.5.40). The second possibility also cannot occur, otherwise, from (4.5.23), we would have j(μ) = lim

n→∞

1 T(φn ) ≥ j(λ2 (ε)) − Cε. 2

360 | 4 Existence and stability of solitary waves From (4.5.40) we obtain λ2 (ε) ≥ μ + λ1 − δ(ε). Since j is an increasing function, we conclude that j(μ) ≥ j(μ + λ1 − δ(ε)) − Cε. Letting ε → 0 we get a contradiction. The last possibility does not occur, otherwise, from (4.5.40), we obtain j(μ) ≥ j(λ) + j(μ − λ), which is in contradiction to Corollary 4.5.2. Hence the conclusion – dichotomy does not occur. Therefore, it is the “concentration” that could only occur. Changing (4.5.27) by 󵄨󵄨 󵄨 󵄨󵄨V(φ̃ n ) − μ󵄨󵄨󵄨 < δ(ε), the same arguments in the proof of Theorem 4.5.1 hold. Step 2. X is nonempty. Let φ be a solution of (4.5.30). Then there exists a Lagrange multiplier λμ such that −Δφ = λμ (bE1 (|φ|2 )φ − a|φ|α φ − ωφ).

(4.5.41)

In order to prove that λμ > 0, we argue as in Theorem 4.5.1. Let ϕ ∈ H 1 be such that ⟨V 󸀠 (φ), ϕ⟩ > 0. Then we have t

V(φ + tϕ) = V(φ) + ∫⟨V 󸀠 (φ + sϕ), ϕ⟩ds, 0

T(φ + tϕ) = T(φ) + tλμ ⟨V 󸀠 (φ), ϕ⟩ +

t2 |∇ϕ|22 . 2

Assuming that λμ < 0 we obtain for t small enough that V(φ + tϕ) > μ and T(φ + tϕ) < T(φ), which is in contradiction with Lemma 4.5.3. Hence λμ > 0. From Corollary 4.5.1 (i) and Lemma 4.5.2 we obtain 2 1 λμ = Iμ− 3 . j

Then we can choose μ0 > 0 such that λμ0 = 1 and, from (4.5.41), we conclude that φ∈X. The conclusion follows exactly as in Steps 3 and 4 in Theorem 4.5.1. Now we consider the stability of standing waves for the generalized Davey– Stewartson system. Consider the following nonlinear Schrödinger equation: iut + Δu + a|u|p−1 u + bE1 (|u|2 )u = 0,

t ≥ 0, x ∈ ℝn ,

(4.5.42)

4.5 The generalized Davey–Stewartson system

| 361

where a, b ≥ 0, 1 < p < 1 + 4/(n − 2), n = 2 or 3, and E1 is the singular integral operator with symbol σ1 (ξ ) = ξ 2 /|ξ |2 , ξ ∈ ℝn . By a standing wave, we mean a solution of (4.5.42) having the form u(x, t) = eiωt φω (x),

(4.5.43)

where ω > 0 and φω is a ground state of the following stationary problem: −Δψ + ωψ − a|ψ|p−1 ψ − bE1 (|ψ|2 )ψ = 0,

{

ψ ∈ H 1 (ℝn ),

x ∈ ℝn , ψ ≠ 0.

(4.5.44)

Before stating our result, we introduce some notations: ω a b 1 |v|p+1 − ∫ |v|2 E1 (|v|2 )dx, Sω (v) = |∇v|22 + |v|22 − 2 2 p + 1 p+1 4 Xω = the set of solutions for (4.5.42)

󸀠 = {ψ ∈ H 1 (ℝn ) : Sω (ψ) = 0, ψ ≠ 0},

Gω = the set of ground states for (4.5.42)

= {φ ∈ Xω : Sω (φ) ≤ Sω (ψ) for all ψ ∈ Xω }.

Assumption (H). We assume that there is a choice φω ∈ Gω such that ω 󳨃→ φω is a C 1 mapping from the interval (0, ∞) into H 1 (ℝn ). Moreover, when the space dimensions n = 2, we assume that |φ|2 = |φω |2

for any φ ∈ Gω .

(4.5.45)

Definition 4.5.1. We say that the standing wave uω (t) = eiωt φω is stable if for any ε > 0 there exists δ > 0 with the following property: If u0 ∈ H 1 (ℝn ) and the solution u(t) of (4.5.42) with u(0) = u0 satisfies ‖u0 − φω ‖H 1 < δ, then 󵄩 󵄩 sup inf 󵄩󵄩󵄩u(t) − φ󵄩󵄩󵄩H 1 < ε. φ∈G

0≤t 0, 1 < p < 1 + 4/n, and n = 2 or 3, then there exists a sequence (ωk ) such that ωk > 0, ωk → 0 and φωk is stable. Theorem 4.5.5. Assume that n = 2 or 3 and that Assumption (H) holds. Let d(ω) = Sω (φω ), 0 < ω < ∞. If d󸀠󸀠 (ω0 ) > 0 then φω0 is stable. We define the following functionals on H 1 (Rn ): T(v) = |∇v|22 , b ω a |v|p+1 − ∫ |v|2 E1 (|v|2 )dx − |v|22 , Vω (v) = p + 1 p+1 4 2

362 | 4 Existence and stability of solitary waves 1 1 Pω (v) = ( − )T(v) − Vω (v), 2 n 1 b a ε(v) = |∇v|22 − |v|p+1 − ∫ |v|2 E1 (|v|2 )dx. 2 p + 1 p+1 4 We note that 1 ω Sω (v) = T(v) − Vω (v) = ε(v) + |v|22 , 2 2 1 Pω (v) = Sω (v) − T(v). n

(4.5.47)

Lemma 4.5.4. Assume that n = 2 or 3, let μω = ( 21 − n1 )T(φω ). (1) Pω (ψ) = 0, ∀ψ ∈ Xω (Pohozaev’s identity); (2) d(ω) = n1 T(φω ) = inf{ n1 T(ψ) : ψ ∈ H 1 (ℝn ), ψ ≠ 0, Vω (ψ) ≥ μω }; (3) d󸀠 (ω) = 21 |φω |22 ; (4) d(ω) = inf{ n1 T(ψ) : ψ ∈ H 1 (ℝn ), ψ ≠ 0, Pω (ψ) ≤ 0}. Proof. For λ > 0, we define ψλ (x) = ψ(x/λ). (1) Take ψ ∈ Xω , then we have Pω (ψ) =

󵄨󵄨 1 1 󸀠 󵄨 𝜕λ Sω (ψλ )󵄨󵄨󵄨 = ⟨Sω (ψ), 𝜕λ ψλ ⟩ = 0. 󵄨󵄨λ=1 n n

(2) See [14, Lemma 3.6]. (3) Since Sω (φω ) = ε(φω ) + (ω/2)|φω |, we have 1 1 󸀠 d󸀠 (ω) = ⟨Sω (φω ), 𝜕ω φω ⟩ + |φω |22 = |φω |22 . 2 2 ̃ (4) Put d(ω) = inf{ n1 T(ψ) : ψ ∈ H 1 (ℝn ), ψ ≠ 0, Pω (ψ) ≤ 0}. From Lemma 4.5.4 (1), ̃ we have d(ω) ≤ (1/n)T(φω ) = d(ω). Conversely, for ψ ∈ H 1 such that ψ ≠ 0, Pω (ψ) ≤ 0, let λ = {T(φω )/T(ψ)}1/n . Then we have 1 1 Vω (ψλ ) = λn Vω (ψ) ≥ ( − )λn T(ψ) = μω . 2 n From Lemma 4.5.4 (2), we have d(ω) ≤

1 1 1 1 T(ψλ ) = λn−2 T(ψ) = ( T(φω )/ T(ψ)) n n n n

1−2/n

1 T(ψ), n

̃ which implies d(ω) ≤ n1 T(ψ). Hence d(ω) ≤ d(ω). Proof of Theorem 4.5.5. Let φω be the ground state of −Δψ + ωψ − a|ψ|p−1 ψ = 0,

{

ψ ∈ H 1 (ℝn ),

x ∈ ℝn , ψ ≠ 0.

(4.5.48)

4.5 The generalized Davey–Stewartson system

| 363

Since 󵄨2 󵄨 ∫ |v|2 E1 (|v|2 )dx = ∫ σ1 (ξ )󵄨󵄨󵄨F (|v|2 )󵄨󵄨󵄨 dξ ≥ 0, for all ∀v ∈ H 1 (ℝn ), where F is the Fourier transform on ℝn , we have 1 1 ω a Pω (φ̃ ω ) ≤ ( − )λn T(ψ) + |φ̃ ω |22 − |φ̃ |p+1 = 0. 2 n 2 p + 1 ω p+1 Here, we have used Pohozaev’s identity for equation (4.5.48). From Lemma 4.5.4 (4), we have d(ω) ≤ n1 T(φ̃ ω ) for all ω ∈ (0, ∞). Moreover, since φ̃ ω (x) = ω1/(p−1) φ̃ 1 (√ωx), we have T(φ̃ ω ) = ωα T(φ̃ 1 ), where α = [2/(p − 1)] − [(n − 2)/2] > 1. Therefore, we have d(ω) ≤

1 α ω T(φ̃ 1 ), n

α > 1, ∀ω ∈ (0, ∞).

(4.5.49)

̂ Furthermore, Here, if d󸀠󸀠 (ω) ≤ 0 in (0, ω)̂ for some ω̂ > 0, then d󸀠 (ω) ≥ d󸀠 (ω)̂ in (0, ω). since it follows from (4.5.49) that limω→0 d(ω) = 0, we have ω

̂ d(ω) = ∫ d󸀠 (s)ds ≥ d󸀠 (ω)ω,

̂ ∀ω ∈ (0, ω).

(4.5.50)

0

We remark that d󸀠 (ω)̂ > 0 from Lemma 4.5.4 (3). Thus, (4.5.50) contradicts (4.5.49). Therefore, there exists a sequence (ωk ) such that ωk > 0, ωk → 0 and d󸀠󸀠 (ω − k) > 0. Hence, Theorem 4.5.3 holds by Theorem 4.5.4. Now we prove Theorem 4.5.4. We assume that n = 2 or 3. Lemma 4.5.5. The sets +

1

n

Aω = {v ∈ H (ℝ ) : Sω (v) < d(ω), Pω (v) > 0}

= {v ∈ H 1 (ℝn ) : Sω (v) < d(ω), −

1

n

1 T(v) < d(ω), v ≠ 0}, n

Aω = {v ∈ H (ℝ ) : Sω (v) < d(ω), Pω (v) < 0}

= {v ∈ H 1 (ℝn ) : Sω (v) < d(ω),

1 T(v) > d(ω), v ≠ 0} n

are invariant regions under the flow of equation (4.5.42). Proof. Let u0 ∈ Aω± and let u(t) be a solution of (4.5.42) with u(0) = u0 . Then from |u(t)|2 = |u0 |2 , ε(u(t)) = ε(u0 ), we have Sω (u(t)) = Sω (u0 ) < D(ω). Therefore, from (4.5.42) and Lemma 4.5.4 (4), we have Pω (u(t)) ≠ 0. Since the function t 󳨃→ Pω (u(t)) is continuous, we have Pω (u(t)) > 0 if u0 ∈ Aω+ and Pω (u(t)) < 0 if u0 ∈ Aω− . This shows that the sets Aω± are invariant under the flow of (4.5.42).

364 | 4 Existence and stability of solitary waves Lemma 4.5.6. If d󸀠󸀠 (ω) > 0, then there exists ε0 > 0 with the following property: For any ε ∈ (0, ε0 ), there exists δ > 0 such that if u0 ∈ Aω+ , ‖u0 − φω0 ‖H 1 < δ and u(t) is a solution of (4.5.42) with u(0) = u0 , then d(ω0 − ε) < (1/n)T(u(t)) < d(ω0 + ε) for all t > 0. Proof. Fix ε > 0 and let ω+ = ω0 + ε, ω− = ω0 − ε. It follows from Lemma 4.5.4 (3) that d(ω− ) < d(ω0 ) < d(ω+ ). Therefore, since d(ω0 ) = n1 T(φω0 ) = n1 T(u0 ) + O(δ), if we take δ small enough, we obtain d(ω− ) < n1 T(u(t)) < d(ω+ ). From Lemma 4.5.5, to conclude the proof, it is sufficient to show that Sω± (u0 ) < d(ω± ). Write Sω± (u0 ) = Sω± (φω0 ) + O(δ) ε = Sωω (φω0 ) ± |φω0 |22 + O(δ) 0 2 = d(ω0 ) + (ω± − ω0 )d󸀠 (ω0 ) + O(δ). Here we have used the definition of d(ω) and Lemma 4.5.4 (3). On the other hand, the Taylor expansion at ω0 gives 1 d(ω± ) = d(ω0 ) + (ω± − ω0 )d󸀠 (ω0 ) + (ω± − ω0 )2 d󸀠󸀠 (ω1 ), 2 where ω1 is a number between ω0 and ω± . By the assumption d󸀠󸀠 (ω0 ) > 0, if ε is sufficiently small, we have d󸀠󸀠 (ω1 ) > 0. Hence, if we take δ small enough, we have Sω± (u0 ) < d(ω± ). Lemma 4.5.7. Suppose Assumption (H) holds. Let {vk } ⊂ H 1 (ℝn ) be a sequence satisfying 1 T(vk ) → d(ω), n Sω T(vk ) → Sω (φω ),

(4.5.51) (4.5.52) (4.5.53)

|vk |2 → |φω |2 .

Then there exists {yk } ⊂ H 1 (ℝn ) such that {τyk vk } has a subsequence {τy 󸀠 vk󸀠 } satisfying k

τy 󸀠 vk󸀠 → φ in H 1 (ℝn ) for some φ ∈ Gω . k

Proof. By assumptions (4.5.51)–(4.5.53), in the same way as in the proofs of Theorems 3.1 and 3.2 in [14], it can be shown that there exists {yk } ⊂ H 1 (ℝn ) such that τy 󸀠 vk󸀠 → φ k

in H 1 (ℝn ),

(4.5.54)

where φ is a minimizer of the minimizing problem in Lemma 4.5.4 (2). When the space dimension is n = 3, the set of minimizers of the minimizing problem in Lemma 4.5.4 (2) is equal to Gω ; see [14, Theorem 3.2]. This concludes the proof in the case of n = 3.

4.5 The generalized Davey–Stewartson system

| 365

When n = 2, for φ in (4.5.54) there exists λ > 0 such that φλ ∈ Gω (see the proof of Theorem 3.1, step 2, [14], p. 983), where φλ = φ(x/λ). From (4.5.53)–(4.5.54) and the definition of φλ , we have 󵄨󵄨 λ 󵄨󵄨 󵄨󵄨φ 󵄨󵄨2 = λ|φ|2 = λ|φω |2 .

(4.5.55)

On the other hand, from (4.5.45) in Assumption (H), we have |φ|2 = |φω |2 .

(4.5.56)

We remark that this is the only place where assumption (4.5.45) is needed. From (4.5.55) and (4.5.56), we have λ = 1. Hence φ ∈ Gω . Then the proof is completed. Proof of Theorem 4.5.5. We prove Theorem 4.5.5 by contradiction. If φω0 is not stable, there exists a sequence of initial data uk (0) and ε0 > 0 such that uk (0) → φω0 in H 1 (ℝn ), 󵄩 󵄩 sup inf 󵄩󵄩󵄩uk (t) − φ󵄩󵄩󵄩H 1 ≥ ε0 , φ∈G 0≤t 0 so that 󵄩 󵄩 inf 󵄩󵄩󵄩uk (tk ) − φ󵄩󵄩󵄩H 1 = ε0 ,

φ∈Gω0

(4.5.57)

and the solution uk existing at least in the time interval [0, tk ]. By conservation laws |u(t)|2 = |u0 |2 and ε(u(t)) = ε(u0 ), Sω0 (uk (tk )) = Sω0 (uk (0)) → Sω0 (φω0 ), 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨uk (tk )󵄨󵄨󵄨2 = 󵄨󵄨󵄨uk (0)󵄨󵄨󵄨2 → |φω0 |2 . Moreover, from Lemma 4.5.6, we have 1 T(uk (tk )) → d(ω0 ). n Therefore, by Lemma 4.5.6, there exist {yk󸀠 } ⊂ ℝn and its subsequence (we also denote it {τyk uk (tk )}) such that τyk uk (tk ) → φ0

in H 1 (ℝn ) for some φω0 ∈ Gω0 .

This contradicts (4.5.57). Hence φω0 is stable.

366 | 4 Existence and stability of solitary waves

4.6 Nonlinear Schrödinger–Kadomtsev–Petviashvili equations We consider the coupled Schrödinger–Kadomtsev–Petviashvili equations: iεt + Δε − βnε = 0, { { { nt + nxxx + np nx − my = −α(|ε|2 )x , { { { 2 {mx = ny , (x, y) ∈ ℝ , t > 0,

(4.6.1)

where α, β are constants. We consider the following solitary wave of (4.6.1): n = n(x − ct, y), { { { m = m(x − ct, y), { { { iωt (i/2)c(x−ct) φ(x − ct, y), {ε = e e

(4.6.2)

where c > 0 and ω ∈ ℝ. We will prove that if p ≥ 4, β ≠ 0 and α, β, ω satisfy αβω < 0,

αβ[(p − 4)β + 3(p + 2)] ≥ 0, αβ[(p − 4)β + 7(p + 2)] ≥ 0,

then (4.6.1) admits no solitary waves. For the special case p = 1, ω > 0, β = −2α and −α > 0, system (4.6.1) admits at least one solitary wave. We first prove the nonexistence of solitary waves. Subtract (4.6.2) from (4.6.1), and then the triple (n, m, φ) satisfies −ωφ + Δφ − βnφ = 0, { { { −cnx + nxxx + np nx − my = −α(|φ|2 )x , { { { {mx = ny ,

(4.6.3)

or let n = 𝜕x ψ and then we have −ωφ + Δφ − βψx φ = 0, { −cψxx + ψxxxx + (ψx )p ψxx − ψyy = −α(|φ|2 )x .

(4.6.4)

Theorem 4.6.1. If p ≥ 4, β ≠ 0 and α, β, ω satisfy αβω < 0,

αβ[(p − 4)β + 3(p + 2)] ≥ 0, αβ[(p − 4)β + 7(p + 2)] ≥ 0,

(4.6.5)

then (4.6.1) admits no solitary waves. Proof. We use the Pohozaev-type identity to prove the claim. Multiplying the second equation of (4.6.3) by xn and ym, respectively, and integrating on ℝ2 , we get 1 3 1 − c ∫ n2 − ∫(nx )2 + ∫ np+2 2 2 p+2 1 + α ∫(xn)x |φ|2 + ∫ m2 = 0, 2

(4.6.6)

4.6 Nonlinear Schrödinger–Kadomtsev–Petviashvili equations | 367

and 1 1 1 1 − c ∫ n2 − ∫ m2 + ∫(nx )2 − ∫ np+2 2 2 2 (p + 1)(p + 2) + α ∫(yn)y |φ|2 − α ∫ n|φ|2 = 0.

(4.6.7)

Testing the second equation of (4.6.4) by ψ, and noting that n = 𝜕x ψ, m = 𝜕y ψ, we have −c ∫ n2 − ∫ m2 − ∫(nx )2 +

1 ∫ np+2 + α ∫ n|φ|2 = 0. p+1

(4.6.8)

Subtracting (4.6.6) from (4.6.7) gives c ∫ n2 − ∫ m2 + 2 ∫(nx )2 −

1 ∫ np+2 (p + 1)(p + 2)

− α ∫[(xn)x − (yn)y ]|φ|2 − α ∫ n|φ|2 = 0.

(4.6.9)

Adding (4.6.8) to (4.6.7) yields −2 ∫ m2 + ∫(nx )2 − α ∫[(xn)x − (yn)y ]|φ|2 = 0.

(4.6.10)

Substituting (4.6.10) into (4.6.8) and (4.6.6), respectively, we obtain −c ∫ n2 − 3 ∫ m2 +

1 ∫ np+2 p+1

− α ∫[(xn)x − (yn)y ]|φ|2 + α ∫ n|φ|2 = 0

(4.6.11)

and 5 1 1 − c ∫ n2 − ∫ m2 + ∫ np+2 2 2 p+2 1 3 + α ∫(yn)y |φ|2 − α ∫(xn)x |φ|2 = 0. 2 2

(4.6.12)

Testing the first equation of (4.6.4) by xφx and yφy , respectively, one gets β ∫(xn)x |φ|2 = −ω ∫ |φ|2 + ∫ |φx |2 − ∫ |φy |2 ,

(4.6.13)

β ∫(yn)y |φ|2 = −ω ∫ |φ|2 − ∫ |φx |2 + ∫ |φy |2 .

(4.6.14)

Substituting (4.6.1) and (4.6.14) into (4.6.11) and (4.6.12), respectively, we get −c ∫ n2 − 3 ∫ m2 +

1 ∫ np+2 p+1

α − 2 ∫[|φx |2 − |φy |2 ] + α ∫ n|φ|2 = 0 β

(4.6.15)

368 | 4 Existence and stability of solitary waves and

1 5 1 − c ∫ n2 − ∫ m2 + ∫ np+2 2 2 p+2 α α − ω ∫ |φ|2 − 2 ∫[|φx |2 − |φy |2 ] = 0. β β

(4.6.16)

Noting −β ∫ n|φ|2 = ω ∫ |φ|2 + ∫ |∇φ|2 , multiplying (4.6.16) by

5 6

and then subtracting (4.6.15), we finally get

2 p−4 c ∫ n2 + ∫ np+2 5 5(p + 1)(p + 2) α 3α 7α − ω ∫ |φ|2 + ∫ |φx |2 + ∫ |φy |2 = 0. 5β 5β 5β

(4.6.17)

On the other hand, it follows from (4.6.8) that 1 ∫ np+2 = c ∫ n2 + ∫ m2 + ∫(nx )2 − α ∫ n|φ|2 . p+1 Combining this with (4.6.17) we have 2 p−4 6αω c ∫ n2 + [c ∫ n2 + ∫ m2 + ∫(nx )2 ] − ∫ |φ|2 5 5(p + 2) 5β(p + 2) α (p − 4)β + 3(p + 2) α (p − 4)β + 7(p + 2) + ∫ |φx |2 + ∫ |φy |2 = 0, β 5(p + 2) β 5β(p + 2) which completes the proof. We discuss the existence of solitary waves in a special case, i. e., ω > 0, β = −2α and p = 1. Still denote −α by α. The equations read as follows: iεt + Δε − 2αnε = 0, { { { nt + nxxx + nnx − my = α(|ε|2 )x , { { { {mx = ny ,

(4.6.18)

where we assume α > 0. If (n, m, φ) is the solitary wave of equations (4.6.18), then we are concerned with the existence of solution to the following equations: −ωφ + Δφ − 2αnφ = 0, { { { −cnx + nxxx + nnx − my = α(|φ|2 )x , { { { {mx = ny ,

(4.6.19)

or letting n = 𝜕x ψ we have −ωφ + Δφ − 2αψx φ = 0,

{

−cψxx + ψxxxx + ψx ψxx − ψyy = α(|φ|2 )x .

(4.6.20)

4.6 Nonlinear Schrödinger–Kadomtsev–Petviashvili equations | 369

Theorem 4.6.2. Equation (4.6.19) admits one non-trivial solution in Y × H 1 (ℝ2 ). It is clear that (4.6.19) is the critical point equation of the functional S(n, φ) =

ω 1 1 ∫ |∇φ|2 + ∫ |φ|2 + ‖n‖2Y + ∫ n3 − α ∫ n|φ|2 , 2 2 3

where ‖n‖2Y = ‖∇c ψ‖2L2 + ‖𝜕xx ψ‖2L2 , |∇c ψ|2 = c|𝜕x ψ|2 + |𝜕y ψ|2 . Without loss of generality, we let c = 1 in the following. Consider the following minimization problem with constraints: Iλ =

inf T(n, φ),

(4.6.21)

(n,φ)∈Σλ

Σλ = {(n, φ) ∈ Y × H 1 : n ≠ 0, φ ≠ 0, V(n, φ) = λ},

(4.6.22)

where ω 1 ∫ |∇φ|2 + ∫ |φ|2 + ‖n‖2Y , 2 2 1 3 V(n, φ) = ∫ n − α ∫ n|φ|2 . 3 T(n, φ) =

Theorem 4.6.2 is a consequence of the existence of a minimizer for problem (4.6.21). In fact, if (n, φ) is a minimizer, then there exists a Lagrange multiplier θ such that −ωφ + Δφ − 2αθnφ = 0, { { { −cnx + nxxx + θnx nxx − myy = αθ(|φ|2 )x , { { { {mx = ny , which implies that ñ = (sgn θ)|θ|n,

m̃ = (sgn θ)|θ|m,

φ̃ = (sgn θ)|θ|φ

is a solution of (4.6.19). In what follows, we prove the existence of solution to the minimization problem (4.6.21) by proving several lemmas. Lemma 4.6.1. Iλ > 0 for any λ > 0. Proof. It follows from the anisotropic imbedding theorem that ‖n‖Lq ≤ C‖n‖Y ,

for any n ∈ Y and 2 ≤ q ≤ 6.

Then we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 3 󵄨 3 󵄨󵄨∫ n dxdy󵄨󵄨󵄨 ≤ C‖n‖Y 󵄨󵄨 󵄨󵄨

for any n ∈ Y.

370 | 4 Existence and stability of solitary waves On the other hand, φ ∈ H 1 (ℝ2 ) implies that ∫ |φ|q dxdy ≤ C‖∇φ‖qL2 ,

for any q < ∞.

Therefore, there is some s > 0 such that for any λ > 0 we have λ=

1 ∫ n3 − α ∫ n(|φ|2 ) ≤ C1 ‖n‖3Y + C2 ‖∇φ‖sL2 . 3

Hence, 0 < λ ≤ C1 Iλ3 + C1 Iλs . This completes the proof of Lemma 4.6.1. Lemma 4.6.2. Σλ ≠ 0 for any λ > 0. Lemma 4.6.3 (Strict subadditivity). We have Iλ < Iλ−α + Iα for any α ∈ (0, λ). Proof. Let nλ = λ1/3 n, φλ = λ1/3 φ. Then (n, φ) ∈ Σ1

⇐⇒

(nλ , φλ ) ∈ Σλ

and Iλ = λ2/3 I1 . The conclusion of Lemma 4.6.3 follows. Lemma 4.6.4 ([18]). Let q be such that 2 ≤ q < ∞. Then there exists a constant C > 0 such that for all f ∈ L1loc (ℝ2 ) with ∇f ∈ L1loc (ℝ2 ) for all R > 0 and for all X0 = (x0 , y0 ) ∈ ℝ2 , one has 1/q

(

󵄨󵄨 󵄨q 󵄨󵄨f (x) − mR (f )󵄨󵄨󵄨 dxdy)

∫

R≤|X−X0 |≤2R

≤ CR2/q (

1/2

∫

|∇f |2 dxdy) ,

R≤|X−X0 |≤2R

where mR (f ) = ∫R≤|X−X

0 |≤2R

f (x)dxdy, X = (x, y) ∈ ℝ2 .

Proof of Theorem 4.6.2. Now we are in a position to prove Theorem 4.6.2 by the concentration–compactness principle. Let (nk , φk ) be the minimizing sequence of problem Iλ for λ > 0, and set ρk = |nk |2 + |mk |2 + |𝜕x nk |2 + 21 ω|φk |2 + 21 |∇φk |2 , where nk = 𝜕x ψk , mk = 𝜕y ψk . We have 1 1 ∫ ρk = ∫(|nk |2 + |mk |2 + |𝜕x nk |2 + ω|φk |2 + |∇φk |2 ) 2 2 = T(nk , φk ) → Iλ .

4.6 Nonlinear Schrödinger–Kadomtsev–Petviashvili equations | 371

(i) If “vanishing” takes place, that is, lim sup ∫ ρk dxdy = 0,

k→∞ X∈ℝ2

∀R < ∞,

X+BR

where BR = BR (0) is a circle with radius R centered at 0, then we are led to a contradiction. In fact, by the estimate 1/2

∫ |n|3 ≤ 3C(sup ∫ (|ψx |2 + |ψy |2 + |ψxx |2 )dxdy) ‖ψx ‖2Y ℝ2

X∈ℝ2

X+B1

1/2

= 3C sup ( ∫ (|n|2 + |m|2 + |𝜕n|2 )) ‖n‖2Y , X∈ℝ2

X+B1

1/3 󵄨󵄨 󵄨 󵄨󵄨 2 󵄨󵄨 3 2 󵄨󵄨 ∫ n|φ| 󵄨󵄨󵄨 ≤ C(∫ |n| ) ‖φ‖L3 . 󵄨󵄨 󵄨󵄨 2 2 ℝ

ℝ

Therefore, nk → 0 strongly in L3 , nk |φk |2 → 0 strongly in L1 . The latter contradicts the fact that (nk , φk ) ∈ Σλ and λ > 0 and so we have ruled out the “vanishing.” (ii) Next we rule out the “dichotomy.” Arguing by contradiction, we may assume lim Q(t) = γ ∈ (0, Iλ ),

t→∞

(4.6.23)

where Q(t) = lim sup

t→∞ X ∈ℝ2 0

∫ ρk dxdy.

(4.6.24)

X0 +Bt

and (nk , φk ) is the minimizing sequence. In order to rule out the “dichotomy”, we need the following lemma, which will be proved in the end. Lemma 4.6.5. Assume that (4.6.24) holds. Then, for any ε > 0, there exists δ(ε) → 0 (as ε → 0) such that we can find n1k , n2k , φ1k , φ2k satisfying 󵄩󵄩 󵄩 1 2 󵄩 1 2 󵄩 󵄩󵄩φk − (φk + φk )󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩nk − (nk + nk )󵄩󵄩󵄩Y ≤ δ(ε), 󵄨󵄨 󵄨 1 1 󵄨󵄨T(nk , φk ) − γ 󵄨󵄨󵄨 ≤ δ(ε), 󵄨󵄨 󵄨 2 2 󵄨󵄨T(nk , φk ) − (Iλ − γ)󵄨󵄨󵄨 ≤ δ(ε), 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨∫((n1 )3 + (n2 )3 − (nk )3 ) − ∫(n1 󵄨󵄨󵄨φ1 󵄨󵄨󵄨2 + n2 󵄨󵄨󵄨φ2 󵄨󵄨󵄨2 − nk |φk |2 )󵄨󵄨󵄨 ≤ δ(ε), k k k󵄨 k󵄨 k󵄨 k󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 1 2 1 2 dist(supp nk , supp nk ) → ∞, dist(supp φk , supp φk ) → ∞.

(4.6.25) (4.6.26) (4.6.27) (4.6.28) (4.6.29)

372 | 4 Existence and stability of solitary waves Taking subsequence if necessary, we may assume V(n1k , φ1k ) → λ1 (ε), V(n2k , φ2k ) → λ2 (ε) as k → ∞. Then 󵄨 󵄨󵄨 󵄨󵄨λ − (λ1 (ε) + λ2 (ε))󵄨󵄨󵄨 ≤ δ(ε) → 0,

as ε → 0.

We distinguish two cases. Case 1, λ1 (ε) → 0 as ε → 0. Choosing ε small enough, we have V(n2k , φ2k ) > 0. Scaling as follows: (ñ 2k , φ̃ 2k ) = ((

1/3

λ2 (ε) ) V(n2k , φ2k )

n2k , (

1/3

λ2 (ε) ) V(n2k , φ2k )

φ2k ),

and since λ2 (ε) → 1, V(n2k , φ2k )

as ε → 0,

we get Iλ2 (ε) ≤ lim inf T(n2k , φ2k ) ≤ Iλ − γ + δ(ε). k→∞

This leads to a contradiction when sending ε → 0, since λ2 (ε) → λ as ε → 0. Case 2, limε→0 |λ1 (ε)| > 0 and limε→0 |λ2 (ε)| > 0. In the same way, we obtain I|λ1 (ε)| + I|λ2 (ε)| ≤ lim inf(T(n1k , φ1k ) + T(n2k , φ2k )) k→∞

≤ Iλ + 2δ(ε), which implies Is + Iλ−s ≤ Iλ ,

for some s ∈ (0, λ).

This violates the strict subadditivity. (iii) The only possibility is that ρk is tight, i. e., there exists a sequence Xk ∈ ℝ2 , such that for all ε > 0, there is a finite number R > 0 and k0 > 0, such that ∫ (|nk |2 + |mk |2 + |𝜕x nk |2 + Xk +BR

ω 1 |φk |2 + |∇φk |2 ) ≥ Iλ − ε. 2 2

(4.6.30)

Since (nk , φk ) is bounded in Y ×H 1 , we may assume that (nk (⋅−Xk ), φk (⋅−Xk )) converges weakly in Y × H 1 to (n, φ) ∈ Y × H 1 . We prove that nk (⋅ − Xk ) → n strongly in Lq+2 ,

∀q < 4,

(4.6.31)

4.6 Nonlinear Schrödinger–Kadomtsev–Petviashvili equations | 373

φk (⋅ − Xk ) → φ strongly in Lq ,

∀q < ∞.

(4.6.32)

We only need to prove (4.6.31), because (4.6.32) can be proved in a similar manner. It follows from (4.6.29) that for all k ≥ k0 , ∫ |nk |2 ≥ ∫ |nk |2 − 2ε, Xk +BR

ℝ2

and hence ∫ |n|2 dxdy ≤ lim inf ∫ |nk |2 dxdy + 2ε. k→∞

ℝ2

Xk +BR

On the other hand, it follows from de Bouard and Saut result (see [18, Lemma 3.3]) that Y 󳨅→ L2loc is compact. Then we may take it for granted that nk → n strongly in L2loc . These two claims yield that nk (⋅ − Xk ) → n strongly in L2 (ℝ2 ). From interpolation and Y ⊂ Lq+2 (∀q ≤ 4), we see that (4.6.31) holds. Therefore V(nk , φk ) → V(n, φ) = λ, and (n, φ) is a minimizer of Iλ . Proof of Lemma 4.6.5. Assume (4.6.23) holds. We can find R0 > 0, Rk ≥ R0 with Rk → +∞ and Xk ∈ ℝ2 such that, for k ≥ k0 , γ−ε ≤

ρk dxdy ≤ γ

∫

and Qk (2Rk ) ≤ γ + ε,

Xk +BR0

where Qk (t) = sup ∫ ρk dxdy. x0 ∈ℝ2

Xk +Bt

It then follows that ρk dxdy ≤ 2ε.

∫ R0 ≤|X−X0 |≤2Rk

Now define n1k , n2k , m1k , m2k , φ1k and φ2k as follows: Choose ξ , η, ζ , σ ∈ C0∞ (ℝ2 ) such that 0 ≤ ξ , η, ζ , σ ≤ 1 and ξ = 1, on B1 ,

supp ξ ⊂ B2 ,

ζ = 1, on B1 ,

supp ζ ⊂ B2 ,

η = 1, on

Bc2 ,

supp η ⊂ Bc1 ,

(4.6.33)

374 | 4 Existence and stability of solitary waves σ = 1, on Bc2 ,

supp σ ⊂ Bc1 .

Let ξk = ξ ( ζk = ξ (

⋅ − Xk ), R1

ηk = η(

⋅ − Xk ), R1

σk = σ(

⋅ − Xk ), Rk

⋅ − Xk ). Rk

Set n1k = 𝜕x (ξk (ψk − ak )),

m1k = 𝜕y (ξk (ψk − ak )), φ1k = ξk φk ,

n2k = 𝜕x (ηk (ψk − bk )),

m2k = 𝜕y (ηk (ψk − bk )),

φ2k = σk φk ,

where ak and bk are real numbers to be chosen later. Now we prove (4.6.25)–(4.6.28). It follows from the definition that 󵄩󵄩 󵄩 󵄩 1 2 󵄩 󵄩󵄩nk − (nk + nk )󵄩󵄩󵄩L2 = 󵄩󵄩󵄩(1 − ξk − ηk )nk − (ψk − ak )𝜕x ξk − (ψk − bk )𝜕x ηk 󵄩󵄩󵄩L2 1/2

≤(

|nk |2 )

∫

R0 ≤|X−Xk |≤2Rk

󵄩 󵄩 + 󵄩󵄩󵄩(ψk − ak )𝜕x ξk + (ψk − bk )𝜕x ηk 󵄩󵄩󵄩L2

󵄩 󵄩 󵄩 󵄩 ≤ √2ε + 󵄩󵄩󵄩(ψk − ak )𝜕x ξk 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩(ψk − bk )𝜕x ηk 󵄩󵄩󵄩L2 . However, 󵄩󵄩 󵄩 󵄩󵄩(ψk − ak )𝜕x ξk 󵄩󵄩󵄩L2 ≤ ‖𝜕x ξ ‖Lp ‖ψk − ak ‖Lq (R1 ≤|X−Xk |≤2Rk ) where get

1 p

+ q1 = 21 . Let ak = ∫R ≤|X−X 1

k |≤2R1

ψk dxdy, apply Lemma 4.6.4 and note (4.6.33) to

󵄩󵄩 󵄩 󵄩󵄩(ψk − ak )𝜕x ξk 󵄩󵄩󵄩L2 ≤ C √ε. Choosing bk = ∫R

k ≤|X−Xk |≤2Rk

(4.6.34)

ψk dxdy, we have 󵄩󵄩 󵄩 󵄩󵄩(ψk − bk )𝜕x ηk 󵄩󵄩󵄩L2 ≤ C √ε.

Therefore, 󵄩󵄩 1 2 󵄩 󵄩󵄩nk − (nk + nk )󵄩󵄩󵄩L2 ≤ C √ε. Similarly, we can prove 󵄩󵄩 1 2 󵄩 󵄩󵄩nk − (nk + nk )󵄩󵄩󵄩Y ≤ C √ε.

4.7 Asymptotic stability of solitary waves for BBM equation

| 375

The inequality 󵄩󵄩 1 2 󵄩 󵄩󵄩φk − (φk + φk )󵄩󵄩󵄩H 1 ≤ C √ε follows by the same method as above with much simpler computations. This completes the proof of (4.6.25). Inequalities (4.6.26) and (4.6.27) follow from the definition of “dichotomy.” Now we prove (4.6.28). The inequality 󵄨󵄨 󵄨 󵄨󵄨 3 1 3 2 3 󵄨󵄨 󵄨󵄨 ∫ ((nk ) − (nk ) − (nk ) )󵄨󵄨󵄨 ≤ δ(ε), 󵄨󵄨 󵄨󵄨 2 ℝ

follows from de Bouard and Saut [18]. For the other term, we have ∫ (nk |φk |2 − n1k |φ1k |2 + n2k |φ2k |2 ) ℝ2

= ∫ (nk |φk |2 (1 − ξk ζk2 − ηk σk2 )) ℝ2

− ∫ ((ψk − ak )|φk |2 ζk2 𝜕x ξk + (ψk − bk )|φk |2 σk2 𝜕k ηk ) ℝ2

= I1 + I2 , where |I1 | ≤

∫ R0 ≤|X−Xk |≤2Rk

|nk ||φ|2 ≤ C‖φk ‖2L4

ρk ≤ δ(ε),

∫ R0 ≤|X−Xk |≤2Rk

and 1/2

|I1 | ≤ C‖φk ‖2L4 (∫ |ψk − ak |2 |𝜕x ξ |2 )

+ C‖φk ‖2L4 (∫ |ψk − bk |2 |𝜕x η|2 )

1/2

≤ δ(ε).

This completes the proof of Lemma 4.6.5.

4.7 Asymptotic stability of solitary waves for BBM equation The H 1 -Lyapunov stability of solitary waves for the following BBM equation: 1 (I − 𝜕x2 )𝜕t u + 𝜕x (u + u2 ) = 0, 2

(4.7.1)

376 | 4 Existence and stability of solitary waves has been discussed in [4] and [6], namely, for any ε > 0, there exists δ > 0 such that if 󵄩 󵄩 inf 󵄩󵄩󵄩u0 (⋅ + s) − uc (⋅)󵄩󵄩󵄩H 1 < δ, s

then for all t > 0 󵄩 󵄩 inf 󵄩󵄩󵄩u0 (⋅ + s, t) − uc (⋅)󵄩󵄩󵄩H 1 < ε, s

where u0 (x) = u(x, 0) and 1 c−1 uc (x) = 3(c − 1) sech2 ( √ x) 2 c

(4.7.2)

is a solitary wave of (4.7.1). Bona and Soyeur have refined the Lyapunov analysis in [7]. For BBM equation they deduced the existence of a real-valued C 1 mapping s(t), with 󵄩󵄩 󵄩 󵄩󵄩u(⋅, t) − uc (⋅ + s(t))󵄩󵄩󵄩L2 < k1 ε and 󵄨󵄨 󸀠 󵄨 󵄨󵄨s (t) + c󵄨󵄨󵄨 ≤ k2 ε, for some constants k1 and k2 . Thus, at each instant, the solution is close (in L2 ) to a solitary wave of speed nearly equal to c, the speed of the unperturbed wave. The results proved here give a more precise description of the solution’s behavior: If u(x, t) is a solution of BBM equation such that u(x, 0) is close to a solitary wave uc (x + γ), γ ∈ ℝ in a suitable sense, then u(x, t) → uc (x − c+ t + γ+ ),

t → ∞,

for some c+ near c and γ+ near γ. To prove asymptotic stability of BBM solitary waves, we choose an exponentially weighted H 1 -norm defined as follows for any a ≥ 0: 󵄩 󵄩 Ha1 = {v | eax ν(x) ∈ H 1 (ℝ)} with ‖v‖Ha1 = 󵄩󵄩󵄩eax ν󵄩󵄩󵄩H 1 . We also define L2a = {ν | eax ν(x) ∈ L2 (ℝ)

󵄩 󵄩 with ‖v‖L2a = 󵄩󵄩󵄩eax ν󵄩󵄩󵄩L2 }.

Solutions of the BBM equation satisfy the following global existence theorem with respect to these norms.

4.7 Asymptotic stability of solitary waves for BBM equation

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Theorem 4.7.1. Let a ∈ ℝ and suppose that u0 (x) ∈ H 2 (ℝ) ∩ Ha1 (ℝ). Then there exists a unique solution u(x, t) of BBM equation with u(x, 0) = u0 (x) such that the map 󵄩 󵄩 󵄩 󵄩 t 󳨃→ 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1 a is continuous for t ∈ [0, ∞). Furthermore, 𝜕t H [u] = 0 and 𝜕t N [u] = 0 for all t ≥ 0, where 1 2 2 1 2 N [u] = ∫[ u + 2

H [u] = ∫[ u +

1 3 u ]dx, 6 1 (𝜕 u)2 ]dx. 2 x

Using estimates in the weighted H 1 -norm, we can prove Theorem 4.7.2. Let uc (x − ct + γ) be a BBM solitary wave, where c > 1 and γ ∈ ℝ. Consider the initial-value problem for BBM equation with initial data u0 (x) = u(x, 0) = uc0 (x + γ0 ) + ν0 (x) ∈ H 2 ∩ Ha1 ,

(4.7.3)

where 0 < a < a∗ (c0 ) = √

c −1 2 √ 0 . c0 2 + √8c0 + 1

(1) There exist C > 0, ε > 0, c∗ > 1, and some b with −ac0 + a/(l − a2 ) < −b < 0, so that if c0 ∈ (1, c∗ ) and ‖ν0 ‖H 1 + ‖ν0 ‖Ha1 < ε, then for all t ≥ 0 we have 󵄩󵄩 󵄩 󵄩󵄩u(⋅, t) − uc+ (⋅ − c+ t + γ+ )󵄩󵄩󵄩H 1 ≤ Cε, 󵄩󵄩 󵄩󵄩 −bt 󵄩󵄩u(⋅ + c+ t − γ+ , t) − uc+ (⋅)󵄩󵄩Ha1 ≤ Cεe , for some c+ > 1, γ+ ∈ ℝ, with |c0 − c+ | < Cε, |γ0 − γ+ | < Cε. (2) The conclusion of (1) holds for all speeds c0 ≥ c+ , except possibly for an exceptional set of values that have no finite accumulation point. Our strategy for proving Theorem 4.7.2 is to seek a solution of the BBM equation in the form of a dominant modulated solitary wave plus a remainder: t

t

u(x, t) = uc(t) (x − ∫ c(s)ds + γ(t)) + ν(x − ∫ c(s)ds + γ(t), t). 0

(4.7.4)

0

Here the speed c(t) and phase γ(t) vary with time, and are unknown functions to be determined by our scheme. Note that if c(t) = c and γ(t) = γ are constant in time, then

378 | 4 Existence and stability of solitary waves t

the solitary-wave component uc(t) (x − ∫0 c(s)ds + γ(t)) reduces to uc (x − ct + γ), an exact solitary-wave solution of BBM. We will make an additional change of variable t

t

τ(t) = ∫ c(s)ds − γ(t),

y(x, t) = x − ∫ c(s)ds + γ(t).

0

(4.7.5)

0

Substitution of (4.7.4) into the BBM equation yields the following evolution equation: ̇ γ(t)). ̇ 𝜕t ν = Aν + F (ν; uc(t) , c(t),

(4.7.6)

Here, the dot denotes differentiation with respect to t, F is a nonlinear term, and A = (I − 𝜕y2 ) 𝜕y Lc0 , −1

Lc0 = −c0 𝜕y2 + c0 − 1 − uc0 ,

(4.7.7)

where c0 > 1 is a fixed speed. We also study the weighted perturbation w = eay ν, which satisfies the equation 𝜕t w = Aa w + G ,

(4.7.8)

Aa w = eay Ae−ay .

(4.7.9)

where G is nonlinear and

Our goal is to show that ‖ν(⋅, t)‖H 1 remains bounded and ‖w(⋅, t)‖H 1 decays to zero as t → ∞. A key ingredient in the proof is a detailed spectral analysis of the operators A and Aa , in the space L2 (ℝ). In particular, it must be shown that the linearized eigenvalue problem Ay = λy (as well as Ac y = λy) has no nonzero eigenvalues in the closed right half-plane. Thus, the only neutral eigenvalue allowable is λ = 0, with eigenmodes derived from symmetries of the equation. To establish this spectral property, we first show that Aa (and therefore A) has no eigenvalues λ in the closed right half-plane except for a generalized eigenvalue of multiplicity 2 at λ = 0. We use different arguments for the cases of small and large |λ|. For large values of |λ|, we use norm estimates on an operator C(λ), which relates the resolvent of Aa to the resolvent of its constant-coefficient part A∞ a . For small values of |λ|, we exploit the convergence of Evans functions for BBM equation to Evans functions for KdV equation. To avoid secular growth of w and ν as t → ∞, we require that Pw ≡ 0, where P denotes the projection onto the (two-dimensional) generalized null space of Aa . This requirement gives rise to a system of ordinary differential equations for c(t) and γ(t) called modulation equations, which are coupled to the infinite-dimensional evolution equation for w. The modulation equations determine the two parameters c(t) and γ(t) uniquely. The final step in the linear spectral analysis is to show that the operator Aa generates

4.7 Asymptotic stability of solitary waves for BBM equation

| 379

a semigroup with exponentially decaying H 1 -norm on the spectral complement of its zero eigenspace. That is, solutions of the linear equation 𝜕t w = Aa w with initial data in the range of Q = I − P decay exponentially to zero in H 1 as t → ∞. To prove this decay estimate, we use a criterion, which requires bounds on the operator norm of the resolvent (λI − Aa )−1 . In particular, we show Re(λ) > −b

󳨐⇒

󵄩󵄩 −1 󵄩 󵄩󵄩(λI − Aa Q) 󵄩󵄩󵄩 < M

(4.7.10)

for some M < ∞, −b < 0. We now consider the details of the spectral theory for the linear operators Aa and A that arise in the equations for w and ν. The linearized equation for a BBM solitarywave perturbation is 𝜕t ν = Aν,

(4.7.11)

where A = (I − 𝜕y2 )−1 𝜕y Lc0 . We expect decay of the weighted perturbation w = eay ν whose linearized equation is 𝜕t w = Aa w,

(4.7.12)

where Aa = eay Ae−ay . We note that the spectral theory of Aa in L2 (or H 1 ) is equivalent to that of A in L2a (or Ha1 ) because λI − Aa = eay (λI − A)e−ay . We have the following theorem: Theorem 4.7.3. Let 0 < a < γ. We define γ = √ c−1 . c (1) The essential spectrum of Aa , denoted by σess (Aa ), is a curve lying in the open left half-plane, with def

−w = max{Re z | z ∈ σess (Aa )} = −ac +

a < 0. 1 − a2

(4.7.13)

(2) There exists γ ∗ ∈ (0, 1) such that for all γ ∈ (0, γ ∗ ) the only eigenvalue λ of A with Re λ ≥ 0 is λ = 0. (3) Let γ ∗ be as in (2). For each γ ∈ (0, γ ∗ ) there exists ε(γ) > 0 such that the only eigenvalue λ of Aa , with Re λ ≥ −ε(γ) is λ = 0. (4) The statements (2) and (3) also hold for γ ∈ (γ ∗ , 1) with the possible exception of a discrete set of γ’s. The proof of Theorem 4.7.3 is handled in the next four subsections. To prove Theorem 4.7.3 (a), note that A = −(I − 𝜕y2 ) 𝜕y (uc ⋅) + A∞ , −1

380 | 4 Existence and stability of solitary waves where A∞ = (I − 𝜕y2 ) 𝜕y (−c𝜕y2 + c − 1). −1

(4.7.14)

Similarly, Aa = −eay (I − 𝜕y2 ) 𝜕y (e−ay uc ) + A∞ a , −1

where ay ∞ −ay A∞ . a =e A e

Because uc (y) decays exponentially to zero as |y| → ∞, A is a compact perturbation of A∞ . By Weyl’s theorem for non-self-adjoint operators, the essential spectra of A and ∞ ∞ A∞ are therefore identical, as are those of A∞ a and Aa . The essential spectrum of Aa ∞ ∞ or Aa is the imaginary axis. Applying Fourier transform to Aa shows that the essential spectrum of Aa is 󵄨󵄨 (−ik − a)(−c(−ik − a)2 + c − 1) 󵄨 , k ∈ ℝ}, σess (Aa ) = {z ∈ ℂ 󵄨󵄨󵄨 z = 󵄨󵄨 1 − (−ik − a)2

(4.7.15)

hence (1) is proved. To prove (2), (3) and (4), the eigenvalues λ of Aa are studied by considering separately different regimes in the complex λ-plane. The two regimes we consider are: Case (1), large λ, i. e., |λ| ≥ Mγ 3 . Here M is a positive constant. For large λ, we expect the resolvent (λI − Aa )−1 to behave like the constant−1 coefficient resolvent (λI − A∞ a ) . We therefore rewrite the eigenvalue problem Aa Y = λY in terms of an operator C(λ), defined by the relation −1

−1

(λI − Aa )−1 = (I − C(λ)) (λI − A∞ a ) ,

(4.7.16)

where C(λ)φ = (Da (−cD2a + c − 1) − λ(I − D2a ) Da uc φ), φ ∈ L2 (ℝ), −1

Da = 𝜕y − a.

(4.7.17)

Proposition 4.7.1. (1) The operator C(λ): L2 → L2 is compact for λ ∉ σess (Aa ). (Hence its spectrum consists solely of eigenvalues, which accumulate at 0.) In particular, C(λ) is compact for all λ with Re λ > −w. (2) For any λ ∈ ℂ \ σess (Aa ), we have that λ is an eigenvalue of Aa if and only if 1 lies in the spectrum of C(λ). (3) Let λ ∈ ℂ \ σess (Aa ). A sufficient condition for λ not to be an eigenvalue of Aa is that the operator C(λ) have norm ‖C(λ)‖L2 →L2 < 1.

4.7 Asymptotic stability of solitary waves for BBM equation

| 381

Proof. Part (2) clearly follows if (1) is assumed. Part (3) follows immediately from (1) and (2). It remains to prove (1). We can prove (1) by the properties of operator C ∗ (λ) : L2 → L2 which is the adjoint operator of C(λ). Proposition 4.7.2. Let δ ∈ (0, 1) be fixed. (1) For any ε1 ∈ (0, 21 ), and C ∗ > 1, there exists M > 0 so that if 1 ≤ c ≤ c∗ ,

Re λ ≥ −ac/4,

and |Im λ| > cMγ 3

or

Re λ > cMγ 3 ,

then ‖C(λ)‖L2 →L2 < 1 − δ, and therefore λ is not an eigenvalue of Aa . (2) The statement in (1) holds with ‖C(λ)‖L2 →L2 replaced by ‖C(λ)‖H 1 →H 1 . Corollary 4.7.1. Let c > 1, a ∈ (0, γ), and δ ∈ (0, 1) be fixed. There exists M > 0 so that if Re λ ≥ 0 and |λ| > Mγ 3 , then λ is not an eigenvalue of A or of Aa . Case (2), small λ – Evans functions. Consider the ordinary differential system dY = B(y, λ, γ)y(y). dt

(4.7.18)

Here Y(s) : ℝ → ℂn is a vector-valued function, and B is an n × n matrix whose entries are functions of y, the complex parameter λ, and possibly the complex parameter γ. Suppose B(y, λ, γ) → B∞ (λ, γ) as y → ±∞. Let μj , 1 ≤ j ≤ n denote the eigenvalues ∞ of B , and vj the corresponding eigenvectors. Suppose that Re μ1 < Re μj ,

j = 2, 3, . . . , n.

(4.7.19)

It is a classical result that if B tends to B∞ , sufficiently quickly as y → ±∞, then (4.7.18) has a one-dimensional subspace of solutions Y(y) satisfying Y(y) = O(eμ1 y ) as y → +∞, and an (n − 1)-dimensional subspace of solutions Y(y) satisfying Y(y) = o(eμ1 y )

as y → −∞.

The angle between these subspaces is given by an analytic function D(λ, γ) known as the Evans function. This D(λ, γ) may be defined as a transmission coefficient: If Y+ is a solution of (4.7.18) that is normalized so that Y+ ∼ eμ1 y v1

as y → +∞,

382 | 4 Existence and stability of solitary waves then Y+ ∼ D(λ, γ)eμ1 y v1

as y → −∞,

the domain Δ of D(λ, γ) is the set of all (λ, γ) such that condition (4.7.19) holds. Thus, if (λ, γ) ∈ Δ and D(λ, γ) = 0, then (4.7.18) has a solution Y(y) satisfying Y+ = O(eμ1 y ) +

Y = o(e

μ1 y

as y → +∞,

) as y → −∞.

(4.7.20)

If in addition Re μ1 < 0 < Re μj ,

j = 2, 3, . . . , n,

then Y(y) satisfies (4.7.20) if and only if Y(y) decays exponentially to zero both as y → +∞ and as y → −∞. The following consequence of results gives conditions for the existence of an Evans function for a given system. Then we sketch the properties of the Evans function for the BBM equation, especially the relations with the eigenvalue λ and with DKdV (λ, γ) and DBBM (λ, γ). Theorem 4.7.4 ([84]). Let Δ be a simply connected subset of ℂ2 . Suppose that system (4.7.18) satisfies the following hypotheses: (i) B : ℝ × Δ → ℂn×n is continuous and analytic in (λ, γ) for each fixed y. (ii) B∞ (λ, γ) = limy→±∞ B(y, λ, γ) exists for (λ, γ) ∈ Δ, and the limit is attained uniformly on compact subsets of Δ. (iii) The integral +∞

󵄩 󵄩 ∫ 󵄩󵄩󵄩B(y, λ, γ) − B∞ (λ, γ)󵄩󵄩󵄩dy

−∞

converges for all (λ, γ) ∈ Δ, and the convergence is uniform on compact subsets of Δ. (iv) For every (λ, γ) ∈ Δ, the matrix B∞ (λ, γ) has a unique eigenvalue of smallest real part, which is simple. Then there exists an Evans function D(λ, γ), analytic on Δ, such that D(λ, γ) = 0 if and only if (4.7.18) has a solution Y(y) satisfying (4.7.20). The function D(λ, γ) is unique in the sense that if both D and D∗ are Evans functions for (4.7.18), then D/D∗ is analytic on Δ. Theorem 4.7.5 ([85]). (1) The Evans function DKdV (Λ) associated with equation 1 AKdV Y = 𝜕y (−𝜕y2 + 1 − 3 sech2 ( y))Y = ΛY 2 is given by DKdV (Λ) = (

2

μ1 (Λ) + 1 ), μ1 (Λ) − 1

4.7 Asymptotic stability of solitary waves for BBM equation

| 383

where μ1 (Λ) denotes the root of μ(−μ2 + 1) − Λ having the smallest (i. e., leftmost) real part. (2) The domain of DKdV (Λ) is the slit complex plane DKdV (Λ) = ℂ \ (−∞, −√

4 ]. 27

(3) The essential spectrum of AKdV : L2a → L2a is a curve contained entirely in the domain {λ | Re λ < −ε} for some ε > 0. Furthermore, if Δ 0 denotes the component of ℂσess (AKdV ) that contains the right half-plane, then DKdV has no zeros in Δ 0 except for a zero of multiplicity 2 at Λ = 0. By Theorem 4.7.4 and Theorem 4.7.5, we can prove Theorem 4.7.6. (1) For every γ ∈ (0, 1), there exists ε(γ) > 0 such that the Evans function DBBM (λ, γ) is defined and analytic on the half-plane {λ | Re λ > −ε}. (2) Furthermore, there exists −λ0 < 0 such that for each γ ∈ (0, 1), the Evans function DBBM (λ, γ) is defined and analytic on the slit half-plane ̃ {λ | Re λ > −λ0 } \ (−λ0 , −Ω(γ)), where ̃ = −γ 3 √4/27(1 + O(γ 2 )). −Ω (3) When Re λ ≥ 0, the following statements are equivalent: DBBM (λ, γ) = 0,

λ is an L2 eigenvalue of A,

λ is an L2 eigenvalue of Aa . In addition, the order of the zero equals the algebraic multiplicity of the eigenvalue. (4) When λ = 0, we have DBBM (0, γ) = 0 and 𝜕λ DBBM (0, γ) = 0, but 𝜕λ2 DBBM (0, γ) ≠ 0. It follows that λ = 0 is an eigenvalue of A of algebraic multiplicity 2. (5) When Re λ < 0 but λ lies to the right of σess (Aa ) in the complex plane, then DBBM (λ, γ) = 0 if and only if λ is an L2 eigenvalue of Aa . In this case, DBBM (λ, γ) = 0 no longer implies that λ is an L2 eigenvalue of A. The Evans function DBBM (λ, γ) associated with the BBM eigenvalue equation is given by 𝜕y (−c𝜕y2 + c − 1 − uc (y))Y(y) − λ(I − 𝜕y2 )Y(y) = 0.

(4.7.21)

Under the scalings λ = cγ 3 Λ,

y = ξ /γ,

(4.7.22)

384 | 4 Existence and stability of solitary waves the eigenvalue equation (4.7.21) becomes, after dropping the tildes, 𝜕ξ (−c𝜕ξ2 − 1 − g(ξ ))Y − ΛY + Λγ 2 𝜕ξ2 Y = 0,

(4.7.23)

where 1 g(ξ ) = 3 sech2 ( ξ ). 2 Notice that, when γ = 0, it becomes the KdV eigenvalue equation. Now we consider |λ| ≤ Mγ 3 . We associate to (4.7.23) an Evans function D∗ (Λ, γ), whose zeros in the closed right half-plane are L2 eigenvalues of (4.7.23). Moreover, D∗ and DBBM are related by D∗ (Λ, γ) = DBBM (λ, γ),

λ = cΛγ 3 .

It suffices to prove that the only zero of D∗ (Λ, γ) with |Λ| ≤ M/c is Λ = 0. As we shall see, D∗ (Λ, γ) is analytic in Λ, continuous in γ, and, for γ = 0, equal to DKdV (Λ). We use Rouché’s theorem to prove that for γ sufficiently small, D∗ (Λ, γ) like DKdV (Λ), has Λ = 0 as its only eigenvalue in the closed right half-plane. This strategy leads to a proof of parts (2) and (3) of Theorem 4.7.3; part (4) is proved by analytic continuation in γ. Proposition 4.7.3. There exists an Evans function D∗ (Λ, γ) for equation (4.7.23) so that the following properties hold: (1) For any M > 0, there exists ε > 0 such that D∗ (Λ, γ) is defined and analytic in Δ ∗ × [−1, 1], where Δ ∗ ⊂ ℂ is a domain containing the scaled slit half-plane 󵄨󵄨 ̃ −λ −Ω(γ) −λ 󵄨 ] {Λ 󵄨󵄨󵄨 Re Λ > 30 } \ [ 30 , 󵄨󵄨 cγ cγ cγ 3 ̃ are as in Theorem 4.7.6 (b), and where −λ0 and −Ω ̃ −Ω(γ) 4 = −√ (1 + O(γ 2 )). 27 cγ 3 (2) D∗ (Λ, γ) = DBBM (λ, γ) for λ = cΛγ 3 . (3) Assume Re Λ ≥ 0 and γ ∈ (0, 1). Then D∗ (Λ, γ) = 0 if (4.7.23) has a solution that decays exponentially as y → ∞, that is, if λ = cΛγ 3 is an L2 eigenvalue of A. Proof of Theorem 4.7.3 (2) and (3). By Proposition 4.7.3, we can choose M∗ > 0 and ε∗ > 0 so that the conditions 1 < c < 2, Re Λ > −ε, |Λ| > M∗

4.7 Asymptotic stability of solitary waves for BBM equation

|

385

imply that λ = cΛγ 3 is not an eigenvalue of Aa . We use the KdV scaling (4.7.22) to find γ∗ ∈ (0, 1), such that if 0 < γ ≤ γ∗ , Re Λ > −ε, and |Λ| ≤ M∗ , then D∗ (Λ, γ) = 0 implies that Λ = 0. Define Δ ε = {Λ | Re Λ ≥ −ε, |Λ| < M∗ },

0 ≤ ε ≤ ε∗ ,

and let 󵄨 󵄨󵄨 m = min{󵄨󵄨󵄨DKdV (Λ)󵄨󵄨󵄨 󵄨󵄨󵄨 Λ ∈ 𝜕Δ ε }. Theorem 4.7.5 states that DKdV (Λ) = D∗ (Λ, 0) has no zeros in except for a zero of multiplicity 2 at Λ = 0. It follows that m is strictly positive. Because D is continuous in γ, there is some γ∗ so small that the inequality 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨D∗ (Λ, γ) − D∗ (Λ, 0)󵄨󵄨󵄨 = 󵄨󵄨󵄨D∗ (Λ, γ) − DKdV (Λ)󵄨󵄨󵄨 < m, holds whenever 0 ≤ γ ≤ γ∗ , Λ ∈ 𝜕Δ ε . Rouche’s theorem implies that for each γ ∈ [0, γ∗ ), the functions D∗ (Λ, 0) and D∗ (Λ, γ) have the same number of zeros (counting multiplicities) inside Δ ε . Because D∗ (Λ, γ) = 0 if and only if DBBM (cΛγ 3 , γ) = 0, from Theorem 4.7.6 we know that for each γ > 0, D∗ (Λ, γ) has a zero of multiplicity 2 at Λ = 0. It follows that for each γ ∈ [0, γ∗ ), D∗ (Λ, γ) has no zeros contained in Δ ε , except for Λ = 0. The conclusion of (2) follows. Finally, let γ ∈ [0, γ∗ ) be arbitrary, and let −ω = max{Re z | z ∈ σess (Aa )} < 0. The conclusion of (3) now follows from Theorem 4.7.6 (b) (relating eigenvalues of A to those of Aa ) if ε(γ) > 0 is chosen so that −ε(γ) > max(−w, −ε∗ ). Proof of Theorem 4.7.3 (4). We only prove that the exceptional set E = {γ ∈ [0, 1) | there exists β ∈ ℝ with |β| ≤ M such that D∗ (iβ, γ) = 0} is a discrete set. Theorem 4.7.3 (4) states that λ = 0 is an eigenvalue of A and of Aa with multiplicity 2. We now find bases for the generalized zero eigenspace Kerg (Aa ) as well as for the adjoint eigenspace Kerg (A∗a ) where A∗a = e−ay A∗ eay is the adjoint of Aa . These bases are used to define projection operators associated with the space Kerg (Aa ). We will use the projection operators in proving the decay of

386 | 4 Existence and stability of solitary waves the weighted perturbation w, which solves the equation 𝜕t w = Aa w + G by requiring w to be orthogonal to Kerg (A∗a ). Differentiating equation (4.7.1) for the BBM solitary wave with respect to y and c, respectively, shows that Lc 𝜕y uc = 0

(hence A𝜕y uc = 0),

A𝜕c uc = −𝜕y uc

(hence A2 𝜕c uc = 0),

where A = −𝜕y (I − 𝜕y2 ) (−c − 𝜕y2 + c − 1 − uc ). −1

The generalized null spaces of A and A∞ a are displayed in the following: Proposition 4.7.4. Let 0 < a < √(c − 1)/c, and let ξ̃1 = 𝜕y uc ,

ξ̃2 = 𝜕c uc ,

y

̃ 1 = θ2 (I − 𝜕y2 )uc + θ1 [𝜕y 𝜕c uc − ∫ 𝜕c uc ], η −∞

̃ 2 = θ3 (I − 𝜕y2 )uc , η where θ1 = (

−1

d N [uc ]) , dc 2

−2

1 d d θ2 = ( ∮[uc ]) ( N [uc ]) , 2 dc dc θ3 = −θ1 , 1 2 2 N = ∫ uc + (𝜕y uc ) , 2 ∮ = ∫ uc . For i = 1, 2, let ξi = eay ξ̃i ,

̃i . ηi = e−ay η

(4.7.24)

Then {ξ1 , ξ2 } and {η1 , η2 } are biorthogonal bases for Kerg (Aa ) and Kerg (A∗a ). That Kerg (Aa ) = span{ξ1 , ξ2 } ⊂ L2 ,

Kerg (A∗a ) = span{η1 , η2 } ⊂ L2 , and ⟨ηi , ξj ⟩ = δij ,

i, j = 1, 2,

where ⟨⋅, ⋅⟩ denotes the L2 inner product. Also, A∗a η1 = 0 and A∗a η2 = η1 .

4.7 Asymptotic stability of solitary waves for BBM equation

| 387

Proof. One can check that 2

A∗ g1 (y) = 0,

(A∗ ) g2 (y) = 0,

where g1 (y) = (I − 𝜕y2 )uc , y

g2 (y) = ∫ 𝜕c uc (z)dz − 𝜕y 𝜕c uc (y). −∞

Furthermore, ⟨g1 , ξ̃1 ⟩ = 0, ⟨g2 , ξ̃1 ⟩ = −𝜕c N [uc ],

y

2

1 d ⟨g2 , ξ̃2 ⟩ = ∫[(𝜕c uc ) ∫ 𝜕c uc ] = ( J [uc ]) , 2 dc −∞

and ⟨g1 , ξ̃2 ⟩ = 𝜕c N [uc ]. ̃ i , ξ̃j ⟩ = δij where ξ̃1 , ξ̃2 are as in the statement of the proposition. The It follows that ⟨η proof may now be completed using the relations (4.7.24). We denote the projection onto the zero eigenspace of Aa by P, and its complement, I − P, by Q. Thus Pφ = ⟨φ, η1 ⟩ξ1 + ⟨φ, η2 ⟩ξ2 , φ = Qφ + Pφ,

for φ ∈ L2 (ℝ). Now we give semigroup and nonlinearity decay estimates. Proposition 4.7.5. Assume that 0 < a < √(c − 1)/c and that λ = 0 is the only eigenvalue of Aa in the closed right half-plane. Let Q denote the projection onto (Kerg (A∗a ))⊥ . Then the initial-value problem 𝜕t w = Aa w,

w|t=0 = w0 ∈ H 1 ∩ range Q has a unique solution w(t) = eAa t w0 ∈ C0 ([0, ∞), H 1 ), with 󵄩󵄩 󵄩 −bt 󵄩󵄩w(t)󵄩󵄩󵄩H 1 ≤ ce ‖w0 ‖H 1 ,

(4.7.25)

388 | 4 Existence and stability of solitary waves for some c > 0, b > 0. Equivalently, Aa generates a C0 -semigroup with exponentially decaying norm on the function space x = H 1 (ℝ) ∩ (Kerg (A∗a ))⊥ . Furthermore, the decay estimate (4.7.25) holds for any −b ∈ (−bmax , 0), where −bmax = inf{−b | λ = 0 is the only eigenvalue of Aa with Re λ ≥ −b ≥ −w}.

(4.7.26)

The proof of Proposition 4.7.5 involves estimates on the resolvent (λI − Aa )−1 . We use the result below, which follows immediately from a theorem of Priiss [86]. Theorem 4.7.7. Let B be the infinitesimal generator of a C0 -semigroup on a Hilbert space Z. Let b > 0. If there exists M > 0 so that 󵄩󵄩 −1 󵄩 󵄩󵄩(λI − Aa ) 󵄩󵄩󵄩Z→Z ≤ M,

Re λ > −b,

then 󵄩󵄩 Bt 󵄩󵄩 −bt 󵄩󵄩e 󵄩󵄩Z→Z ≤ e . We first prove nonlinearity decay estimates. Suppose u(x, t) is a solution of the BBM equation (I − 𝜕x2 )𝜕t u + 𝜕x (u +

u2 ) = 0. 2

If we make the substitutions u(x, t) = uc(t) (y(x, t)) + ν(y(x, t), t), t

(4.7.27)

y(x, t) = x − ∫ c(s)ds + γ(t), 0

and use the fact that each uc(t) is a solitary-wave solution of BBM equation, we obtain an equation for the perturbation ν, ̇ (I − 𝜕y2 )𝜕t ν = 𝜕y Lc(t) ν + F1 (ν, uc(t) , c,̇ γ),

(4.7.28)

where Lc(t) = (−c(t)𝜕y2 + c(t) − 1 + uc(t) ), 2

2

1 2

2

̇ c uc ) − γ(I ̇ y uc + c𝜕 ̇ − 𝜕y )𝜕y ν − 𝜕y (ν ). F1 = −(I − 𝜕y )(γ𝜕 Choosing t

τ = ∫ c(s)ds − γ(t), 0

(4.7.29)

4.7 Asymptotic stability of solitary waves for BBM equation

| 389

the equation for ν(y, τ) can be written as 𝜕τ ν =

1 Aν + F (ν), c0

F (ν) =

c − c0 − γ̇ 1 1 −1 ̇ c uc ) + (I − 𝜕y2 ) 𝜕y (uc ν − uc0 ν) ̇ y uc + c𝜕 (−1 + )(γ𝜕 c0 c − γ̇ c0 1 c − c0 − γ̇ 1 1 −1 −1 (I − 𝜕y2 ) 𝜕y (uc ν − ν) − (I − 𝜕y2 ) 𝜕y ( ν2 ), + c0 c − γ̇ c − γ̇ 2

where the dot still denotes differentiation with respect to t, and A = (I − 𝜕y2 ) 𝜕y Lc0 , −1

Lc0 = −c0 𝜕y2 + c0 − 1 − uc0 . The weighted perturbation w(y, τ) = eay ν(y, τ) satisfies 𝜕t w =

1 ̇ A w + G (w, ν, uc , c,̇ γ), c0 a

where ay

G =e F =

c − c0 − γ̇ ay 1 ̇ c uc ) + G̃ ̇ y uc + c𝜕 (−1 + )e (γ𝜕 c0 c − γ̇

G̃ = G1 + G2 + G3 ,

1 −1 (I − D2a ) Da (uc w − uc0 w), c0 1 c − c0 − γ̇ −1 G2 = (I − D2a ) Da (uc w − w), c0 c − γ̇

G1 =

G3 = −

(4.7.30)

1 1 −1 (I − D2a ) Da ( wν). c − γ̇ 2

Recall that P is the projection onto the (two-dimensional) zero eigenspace of Aa and Q = I − P. In order to have ‖w(⋅, t)‖H 1 → 0 as t → ∞, we impose 𝜕t w =

1 A w + QG , c0 a

P G = 0,

Pw(0) = 0.

(4.7.31)

The requirement that P G = 0 leads to a system of equations, called modulation equations, which prescribe the evolution of the parameters c(t) and γ(t) appearing in (4.7.27). The condition P G = 0 is, by Proposition 4.7.4, equivalent to the conditions ⟨ηi , G ⟩ = 0,

i = 1, 2,

(4.7.32)

390 | 4 Existence and stability of solitary waves where {η1 , η2 } is a basis for the adjoint null space Kerg (A∗a ). If we use the definitions of η1 , η2 from Proposition 4.7.4, we find that condition (4.7.32) holds if and only if γ̇ ċ

A (t) [ ] = c0 (1 −

c − c0 − γ̇ ⟨η1 , G̃⟩ )[ ], ⟨η2 , G̃⟩ c − γ̇

where A (t) = [

̃1 , 𝜕y uc ⟩ ⟨η ̃2 , 𝜕y uc ⟩ ⟨η

̃1 , 𝜕c uc ⟩ ⟨η ]. ̃2 , 𝜕c uc ⟩ ⟨η

Let e1 (y, t) = 𝜕y uc(t) (y) − 𝜕y uc0 (y) and e2 (y, t) = 𝜕c uc(t) (y) − 𝜕c uc0 (y). Then, with the use of the biorthogonality relations ⟨ηi , ξj ⟩ = δij , i, j = 1, 2, we have ̃1 , e1 ⟩ 1 + ⟨η ̃ ⟨η2 , e1 ⟩

A (t) = [

1 = [ 0

̃1 , e2 ⟩ ⟨η ] ̃2 , e2 ⟩ 1 + ⟨η

0 󵄨 󵄨 ] + O(󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨). 1

Therefore, whenever |c(t) − c0 | is small, the matrix A is invertible and the condition P G = 0 is equivalent to the modulation equations γ̇ ⟨η , G̃⟩ [ ] = B (t) [ 1 ̃ ] , ċ ⟨η2 , G ⟩

(4.7.33)

where ‖B (t)‖ ≤ C. By means of the implicit function theorem, we can prove the local existence of a decomposition u(x, t) → (ν(y, t), γ(t), c(t)) and continuation principles. Proposition 4.7.6. Let 0 < a < √(c0 − 1)/c0 . Let s be real, and t1 ≥ 0. Then there exist δ0 , δ1 > 0 such that the following holds: For any real γ0 , if u(x, t) is such that eax u ∈ C([0, t1 ], H s ),

(4.7.34)

with 󵄩 󵄩 sup 󵄩󵄩󵄩ea(⋅+γ0 ) (u(⋅, t) − uc0 (⋅ − c0 t + γ0 ))󵄩󵄩󵄩H s < δ0 ,

0≤t≤t1

then there exists a unique function t 󳨃→ (γ(t), c(t)), 2 {(γ, c) ∈ C([0, t1 ], ℝ ), { sup 󵄨󵄨󵄨󵄨γ(t) − γ0 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨󵄨 < δ1 , { 0≤t≤t1

such that +∞

ay

Fk [u, γ, c](t) = − ∫ [u(x, t) − uc(t) (y)]e ηk (y)ds = 0, −∞

(4.7.35)

4.7 Asymptotic stability of solitary waves for BBM equation

| 391

t

for k = 1, 2, 0 ≤ t ≤ t1 , where y = x − ∫0 c(s)ds + γ(t). The number δ0 may be chosen as a decreasing function of t1 . The map u 󳨃→ (γ, c), from the set defined in (4.7.34) to that defined in (4.7.35), is analytic; moreover, if eax u ∈ C m ([0, t1 ], H s ) for some integer m > 0, then (γ, c) ∈ C m ([0, t1 ), ℝ2 ). Proposition 4.7.7. There exist δ0 , δ1 > 0 such that for any t0 > 0, if eax u ∈ C([0, t0 ], H s ), 󵄩 󵄩 sup 󵄩󵄩󵄩eaν ν(⋅, t)󵄩󵄩󵄩H s ≤ δ0 /3,

0≤t≤t0

where t

ν(y, t) = u(x, t) − uc(t) (y),

y = x − ∫ c(s)ds + γ(t), 0

and if (γ, c) ∈ C([0, t0 ], ℝ2 ), F [u, γ, c](t) = 0,

󵄨 󵄨 sup 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 ≤ δ1 ,

0≤t≤t0

0 ≤ t ≤ t0 ,

then a unique extension of (γ, c) in C([0, t0 + t∗ ], ℝ2 ) exists for some t∗ > 0, with F [u, γ, c](t) = 0,

0 ≤ t ≤ t0 + t∗ .

Furthermore, if eax u ∈ C m ([0, ∞], H s ), then (γ, c) ∈ C m ([0, t0 + t∗ ], ℝ2 ). Proposition 4.7.8. Suppose u(y, t) is a solution of BBM equation with initial data u0 (x) = u(x, 0) = uc0 (⋅ + γ0 ) + ν0 (x) ∈ H 2 ∩ Ha1 .

(4.7.36)

Suppose also that a decomposition u(x, t) → (ν(y, t), c(t), (t)) exists on the interval 0 ≤ t ≤ T, with |c(t) − c0 | < M sufficiently small for t ∈ [0, T]. Let z = uc(t) − uc0 + ν. Then there exist constants C, D > 0 so that for 0 ≤ t ≤ T we have 󵄨 󵄨2 ‖ν‖2H 1 (1 − D‖ν‖H 1 ) ≤ C(|δε| + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + ‖w‖2L2 ), where δε = ε[u(⋅, t)] − ε[uc0 (y)] = ε[u(⋅, 0)] − ε[uc0 ].

392 | 4 Existence and stability of solitary waves Proof. Recall that Lc0 = −c0 𝜕y2 + c0 − 1 − uc0 . We have δε = ε[uc0 + z] − ε[uc0 ] 1 1 = − ∫ zLc0 z + ∫ z 3 2 6 1 = − ∫ νLc0 ν − ∫(uc(t) − uc0 )Lc0 ν 2 1 1 − ∫(uc(t) − uc0 )Lc0 (uc(t) − uc0 ) + ∫ z 3 . 2 6

(4.7.37)

We now estimate each of the quantities on the right-hand side of (4.7.37). First, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 −ay 󵄨󵄨− ∫(uc(t) − uc0 )Lc0 ν󵄨󵄨󵄨 = 󵄨󵄨󵄨∫[e Lc0 (uc(t) − uc0 )]w󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 ≤ C(|c(t) − c0 | + ‖w‖L2 ).

(4.7.38)

Also 1 󵄨 󵄨2 − ∫(uc(t) − uc0 )Lc0 (uc(t) − uc0 ) ≤ C(󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 ) 2

(4.7.39)

1 1 1 − ∫ νLc0 ν = − ∫[c0 νx2 + (c0 − 1)ν2 ] + ∫ uc0 ν2 . 2 2 2

(4.7.40)

and

Substituting (4.7.40) to (4.7.37) gives 1 ∫[c0 νx2 + (c0 − 1)ν2 ] = −δε − ∫(uc(t) − uc0 )Lc0 ν 2 1 − ∫(uc(t) − uc0 )Lc0 (uc(t) − uc0 ) 2 1 1 + ∫ z 3 + ∫ uc0 ν2 . 6 2

(4.7.41)

We have 1 1 ∫ uc0 ν2 = ∫ e−2ay uc0 w2 , 2 2 and therefore 󵄨󵄨 1 󵄨 󵄨󵄨 2 󵄨󵄨 2 󵄨󵄨 ∫ uc0 ν 󵄨󵄨󵄨 ≤ C(a)‖w‖L2 , 󵄨󵄨 2 󵄨󵄨 because e−2ay uc0 (y) ∈ L∞ . Applying inequalities (4.7.38), (4.7.39) and (4.7.41) to (4.7.42) gives 1 󵄨 󵄨2 ∫[c0 νx2 + (c0 − 1)ν2 ] ≤ |δε| + C(‖w‖2L2 + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 ) 2 󵄨󵄨 1 󵄨󵄨󵄨 󵄨 + C‖w‖2L2 + 󵄨󵄨󵄨 ∫ z 3 󵄨󵄨󵄨, 󵄨󵄨 6 󵄨󵄨

(4.7.42)

4.7 Asymptotic stability of solitary waves for BBM equation

| 393

and therefore, because ‖z‖L3 ≤ C‖z‖H 1 , 󵄨2 󵄨 ‖ν‖2H 1 (1 − D‖ν‖H 1 ) ≤ C(|δε| + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + ‖w‖2L2 ).

(4.7.43)

Proposition 4.7.9. Let 0 < a < a∗ (c0 ), T > 0, and let b ∈ (0, bmax ). Then there exist δ∗ > 0 and ε∗ > 0, so that if u(x, t) ∈ C([0, T], H 1 ∩ Ha1 ) is a solution to BBM equation, and if the following conditions hold: (i) a decomposition u(x, t) → (ν(y, t), c(t), (t)) exists for t ∈ [0, T]; (ii) for t ∈ [0, T] we have 󵄨󵄨 ̇ 󵄨󵄨 √δε + 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩 1 + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄨󵄨󵄨 c − c0 − γ 󵄨󵄨󵄨 + 󵄩󵄩󵄩ν(⋅, t)󵄩󵄩󵄩 1 ≤ δ∗ , 󵄩 󵄩H 󵄨 󵄨 󵄨󵄨 c − γ̇ 󵄨󵄨 󵄩 󵄩H 󵄨 󵄨

(4.7.44)

where δε = ε[u(y, 0)] − ε[uc0 (y)]; (iii) |c(0) − c0 | + √δε + ‖w(⋅, 0)‖H 1 ≤ ε∗ ; then there is C > 0 so that for t ∈ [0, T] we have 󵄨 ̇ 󵄨󵄨󵄨 󵄩 󵄩 󵄩 󵄨 󵄨 󵄨󵄨 c − c0 − γ(t) 󵄨󵄨 + 󵄩󵄩ν(⋅, t)󵄩󵄩󵄩 1 ≤ Cε∗ , ekbt 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩H 1 + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄨󵄨󵄨 󵄨󵄨 󵄩 󵄩H 󵄨󵄨 c − γ̇ 󵄨

(4.7.45)

with k = 1 + 2δ∗ /c0 . Proof. If δ∗ is sufficiently small, then the assumption 󵄨󵄨 1 󵄨󵄨󵄨 1 󵄨󵄨 − 󵄨󵄨󵄨 < δ∗ 󵄨󵄨 󵄨󵄨 c(t) − γ(t) ̇ c0 󵄨󵄨 t

implies that the change of variable t → τ = ∫0 c(s)ds − γ(t) is valid for t ∈ [0, T]. We use equations (4.7.31) for w(y, τ) to obtain the required estimates. We first estimate |c|̇ + |γ|̇ with the use of the modulation equations (4.7.33), γ̇ ⟨η , G̃⟩ [ ] = B (t) [ 1 ̃ ] , ċ ⟨η2 , G ⟩ ̇ where B (t) = I +O(|c(t)−c0 |+|γ(t)|). Choose δ so small that A (t) is invertible whenever (4.7.44) holds. Then 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨γ(t)󵄨󵄨 + 󵄨󵄨c(t)󵄨󵄨 ≤ c‖G̃‖L2 ≤ C(‖G1 ‖ + ‖G2 ‖ + ‖G3 ‖), where C depends on δ∗ . Directly from (4.7.30), the terms ‖Gj ‖ can be estimated as 󵄨 󵄨 ‖G1 ‖ ≤ C 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨‖w‖H 1 , 󵄨 󵄨 󵄨 ̇ 󵄨󵄨 ‖G2 ‖ ≤ C(󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄨󵄨󵄨γ(t) 󵄨󵄨)‖w‖H 1 , ‖G3 ‖ ≤ C‖ν‖H 1 ‖w‖H 1 .

(4.7.46)

394 | 4 Existence and stability of solitary waves It follows that 󵄨 󵄨 ̇ 󵄨󵄨 󵄨 ‖G̃‖L2 ≤ C(󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄨󵄨󵄨γ(t) 󵄨󵄨 + ‖ν‖H 1 )‖w‖H 1 . Therefore, whenever (4.7.44) holds, we have 󵄩 󵄩 󵄩 󵄨 󵄩 󵄨 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨γ(t)󵄨󵄨 + 󵄨󵄨c(t)󵄨󵄨 ≤ C(󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄩󵄩󵄩ν(⋅, t)󵄩󵄩󵄩H 1 )󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩H 1 ≤ Cδ∗2 .

(4.7.47)

To estimate ‖w(τ)‖H 1 , we first write system (4.7.31) as an equivalent integral equation: w(τ) = e

Aa τ/c0

τ

w(0) + ∫ eAa (τ−s)/c0 QG (s)ds.

(4.7.48)

0

If we apply the semigroup decay estimates of Proposition 4.7.5, we have e

τ

(b/c0 )τ 󵄩 󵄩

󵄩 󵄩 󵄩 󵄩 (b/c )s 󵄩 󵄩󵄩w(τ)󵄩󵄩󵄩H 1 ≤ C 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 + C ∫ e 0 󵄩󵄩󵄩QG (s)󵄩󵄩󵄩H 1 ds. 0

We next obtain bounds on ‖G (s)‖H 1 . By (4.7.30), 󵄨󵄨 c(t) − c − γ(t) ̇ 󵄨󵄨󵄨 󵄨 0 󵄨󵄨)(|γ|̇ + |c|) ̇ ‖G ‖ ≤ C(1 + 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 c(t) − γ(t) ̇ 󵄨 + C(‖G1 ‖ + ‖G2 ‖ + ‖G3 ‖).

(4.7.49)

Combining (4.7.49) with (4.7.44), (4.7.47) and (4.7.46) yields ‖G ‖ ≤ Cδ∗ ‖w‖H 1 . From the integral equation (4.7.48) for w, we then obtain, for 0 ≤ τ ≤ τ(T), τ

󵄩󵄩 󵄩 −(b/c0 )τ 󵄩 󵄩󵄩w(0)󵄩󵄩󵄩 1 + Cδ∗ ∫ e−(b/c0 )(τ−s) ‖w‖H 1 ds. 󵄩󵄩w(τ)󵄩󵄩󵄩H 1 ≤ Ce 󵄩 󵄩H

(4.7.50)

0

Choose b ∈ (b, bmax ) and note that (4.7.50) holds with b replaced by b󸀠 . Let 󸀠

󵄩 󵄩 󵄨 M∗ (T) = sup{e(b/c0 )τ 󵄩󵄩󵄩w(τ)󵄩󵄩󵄩H 1 󵄨󵄨󵄨 0 ≤ τ ≤ τ(T)}. Then for 0 ≤ τ ≤ τ(T) we have 󸀠 󵄩 󵄩 󵄩 󵄩 e(b/c0 )τ 󵄩󵄩󵄩w(τ)󵄩󵄩󵄩H 1 ≤ Ce((b −b)/c0 )τ 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1

τ

󸀠

+ Cδ∗ M∗ (T) ∫ e−((b −b)/c0 )(τ−s) ds 0

󵄩󵄩 󵄩󵄩w(0)󵄩󵄩H 1 󸀠 c + Cδ∗ M∗ (T)( 󸀠 0 )(1 − e−((b −b)/c0 )τ ). b −b

= Ce

((b󸀠 −b)/c0 )τ 󵄩 󵄩

4.7 Asymptotic stability of solitary waves for BBM equation

| 395

Taking sup0≤τ≤τ(T) of both sides shows that 󵄩 󵄩 M∗ (T) ≤ C 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 + Cδ∗ M∗ (T)(c0 /(b󸀠 − b)). Therefore we may choose δ∗ so small that 󵄩 󵄩 󵄨 M∗ (T) = sup{e(b/c0 )τ 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 󵄨󵄨󵄨 0 ≤ τ ≤ τ(T)} 󵄩 󵄩 ≤ 2C 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 . Because the transformation t → τ is bounded, it follows that 󵄩 󵄩 󵄩 󵄩 sup eκbt 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩H 1 ≤ C 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 ,

(4.7.51)

0≤t≤T

for some C > 0, where κ = 1 + 2δ∗ /c0 . Furthermore, substituting (4.7.51) into (10) now gives 󵄩 󵄩 ‖ν‖H 1 ≤ C(√δε + |c − c0 | + 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 ).

(4.7.52)

To establish inequality (4.7.45), it remains to estimate |c − c0 | and 󵄨󵄨 1 1 󵄨󵄨󵄨 󵄨󵄨󵄨 c − c0 − γ̇ 󵄨󵄨󵄨󵄨 󵄨󵄨 − 󵄨󵄨 = 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 c − γ̇ c0 󵄨󵄨󵄨 󵄨󵄨󵄨 c − γ̇ 󵄨󵄨󵄨 in terms of the initial data, that is, in terms of √δε + |c(0) − c0 |2 + ‖w(0)‖H 1 . To do so, ̇ we return to the transformed time variable τ. Because dτ/dt = c(t) − γ(t), we have τ

τ

0

0

dc 1 ̇ c(τ) = c(0) − ∫ (s)ds = c(0) − ∫ c(s)ds. ds c − γ̇ Therefore, for 0 ≤ τ ≤ τ(T), we have τ 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨c(0) − c0 󵄨󵄨 ≤ 󵄨󵄨c(0) − c0 󵄨󵄨 + ∫󵄨󵄨󵄨 󵄨|c|ds 󵄨󵄨 c − γ̇ 󵄨󵄨󵄨

󵄨 󵄨 ≤ 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨

0

τ

󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨 + sup 󵄨󵄨󵄨 󵄨󵄨C ∫(√δε + |c − c0 | + ‖w‖L2 )‖w‖H 1 ds 0≤τ≤τ(T)󵄨󵄨 c − γ̇ 󵄨󵄨 0

󵄨󵄨 1 󵄨󵄨 τ 󵄨 󵄨 󵄨󵄨 −bs 󵄨 ≤ 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + Cδ∗ M∗ (T) sup 󵄨󵄨󵄨 󵄨󵄨 ∫ e ds 󵄨 󵄨󵄨 ̇ c − γ 0≤τ≤τ(T)󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + Cδ∗ (1 + 2δ∗ )M∗ (T)/b 󵄨 󵄨 󵄩 󵄩 ≤ 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + Cδ∗ (1 + 2δ∗ )󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 /b 󵄨 󵄨 󵄩 󵄩 ≤ 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + C 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 .

0

396 | 4 Existence and stability of solitary waves Finally, 󵄨󵄨 1 1 󵄨󵄨󵄨 |c − c0 | + |γ|̇ 󵄨󵄨 − 󵄨󵄨󵄨 ≤ , 󵄨󵄨 󵄨󵄨 c − γ̇ c0 󵄨󵄨 1 − |γ|̇ but 󵄩 󵄩 |γ|̇ ≤ C(|c − c0 | + ‖ν‖H 1 )‖w‖H 1 ≤ Cδ∗ 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 ≤ Cδ∗2 , so for δ∗ sufficiently small we find 󵄨󵄨 1 1 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 ̇ − 󵄨󵄨󵄨 ≤ C(󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + |γ|) 󵄨󵄨 󵄨󵄨 c − γ̇ c0 󵄨󵄨 󵄨 󵄩 󵄩 󵄨 ≤ C(󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 ) + C|γ|̇ 󵄨 󵄨 󵄩 󵄩 ≤ C(󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 ) + C(|c − c0 | + ‖ν‖H 1 )‖w‖H 1 󵄨 󵄨 󵄩 󵄩 ≤ C(√δε + 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 + 󵄩󵄩󵄩w(0)󵄩󵄩󵄩H 1 ), completing the proof of the proposition. Proposition 4.7.10. Let u(y, t) be a solution of BBM equation with initial data u0 (x) = u(x, 0) = uc0 (x + γ0 ) + ν0 (x) ∈ H 2 ∩ Ha1 .

(4.7.53)

There exists ε > 0 such that if 󵄩󵄩 󵄩 󵄩󵄩u(⋅, 0) − uc0 (⋅)󵄩󵄩󵄩H 1 < ε a and √δε + 󵄩󵄩󵄩u(⋅, 0) − uc (⋅)󵄩󵄩󵄩 1 < ε, 󵄩 󵄩H 0 then the following claims hold: Claim 1. There exists t∗ > 0 such that (1) A decomposition u(x, t) → (ν(y, t), c(t), (t)) as in Proposition 4.7.4 exists for 0 ≤ t ≤ t∗ . (2) Inequality (4.7.44) from Proposition 4.7.9 holds for 0 ≤ t ≤ T∗ . Claim 2. The set M = {T | (1) and (2) from Claim 1 hold with t∗ = T} is closed in [0, ∞). Claim 3. The set M defined in Claim 2 is open in [0, ∞). (Therefore M = [0, ∞).) Proof. Claim 1. If u0 ∈ H 2 ∩ Ha1 , then the map 󵄩 󵄩 󵄩 󵄩 t → 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1 + 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩H 1

a

is continuous for t ∈ [0, ∞). Therefore, there exist ε1 > 0 and t1 > 0 such that if 󵄩󵄩 󵄩 󵄩󵄩u(⋅, 0) − uc0 (⋅)󵄩󵄩󵄩H 1 < ε1 , a

4.7 Asymptotic stability of solitary waves for BBM equation

| 397

then 󵄩 󵄩 sup 󵄩󵄩󵄩u(⋅, t) − uc0 (⋅ − c0 t)󵄩󵄩󵄩H 1 < δ0 , a

0≤t≤t1

where δ0 is as in Proposition 4.7.4. It follows from Proposition 4.7.4 that a decomposition u(x, t) → (ν(y, t), γ(t), c(t)) exists for 0 ≤ t ≤ t1 . The quantity √δε + 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩 1 + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄩󵄩󵄩ν(⋅, t)󵄩󵄩󵄩 1 , 󵄩H 󵄨 󵄩 󵄩H 󵄨 󵄩 from the left-hand side of (4.7.44), is continuous in t for 0 ≤ t ≤ t1 . Therefore, there exist ε2 > 0 and t∗ ∈ (0, t1 ] such that if √δε + 󵄩󵄩󵄩u(⋅, 0) − uc (⋅)󵄩󵄩󵄩 1 + 󵄨󵄨󵄨c(0) − c0 󵄨󵄨󵄨 < ε2 , 󵄩 󵄩H 󵄨 󵄨 0 then (4.7.44) holds for 0 ≤ t ≤ t∗ . Claim 2. This is clear from the continuity of (γ, c), Fk , k = 1, 2, and the left-hand side of (4.7.44) as functions of t. Claim 3. Choose ε∗ in Proposition 4.7.9 so that Cε∗ < min{δ0 /3, δ1 } where C is as in Proposition 4.7.9 and δ0 , δ1 are as in Proposition 4.7.7. Let T ∈ M. If |c(0) − c0 | + √δε + ‖w(0)‖H 1 < ε∗ , then 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄨󵄨 c − c0 − γ̇ 󵄨󵄨󵄨 ekbt 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩H 1 + 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 + 󵄨󵄨󵄨 󵄨 󵄨󵄨 c − γ̇ 󵄨󵄨󵄨 󵄩 󵄩 + 󵄩󵄩󵄩ν(⋅, t)󵄩󵄩󵄩H 1 < Cε∗ , t ∈ [0, T]. In particular, 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩eay ν(⋅, t)󵄩󵄩󵄩H 1 = sup 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩H 1

0≤t≤T

0≤t≤T

󵄩 󵄩 ≤ sup ekbt 󵄩󵄩󵄩w(⋅, t)󵄩󵄩󵄩H 1 0≤t≤T

< Cε∗ ≤ δ0 /3 and 󵄨 󵄨 sup 󵄨󵄨󵄨c(t) − c0 󵄨󵄨󵄨 < Cε∗ ≤ δ1 .

0≤t≤T

By Proposition 4.7.5, the decomposition (γ(t), c(t)) can be extended to the interval [0, T∗ ] for some T∗ > T. Also, if √δε < δ∗ /2 and Cε∗ ≤ δ∗ /2, then the strict inequality (4.7.44) holds on the interval 0 ≤ t ≤ T, and, by continuity, on 0 ≤ t ≤ T∗∗ for some T∗∗ with T < T∗∗ ≤ T∗ . Therefore [0, T∗∗ ) ⊆ M and M is open. The proof of Theorem 4.7.2 may now be completed. The crucial observation is that ̇ ̇ and c(t), inequalities (4.7.47) and (4.7.45) imply exponential decay of γ(t) and thus the existence of the limits γ+ = lim γ(t), t→∞

c+ = lim c(t). t→∞

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Index absorbing set 26, 27, 61, 66 attractor – discrete 1 – global 147, 152, 162, 180, 188 attrator 1 BBM equation 375 bilinear form 33, 34 Burgers equation 107 center manifold 268 Davey–Stewartson system 348 discrete infinite-dimensional dynamical system 40, 50 discretization 1, 99 – fractional step 69 – spatial 70 – time 36, 106 equilibria 194, 195, 199 essential spectral 315 essential spectrum 379 finite difference – method 2, 99, 102, 107, 108 – scheme 18 fixed point 216, 220, 243, 266 – stable center fixed point 196 – unstable saddle fixed point 196 fractal dimension 2, 17, 36, 81, 158 fractional step method 49, 62, 69 Fréchet derivative 282 Gevrey class 148, 162, 171, 175, 180 Gevrey operator 148 Ginzburg–Landau equation 1, 60, 130, 152, 160, 174, 193, 210 Gronwall lemma 8, 120 Hausdorff dimension 2, 17, 36, 81 heteroclinic orbit 206, 207 homoclinic orbit 207, 219, 227 hyperbolic fixed point 188, 189

inertial manifold 36–38, 40, 50, 63, 69 – approximate 142, 145 – stationary approximate 142 Kadomtsev–Petviashvili equations 334 KdV–Burgers equation 65 Kuramoto–Sivashinsky equation 133, 147, 170 Landau–Lifschitz equation 70 level set 174, 184 long wave–short wave 324 Navier–Stokes equation 108 Newton–Boussinesq equation 116 nonlinear Galerkin method 82, 84, 94, 108, 114–116, 129 orbital stability 282 oscillation 170 phase plane 131 Poincaré inequality 155 saddle 204 saddle-focus 204 Schrödinger equation 130, 131, 193, 219, 308 Schrödinger–Kadomtsev–Petviashvili equations 366 semigroup 29, 66, 77, 78, 188, 189 skew-symmetric 282 skew-symmetry 37, 61, 78 solitary wave 281, 309, 312, 325, 334, 366, 375, 377 spatial difference operator 70 spectral gap condition 63 the first Chern number 193, 215, 216 the winding number 216 traveling wave solutions 210 unitary operator 282, 310, 326 winding number 160, 168 Zakharov system 17