Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory (Lecture Notes in Mathematics, 1756) 9783540418337, 3540418334

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1756

3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Peter E. Zhidkov

Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory

123

Author Peter E. Zhidkov Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research 141980 Dubna, Russia E-mail: [email protected]

Cataloging-in-Publication Data applied for

Mathematics Subject Classification (2000): 34B16, 34B40, 35D05, 35J65, 35Q53, 35Q55, 35P30, 37A05, 37K45 ISSN 0075-8434 ISBN 3-540-41833-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10759936 41/3142-543210 - Printed on acid-free paper

Contents

Page

Introduction

I

Notation

5

Chapter

1.

Evolutionary

(generalized)

1.1

The

1.2

The nonlinear

1.3

On the

1.4

Additional

Chapter

2.

equations.

Results

on

existence

Vries equation Korteweg-de equation Schr6dinger (NLSE)

blowing

36 37

problems

Stationary

39

Existence

of solutions.

An ODE approach

Existence

of solutions.

A variational

2.3

The concentration-

2.4

On basis

2.5

Additional

3.3 3.4

Additional

3.2

Chapter

4.

of P.L.

49

Lions

56

of solutions

62

remarks

76

of solutions

79

of soliton-like

of kinks

for

of solutions

Invariant

42

method

method

compactness of systems

properties

Stability

Stability Stability Stability

3.1

10 26

up of solutions

2.2

3.

(KdVE)

remarks.

2.1

Chapter

9

solutions

the

80

KdVE

of the

90

NLSE

nonvanishing

as

jxj

remarks

oo

94 103

105

measures

4.1

On Gaussian

measures

4.2

An invariant

measure

4.3

An infinite

series

4.4

Additional

remarks

in Hilbert

for

the

of invariant

107

spaces

NLSE measures

118

for

the

KdVE

124

135

Bibliography

137

Index

147

Introduction

that

field

leading

are

approach properties makes more

possible

it

general

study).

the

present

In

qualitative

results

studies

dealt

with

on

of

travelling

investigate book,

author

to

the

problems

waves)

dynamical

stability

systems

twenty

substituted

solitary

of

in

by

following

(generalized)

main

material

is

These

topics.

these

equations,

(for example, consideration,

under

of invariant

construction

Vries

for

kinds

special

equations

Korteweg-de

generated

the

four

are

of

the

and the

waves,

of the

problems

initial-value

for

when solutions

arising are

There

a

evolutionary,

and

stationary

So, the selection

years.

interests.

of solutions

existence

both

in

of the

methods

and

problems

approach

(maybe

problems

of

class

some

surveys

scientific

author's

the

standing

of the

about

during

wider

blowing-up,

or

and this

etc.,

is

prob-

of various

stability

as

there

The latter

equations.

equations,

these

consideration,

under

equations

such

of solutions

hand,

other

well-posedness

the

subject,

this

known nonlinear

the

on

of differential on

by generated an essentially

the

and,

narrow

(on

problem

that

equations

these

currently

of

class

theory

behavior

systems

stationary or

problems for

of

related

mainly are

theory

he has

that

to

sufficiently

investigations

the

dynamical

of

the

time,

the

is

methods

by

problem;

scattering

inverse

qualitative

the

equations,

these

same

is

includes

particular

in

for

lems

called

approach,

another

At the

method

by this

PDEs solvable

of the

method

the

[89,94]).

example

for

see,

by

solvable

called

from

mono-

discoveries

field

this

problems

in the

example,

for

scattering

inverse

physicists

a

in view

Physical

mathematical

related

equations

nonlinear certain of studying possibility the to were quantum analyze developed

has grown into

-

and

observed,

are

One of the

partial

of nonlinear

kind

problems.

of the

novelty

consideration

[60].

special

a

mathematicians

of both

and of the

Makhankov

of

theory

the

-

solutions

attention

under

equations

V.G.

by

the

applications

the

to

graph

(PDEs) possessing

attracts

important

of its

of solitons

theory

the

30 years

equations

differential

large

last

the

During

measures

Schr6dinger

and nonlinear

equations. the

We consider

Ut

and the

Schr6dinger

nonlinear

+

Korteweg-de

f (U)U.,

+ UXXX

i is the

imaginary

and

complex

the

NLSE with

in the

unit,

second),

u

=

t E

u(x, t) R,

x

0

(NLSE)

equation

iut + Au + f where

=

(KdVE)

equation

Vries

is

(Jul')u an

0,

unknown function the

E R in

=

case

of the

(real

in the

KdVE and

first

x

E

A

=

case

R' for N

a

positive

integer

N, f (-)

is

a

smooth

real

function

and

E k=1

P.E. Zhidkov: LNM 1756, pp. 1 - 4, 2001 © Springer-Verlag Berlin Heidelberg 2001

82 aX2

k

2

Laplacian.

is the

As 2) respectively,

Chapter

u

qonsider

the

x) (it

equation

what

(as JxJ

oo

NLSE,

for

following A0

which

we

0

if

the

in

NLSE,

In what

is

supplement

with

nonlinear

Loo

_

elliptic

boundary

some

problem

A similar

problem

0(k)(00)

type

Difficulties

=

nontrivial

solutions

the

kinds).

In this

integer

any

argument

r

I > 0 there

exists

Ix I,

method

are

the

and

of

in finite

intervals

Chapter Lyapunov set

of the

qualitative

method.

results

sense.

X, equipped

to

3 is devoted

to the

Omitting a

those

some

distance

of

of P.L.

Lions.

being

differential

indicated

stability

2).

Chapter

of

functions

argument

is

the

as

a

f r

=

following:

of

W. for

function

of the

waves.

These

addition,

briefly

(ODEs)

in

consider

this

(for example,

the

chapter

we

L2)

in

Sturm-Liouville-type

one-dimensional

1,

==

of the

equations

we

basis

a

for

solitary

of

latter, In

N

0.

>

r

us

which,

existence

of the

of

property

the

non-uniqueness the

on

for

ordinary

example

of nonlinear

similar

with

the

half-line the

(see

solved

only

problem

our

the

into

order:

conditions

(for example,

f interesting of

00

NLSE with

speaking,

exist

proving

an

easily

depending

theory As

on

on

of

method

eigenfunctions

systems

solution

the

satisfying

waves

generally

functions

waves

second

will

--+

x

example,

for

KdVE and

solitary

case,

standing

kinds as

le,

Ej

X

sufficiently

solutions

I roots

methods

compactness recent

upon

a

exactly

the

solutions for

two

variational

concentrationtouch

has

this

In

when such

result

=

be

can

consider

us

typical

We consider the

0, 1, 2)

=

occurs

Let

the

case,

(k

0

of

uniqueness

of these

of the

into

if necessary,

0.

=

KdVE. For

the

and

when N > 2.

arise

above

for

arises

of existence

for

waves

function,

limits

possessing

conditions,

061--

the

0,

real

a

expression

standing

specifying,

equation

=

is

solutions

expression

f(1012)0

+

the

function

the

0

notation,

and

Chapter

In

the

KdVE and

c R and

follows,

bounded

a

w

this

just

substitute

we

of the

case

where

NLSE). Substituting

the

and

problem

Cauchy

of the

when

arises

introduce

with).

waves

the

f (s)

functions

constants).

positive

are

v

well-posedness It

wt)

-

to

dealt

solitary

obtain

we

x

of the

case

being

of the

for the KdVE and the NLSE used further.

O(w,

=

the

on

and

a

problem.

is convenient is

the -+

u

in the

equation

be called

results

stationary

waves

e `O(w,

=

the

(where

problems

value

travelling

for

e-a.,2

+S21

1 contains

initial-boundary we

2

as

1

physics,

for

important

following:

the

are

Isl"

2,

examples,

Typical

for

problems

above. of

details,

this

R(., .),

there

solitary means

exists

waves,

that, a

which is understood if for

unique

an

arbitrary

solution

u(t),

in the

uo

from

t >

0,

a

of

3

the

under

equation

T(t),

belonging

R, if for

that

by

obtained

equation

(in

the

us

introduce

Let

of the

functions

(in particular)

call

we

distance

special

a

argument

by

x,

functions

two

some

p becomes

stability

of

a

solitary

family

two

t

0 in the

=

metric

W2.

time,

they

can

usual

the

stability

of

many authors

For

taken

and

solitary in the

wit)

-

we

shall

with

s

=

of

following

if

they

H'

is

remark

the

here

that

family

two-parameter ob w

now

> 0

arbitrary

(x, t) and

=

b

are

close

at

t

all

for

the

=

approach

the

at

Sobolev

or

point spaces,

velocities

0 in the

wi

sense

of the

same

sense.

solitary

waves

developed

was

of

stability

this

He called

p.

his

(a, b).

other

t > 0 in the

stability

distance

possesses

Therefore

E

of

the

because

usually

non-equal

have

close

proved Later,

Lebesgue

as

are

to the

their

NLSE,

such

w

each

to

the

with

first,

p;

KdVE

r)

-

investigate

to

distance

the

close

to be close

has

wave.

consider

of the

waves

of

v(x

=_

functions

parameter

L02t),

> 0

respect

solitary

a

-

waves

[7]

paper

the

on X

t

verified

easily

to this

by

results.

distance

the

p should

be modified.

should

It

form:

(u,vEH')

T"Y

family,

0

X

u(x)

condition

it is natural

d(u,v)=infllu(.)-e"yv(.--r)IIHI

where

0' (w, x) 0

if

H1 consisting

space

equivalent

of

second,

x;

spaces

all

solitary

if two

f (s)

O(W2,

for

sense

the

respect

depending and

Sobolev

real

reasons,

in

functional

be

form

the

For several

KdVE with

pioneering

KdVE with

the

only

our

rule:

of classes

set

translations

same

same

in his

the

waves

x

the

in

then

p,

Benjamin

where

kink

a

was

in

diffusion

nonlinear

a

is called

H1, satisfying

from

v

of standard

sense

At the

distance T.B.

waves

[48]:

Piskunov

for

to

for

C

JJu(-)-v(-+,r)JJHi.

inf

space.

solitary

O(wl,

in the

,ERN

of the

up to

of

be close

cannot

and

solitary

of

N.S.

kink

a

wave

following

the

then

waves

solutions

any

be

a

KdVE is invariant

smooth

and

u

R, equivalent,

E

7-

distance

the

stability

of

solitary

a

has

one

distance

u(t), belonging R(T (t), u(t))


with

any

R(T (0), u(O)) first

one-dimensional

for


historically

the

called

t > 0 is

b

exists

t > 0 and

Probably

t > 0.

any fixed

0 there

>

e

any fixed

X for

all

any

to X for

belonging

consideration, X for

to

complex

space,

usual

the

-r

E

R'

and

one-dimensional

7

E R.

To

NLSE with

cubic

fact,

this

clarify

f (s)

=

we

has

s

a

of solutions

V-2-w real at

t

exp

I i [bx

-

parameters. =

0 in the

(b

2 _

W)t]

Therefore, sense

of the

cosh[v/w-(x two

-

arbitrary

distance

p,

2bt)] solutions

cannot

from

be close

for

this all

4

t > 0 in

the

any two

standing

family

NLSE, of the

sense

of the

distance

40(x, t)

above

satisfy

in the

the

two values

to

correspond

of the

waves

corresponding close

they

if

sense

same

different

to

close

at

t

parameter all

p for

At the

to same

stability

of

By analogy,

W.

-

distance

of the

sense

nonequal

w,

t > 0.

definition

the

of V

values

0 in the

=

each

other,

time,

the

in the

p and

of the

sense

be

cannot

functions

of

distance

d. In the two

necessary")

O(x)

>

for

nonlinearity

a

0, that

Next,

for

In

Chapter

theory

For the energy

we

and, for

the

higher

problem,

tific

contacts

appearance

and of the

Roughly

literature.

(with respect d Q(0) > 0 dw

condition

opinion

that

stability

3 is

devoted

oo.

We present

that

"almost 0 and

=

speaking,

to the

distance

is satisfied. to the

respect

kinks

of kinks

distance

always

are

under

stable.

assumptions

objects

many

on

dynamical the

If

recurrence

construct

KdVE in the we

wishes

present

theorem

explains

measure

associated

when it is solvable

infinite

corresponding for

by

our

phenomenon with

the

the

equa-

for

dynamical partially.

conservation

method

of the

of invariant

measures

colleagues

and friends

for

that

contributed

sequence

to

observed

was

measure

in

according

stability

the

this

con-

is well-known

phenomenon

invariant

theory

application

which

by

the

of

inverse

associated

laws.

to

thank

discussions

present

case an

generated

measures

in

such

the

means

bounded

invariant

an

it

system

a

one

phenomenon

Fermi-Pasta-Ularn

have

we

on

direction.

invariant

applications

important

interesting

in this

open

of the

waves

and

new

a

constructing

attention

speaking,

Roughly

a

simulations, Poincar6

our

of

remain

of

Fermi-Pasta-Ulam

waves.

of

questions

solitary

of

stability

a

problem

the

have

stability

the

to

many

with

It is the

conservation

The author

lim 1XI-00

for the KdVE with

the

prove

We concentrate

equations. the

NLSE,

scattering

deal

trajectories

then

-+

however

we

of nonlinear

many "soliton"

system,

JxJ

These

By computer

tion.

with

4,

if the

widespread

we

satisfying

is stable

(and O(x)

sufficient

a

type.

as

physics.

of all

view,

present

physical

wave

of kinks

a

Chapter

systems.

with

Poisson

of

equations.

dynamical

the

is

be said

should

nected

general

part

non-vanishing

It

our

there

we

waves

in the

NLSE)

the

NLSE,

solitary

of

solitary

a

stability

of

point of

f

The last

type.

type

the

consider

this

function

NLSE

stability Q-criterion

general

of

we

Confirming the

KdVE and the

the

the

is called

Among physicists

p.

of

for

KdVE and to d for

the

p for

of the

cases

condition

with book.

all

his

them

have

the

useful

importantly

sciento

the

Notation

the

of the

case

otherwise,

stated

Unless

KdVE and

I for

(X1)

X

N

8'Xi

i=1

R+

[0,

=

For

E

=

positive

denote

positive

a

always

are

constants.

for

integer

NLSE.

the

Laplacian.

Q C RN

domain

defined

on

Lp(RN)

and

D with

Mp

=

the

L,(Q) (p : 1) is the usual i JUIL P(o) ff lu(x)lPdx}p.

norm

Lebesgue

lUlLp(RN).

f g(x)h(x)dx

for

(ao,aj,...,a,,...)

=

g, h E

any

L2(9)-

11al 1212

R,

E

an

:

00

1: a2


0 the

any

T); Hpn,, ,,(A))

that there

This that

global

of

solution

Hn-solution for

the

in

all

t

R).

E

periodic

case

following.

problem sense

continuation

a a

problem

of the

book is the

> 2

about

about

and

well-posedness

in this

Theorem any

is correct

it

wider

a

Hpn,,,

the

standard

global

unique

a

-3

i

u(-, t)

)

(A)).

KdVE.

periodic

depends

continuously

map uo

T, T);

with

is

on

there

for

H'-solution the

initial

from

continuous

In addition

Then

exists

a

data

Hpn,,(A) sequence

of quantities A

Eo (u)

A

I u'(x)dx,

Ej(u)

0

I

2

U2(X) X

6

3(X)

dx,

U

0

A

E.,,

(u)

12

[U(n)12 X

+ CnU

[U(n-1)]2 X

qn(U7

...

(n-2))

dx,

)U X

n

=:

2,3,4,...,

0

where

periodic

Cn

are

constants

Hn-solution

and qn

u(.,t)

of

are

polynomials,

the

problem

such

(1.1.1),(1.1.2)

that

for

(with

any

integer

f(u)

=

n

u)

> 2

the

and

quanti-

a

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

12

E,,(u(.,

Eo(u(.,

ties

regularization

following

Wt

1.1.3

f(W)W.'

consists

Proposition

(1.

3).

1.

At the

fact,

1.1.6

global

unique

(4)

+ WXXX+

step,

6W

first,

At

steps.

xER,

0,

=

X

E,,

consider

we

t>O,

are

the

(1.1.4)

e>O,

(1.1.5)

(x)

Wo

=

take

f(-)

to

the

limit

E

uo

function

satisfying

the

(0, 1] the problem (1. 1-4), (1. 1. 5) has n 1, 2,3,.... ([0, n); S) for an arbitrary

c

E

=

is,

of course,

of

exists

independent

an

differentiable

S there

(1. 1.4),(1.1.5).

problem

+0 in the

--4

infinitely

an

any

differentiable E

Coo

which

be

for

Then,

S and

belongs

statement

Let

infinitely

an

uo E

any

we

1.1.7

(1.1.3).

be

which

following

Proposition estimate

for

Then,

second

f(-)

Let

solution

the

get

we

Eo,...'

following:

the

estimate

junctionals

the

(1.1.1),(1.1.2):

problem

(x, 0)

a

e.

of several

W

and prove

i.

t,

on

Hn-solutions.

of the

+

depend

do not

of Theorem

proof

Our

t))

for periodic

laws

conservation

function a

satisfying

the

u(.,t)

solution

unique

In

interest.

00

U C-((-n,

n); S) of

the

(1.

problem

1.

1), (1. 1.2).

n=1

At the

third

Now

using

step,

II ( 00

P1,0(u)

=

)

dx1

2

dx

00

the

generates Proof

of

The system

Lemma 1.1.8

topology

follows

in

from 00

2

PM 1(u)

x

,

21

the the

we

Proposition

proving

to

turn

we

1.1.7,

Proposition

I

I

1

2

and

following:

the

po,,(u)

1

00

x21u2(x)dx -00

I

00

2

u(x)

dx

dxm

dm

dx-

[X2,dmu(x) ] dxm

-00

k=O

0, 1, 2,...

S.

space

(dM ) E

=

relations

U(X)

min m;211

Cl""

with

seminorms

00

:5

We begin

1.1.6.

1.1.3.

Theorem

prove

j -.

I

x

2(21-k)

u

2

(x)dx

+

d2m-kU) (dX2m-k

2

f

dx.

0

dx




n

(1.1-6)

2,3,4,...,

=

U0(X)j

=

Fourier

(1.1.7)

transform,

one

easily

can

show that

00

U

Wn E

([0, m); S),

C-

n

2,3,4,....

=

M=1

Taking

into

(1.1.3)

account

applying

and

Sobolev

embedding

inequalities,

we

get from (1.1.6):

[Wn ( ) 21

00

I

1 d 2 dt

2

-00

00

192Wn

+

-WX2

(194 ) Wn

dX

I (Wn

OX4

+

a4

00

Wn) f(Wn-1)

19Wn-1

dx


+

1)(IW(4) 12

Gronwell's 0 such

Let

us

now

JWn 12)

+

nx

obtain

21luol 122

< -

the

I

-,E

YX2

C2(f)(1

+

) (,94 )2 2

Wn

+

a IWn axl

9X4

'9X

2

(1.1.10)

estimates

let

112(p+l)

dx+

immediately

2,3,4,...,

t E

11 W j 122).

+

2

implies

the

n

,

[01to]-

us

(1-1.8) existence

(1.1.9)

2

< -

c(E,

I

=

3, 4, 5, induction

in

1, equation

00

2

nx

Now,

I JWn-1

(1.1.8)

....

W(1+2)

n

dx-

2

E

00

m,

a2Wn

[0, to] and n 2, 3, 4, By using the and we get: (1. 1.9) embedding theorems,

t

estimate

=

0XI

estimates

ax,

and the


0 is

trivial.

large the

also

d=-oo

I

X2mW2dx + C, (c, m), n

r

I

=

I and

=




For

the

for

W2xx]

K

large.

arbitrarily

example,

6

K)W2 + n

n

dx +

The terms

Pi of the

terms

C22

Pi of other second

kinds

kind

we

can

be

have

00

Pj :5 C + C

f

X2m(W2_,

+

large

the term

n

W2 n)dx.

-00

00

So,

we can

choose

the

K> 0

constant

so

that

e

f

2mW2x.,dx

X

n

becomes

-00 00

larger

than

the

sum

of all

terms

of the

kind

f

2z K

2mW2x.,dx.

X

Therefore,

n

we

get

-00

(X,

I d

2 dt

I 00

00

X2mW2dx n

< -

C(c, m)

1 + -

I -00

2m(W2-,

X

n

+

2)dx

W n

(1.1.12)

(GENERALIZED)

THE

1.1.

KORTEWEG-DE VRIES EQUATION(KDVE)

(1.1.11) follows from (1.1.12). Inequalities (1.1.9)-(1.1.11) immediately yield in the space C([O, ti(r-)]; fWn}n=1,2,3,... S). Also, the

15

The estimate

00

Tt

2

of the

compactness

sequence

00

f

I d

the

estimate

gndx

f

C3 (E)




dx
0.

some

5)

t

(1.

uniform

of the

1.

with

of

Therefore,

solution

a

for

for

an

C([O, T']; S) (where T' the condition Jjw(-,0;c)jjj

C

E

Then

C

any

these

with

respect

L 1. 6 be valid

(1.1.5).

Also,

that

3)

[0, T].

E

of the

to

E

c

(0, 1],

for

C([O, T]; S).

class

(1-1.4),

> 0.

t

uniqueness

of Proposition

0 such

>

and the

estimates, 1.

E R and

x

proved.

is

problem

condition

satisfying

t E

the

argument

exists

of two

[W(X, t; 6)] 2dx,

on

lemma,

assumptions

solution

of

T > 0 there

function and

a

function

creasing and

be

some

(1. 1.4), (1.

problem

depend

not

(1.1.4),(1.1.5)

Lemma 1. 1.9 Let the

C([O, T]; S)

0 does

Gronwell's

make

to

with

existence

00

problem

want

we

the

suppose

us

00

[W(X, t; 6)]2

the

to

of the

S). Thus, taking the limit problem (1-1-4),(1.1.5)

Q0, T]; S) (1.1.4):

equation

-00

1 W

se-

L2) S),

of the

let

class

00

d

where

the

Q0, t'(c)]; C([O, ti(c)];

in

compactness

Q0, t'(c)];

space

local

Therefore, space

S).

uniqueness

w'(x, t;,E)

solutions

oo,

-+

to

w(x, t; c)

to

converges

the

and

fWn}n=1,2,3.... converges E (0, ti] is sufficiently

(1.1.6),(1.1.7)

in the

(1.1.6)

by equation

>

w(x, t; 6)

and let

lw(., t; 6)12 p E (0, 4), is

nonin-

a

0,

R,

> 0

arbitrary

infinitely

differentiable

constants

C and p,

any

(0, T]


term

to the

we come

be estimated

can

'X,

Tt

2

after

an

00

I

1 d

by analogy

estimate

2mW2dX

f

C3 + C4




whose

[0,T*)

point T*-O

t

t

0, i.

>

us

way

for

as

belonging above

to

Now,

the

we

turn

I,

any

m

-+

oo.

of the

infinitely

1.1.7

For each

n

=

and

Let

us

sequence

1, 2, 3,

...

9, is bounded

the

and let

R2

=

with ....

sup

us

,

where

R2 (C7

Pi

=

Un

f (.)

estimate

2 E H and 2

clearly

W3 E (0, oo).

We set

> 0

be

same

solution with

the

in H and we

>

R2 0

-

>

Let

is

with

x

(- 1, 1); R)

>

0 and

let

in H'

strongly

denote It

=

function

continu-

If,,(')jn=1,2,3.... (1.1.3)

T

the

as

solution

clear

that

0 is

given

by

also

R3

Sup

n

Then,

a

twice

C2((_M, M)

uo

T)

the

of

and let the

in

weakly

1,

T

uniqueness

by analogy

((- T, T); S) Un. fn and uo 0

the

some

any

in

arbitrary

an

(1-1.3)

E C-

f

proved

uniqueness

satisfying

to uo

with

we

can

proved

take

estimate

to

(x),

ul

too.E1

arbitrary

take

converging by Un (X, t)

JjUnjjj,T)jn=1,2,3

Let

converging also

be

can

6)

The

be

can

and

proved,

functions

taken

JR2(CIP)

t < 0

considering =

for

(1. 1.4),(1.1.5).

T > 0

limit

a

problem

existence

1.1.3.

f () satisfying

problem

sequence

Theorem

p

of the

any

is

T* +

of

proved.0

Lemmas 1.1.9-1.1.12

+0 in the problem

differentiable

1,2,3,.... a

C([O, T); S)

E

c --+

is

exists

Thus,

[T*,

of time 1. 1.6

Due to

Proposition

C and

C S be

1.1.7.

domain

function

=

an

half-interval

w(x, T*; E)

data

interval

of

(1. 1.4),(1.1.5),

the

there S.

space

initial

So, Proposition

t in the

proving

to

on

the

C([O, T); S) for The (1.1.4),(1.1.5).

any fixed

constants

same

as

Thus,

of

sequence

Un 0 1 n=1,2,3....

1. 1.

class

differentiable

a

for

of the

with

problem

u(x, t)

limit

construction.

(1. 1.4)

existence

half-neighborhood

results,

of the

sense

0, solv-

t >

the

onto

right

above-indicated in the

Proposition

solution the

S for

the

to

contradiction.

a

problem

the

due

all

Suppose

arbitrary

an

for

t; c) of the problem

be continued

can

on

global,

the E S.

uo

w(x,

proved,

be continued

equation

prove

a

been

of this

let

S-solution

E S understood

for

by taking solution

ously

get

of

of this

n

Then, ul

=

now

existence

obtained

f

T*.

solvability we

e.

Let

the

be

cannot

=

get the local 6

and

Cauchy problem

the

already

imply

Indeed,

corresponding

has

El

immediately

(1.1.4),(1.1.5).

the

w(x, t; E)

lim

1.1.9-1.1.12

problem that

uniqueness

of time the

and

proved.

is

the

Lemma

I I Un0 1 12-

n

R4

=

R4(R3)

where

the

function

R4

R4(R3)

> 0

is

(GENERALIZED)

THE

1.1.

by

given

Then,

Lemma I.1.10.

jju,,(-,t)jjj For x

t

[-T,O)

E

these

and t

-x

--

(I.1.1).

equation

t)112

5

W4)

(-T,T).

t E

Therefore,

have

we

21

1.1.10,

and

(1.1.25)

change

by the simple

be obtained

can

for

of variables

t > 0

00

0"

I

JjUn(')

and

estimates

in

-t

--+

due to Lemmas 1.1.9

R2

!5

EQUATION(KDVE)

KORTEWEG-DE VRIES

d

(Un

2 dt

Um) 2dx

-

=

(Un

-

Um)(fn(Un)Unx

-

fm(um)um.,)dx

-

-00

00

I

J(Un

Um)[f(Un)(Unx

-

Umx)

-

+

Umx(f(Un)

f(Um))+

-

0.0

+(fn(Un)

f(Un))Unx

+

Umx(f(Um)

fm(um)jjdx

-

00

f

C(T)

(Un

um)2dx

-

+ an,m)

00

where

an,m

--->

+0

n,

as

convergence

of the

Due to the

estimates

m

sequence

let

us

take

fUn(*) t)}n=1,2,3....

+oo and

by analogy

fUn}n=1,2,3....

in the

for

t < 0.

These

TI; L2)

C([-T,

space

yield

estimates to

the

u(x, t).

some

(1.1.25),

u(-, t) Indeed,

---

weakly

t E

[-T, T].

2

E H

loss

of the

(1. 1.25),

Due to

generality

I JU('7 t) 112:5

and

tE[-T,T].

H2 , hence

in

compact

the (without Therefore, JUn(') t)}n=1,2,3_.).

subsequence

u(-, t)

JjU(',t)jj2

exist

oo

in

E

>

H2).

Q-T,T]; 0 and

sequence

a

as

k

as

k

--+

and

oo

-*

oo

6-

Itn}n=1,2,3...

sequence

in H2

from space

Lemma 1. 1. 16 that

H-1 for

solution.

Let

defined and

U'

=

ul

in U1

I

'0

-

t

any

E

of the

solution

00

1 d

-4

in H2

one

So,

converging can we

easily

get

a

to

some

prove

as

in

contradiction,

proved.0

generalized

(1.1.1),(1.1.2) t E [0, T2)

n

there

U(',tn)112

-

-+

7

is

follows

easily

It

and

invalid

of the

subsequence

a

as

that

JjUn(* tn) Let

u(.,t)

(., t) an

-

7

problem

and

interval

f (u (-, t)) u', (., t) uxxx (., t) (. t) [-T, T] and, thus, we have proved the ut

U2

('7 t)

Let

be two

of time

(-Tl,

generalized

T2)

where

U2: 00

2 W

(X, t)dx

W(f(U1)U1x -00

-

f(U2)U2x)dx

us

prove

solutions

T1, T2

the

of the > 0.

We

(GENERALIZED)

THE

1.1.

jA j (

)2

d Xn

2

+ CnU

KORTEWEG-DEVRIES

(

)2

q,,

u(x),...'

polynomials,

such

d Xn-1

-

EQUATION(KDVE)

dx

dx,

n-2

25

n>2,

0

where

Cn

real

are

differentiable

period oft, of

i.

e.

the

the

quantities

the

junctionals

statement

with

f(u)

readers

consisting

weakly

to uo

infinitely period

and t E

Then,

all

strong

I

integer

Hpnr (A)

in

I

as

2,

result

(1.1.1),(1.1.2)

questions,

(see

is obtained

of the

Hn,,(A)

uo E

and

periodic

(x, t),

Let ul

oo.

--+

proof

the

in

as

1, 2, 3,

=

in

I

a

with

x

1, 2, 3,

=

we

refer

Additional

1.1.3,

one

can

fU0(1)}1=1,2,3. .

sequence

period

the

A converg-

corresponding

be the

...,

periodic

in

show that

for

problem

of Theorem

and

...

JU(*) 0)12

=

that

and

ul 0

now

--+

is

and,

A

I

A

3(x)

dx

U

f

and

0

weakly

on

En (u (.,

addition,

here in

Hpnr(A)

Suppose

the as

this

0))

I

---+

any

R,

strongly

and

continuous

CnU (dn-lU)

space,

=

we

of the

with

x

the

any T > 0

to

2 _

gn

I-00

Hpner(A);

in addition

:

0 be

(U'. .'

arbitrary.

and the

dn-2

u

dXn-2

Since

the

functionals

dx

have

En (ul (., 0))

lim inf

f U1 (X, t) 1 1=1,2,3....

5 C1-

Hpnr(A)

on

:5 C1

sequence

weak in

JjU&)t)jjHPn r(A)

I-00

equality

strong

(x, t)

dXn-1

0

continuous

u

t E

:5 liminf

& are obviously

Eo,...,

limit

a

for

HPn, ,r(A)

Uo in

jjuj(-,t)j1Hpn,,(A)

max

tE[-T,71

there

T); Hpn,-r'(A))

C((-T,

in

functionals

strongly

this

other

to

(-T, T)

Let

in

the

H' -solutions

problem

of the

devoted

is

functions

1JU(*)t)11HPn_(A)

are

book

>

n

solutions

U(*) t)12 for

with

independent

and

for periodic

laws

integrability

where

differentiable

differentiable A.

complete present

determined

are

infinitely x

u.

=

literature

arbitraiy

infinitely

of

the

t)),...

in

chapter).

take

us

since

arbitrary

an

periodic

conservation

are

f (u)

with

for

that

problem

E,(u(.,

to the

corresponding

this

to

Let

and,

u

the

to

1), (1, 1.2)

1,

the

&,...

Eo,...,

is related

=

remarks

ing

(1.

problem

This

are

of u(x,t) Eo(u(.,

solution

A,

and qn

constants

takes

place

if

En (u

(., to));

and

only

if

u&, to)

--+

u(., to)

oo.

that

En (u (.,

0))

>

En (u (, to)). 7

(1-1-31)

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

26

IUI,}I=1.2,3....

Let

argument

strongly

Hp' JA).

Hpn,,(A)

in

(-, t), (1.1.31),

of the

I

ul

of

...

get

as

differentiable

converging

functions to

of the 1

as

-4

(1.

1.

infinitely

differentiable

1) satisfying

ul

(., to)

lim inf &(u,

(-, 0))

periodic

ul, (-).

=

solutions

Therefore,

due to

have:

we

we

e.

infinitely A and

of

sequence

equation

E,, (u (-, 0)) i.

period

00 u(-, to) autonomy of equation (1.1.1), the function CQ-T, T]; Hpn ;'(A)) and, for any fixed t E [-T, T], weak

corresponding

1, 2, 3,

=

the

in view of the

in

strong

of

sequence

with

Then,

limit,

is the

arbitrary

an

periodic

E R

x

in

u(-, t)

be

E,, (u (-, to))

>

contradiction.

a

Thus,

=

1-00

for

ul(.,

any t E R

t)

(u (., 0)),

! E,,

u(., t) strongly

--+

HPn,

in

oo.

One

that,

if

again

then

ul(.,

t)

t)

ul(.,

complete analogy that u(-, t) E C([-T, T]; Hpn,,(A)) and u. C Hpnr(A) in Hpn,,,(A) I fUoj1=1,2,3.... as uo strongly in T > 0 u(., t) strongly C([-T, T]; Hpn,(A)) for an arbitrary the

-+

corresponding

are

proof

periodic

En(u(-,

t))

Hpn r(A).

I 1 .5 about

follows

The

In this

the

-

from

Theorem

1.2

the

1.1.5

time-independence

continuity

completely

is

nonlinear

section,

we

prescribed

consider

We shall

results

iUt + AU +

f(IU12)U

initial

complex

plane

f(JU12)U

tion

for

the

is

k times

the

complex

n)

x

En

t)),

the

on

...,

space

=:

0,

=

Uo

(NLSE)

equation of solutions

existence

N,

of the

NLSE

(1.2.1)

t E R

E R

x

(X, 0)

(X).

(1.2.2)

operator

A in

equation

operator

-D.

Here

function

continuously

f(JU12)U

:

linear

differentiable

00

U C'((-n,

Eo,...'

The state-

Eo (u (.,

quantities

functionals

the

on

two-dimensional

the

as

Laplace with

it

the

C

H'-solution uo.

data

identifying

smoothness

of

the

of the

Schr6dinger several

understand

sense

of

where

proved.0

U

ized

a unique u(-,t) global periodic depending on the initial data

1.1.3,

oo,

--+

problem

of the is

continuously

of Theorem

ment

Hn-solutions

of Theorem

problem

of the

and

-->

As in the

with

by

prove

can

R2)))

(-m, m);

if

f(JU12 )u

as

a

(1.2.1)

we

C

1

space

R2,

(we

write

map from

shall

conditions

accept

Considering

C.

)

we

say

this

in

2

R

general-

the

in

into

that

the func-

the

f(I

case 2

R

is

U

12 )U

E

k times

m,n=l

continuously

differentiable.

To formulate

jxj

--+

oo,

we

need

a

the

result

on

following

the two

existence

of solutions

assumptions.

of the

NLSE

vanishing

as

THE NONLINEARSCHR6DINGER EQUATION(NLSE)

1.2.

(fl) f (s),

and

Co N

where

such

2,

>

s

be

0,

-

differentiable

continuously

a

a

real-valued

1),

where

be

(0, p*

and p E

> 0 =

f(JU12 )u

Let

function p*

if

N-2

of

in the

3 and

N

the

p*

argument

(9

Co(I

i)u

>

Let there

Remark u

C

exist

0 and pi

>

a

Under

1.2.1

iU2

U1 +

-=

exist

arbitrary

1 is

(0,

E

+ '

N

)

JUIP), such

U

for

(1.2-3)

E C.

f(S2)

that


0

only

Iluolli,

on

and therefore

such

T,

constant

a

exists.

Since we

have

that

for

the

interval

of

fixed

a

proved, any

E

H' there

uo

the

on

(-T, T)

(1.2.4).

Since

the

not

length

by

and

of the

I Ju(-, t) 111

limsup

T >

there

t

is

of

of the

TF

existence

of

again uo H'-solution

unique

Tk+,

be the

and

0
0

case

t < 0

00

Gtuo

=

(47ri)-l'

I

uo e

(x

By setting can

+ 2 v"t_z)

relation

yields

at

+ uo

(x

and

by analogy), -

2

vft-z)

dz.

formally 00

-[Gtuo]

4t

N/"Z-

0

This

z

be considered

=

(4ri)-12

f 0

e' 'u'(x

0

+ 2 vft-z)

-

VIt-

u' 0 (x

-

2

vltz).dz.

accepting we

get

for

THE NONLINEARSCHR6DINGEREQUATION(NLSE)

1.2.

For

> 0

c

I

Ci-. e

we

have

U0(x

+

2v t_z)

u' 0 (x

-

2v'-t-z)

-

dz

-\/t-

icu'(x0

-ie

=

2 v'tc)

+

,

35

u'(x -0

-

2v1t_`c)

-

+

-

Vt_

0

I

C

j

'(x+2v' t_z)+u0'(x-2-\/t_z)

uo

_.

dz,

V"Z-

0

where

uo(x)

of this

equality

interval. is

an

0

--->

tends

to

zero

integral

improper zero,

for

consequently,

and,

oo

--

which

(1.2.8)-(1.2.10)

t > 0 and

any

,t[Gtuo]

i(4ri)

=

to

right-hand

any bounded

58-t [Gtu0j

uo'(x VZ_

2vft-z

-

side

interval

not

and

is determined

at

+

side

any bounded

t from

in the

term

in t from

2vlt-z)

u"0 (x +

2

respect

derivative

00

right-hand

in the

term

second

the

R the

E

x

with

uniformly

converges

first

the

uniformly

oo

c --+

as

due to estimates

Since

containing

jxj

as

)dz

0 00

i(47rit)-

e

JEWL, U/1 (y)dy

=

0

a2

-[Gtuo] X2

i

Co

for

t < 0

at

t=o

=

a

[Gtuo]

=

there

lim

exists

These

0.

L

first

statement

(1.2.14),(1.2.15) Now

equation

is we

homogeneous

any

X3_SolUtion

that

In view

any

of the

X'-solution

only

X'-solution

the

=

of

iu".0

Hence,

imply

also

arguments

of

arguments there

exists

that

=-g(u(xt)). (1.2.7)

equation problem

the

above

arguments,

the

satisfies

problem

Lu=O,

xER,

it

suffices

(1.2.14),(1.2.15) to

that

prove

satisfies the

linear

problem

U(X, 0) has

at

[Gtuo]

above

proved.

prove

(1.2.7).

that

a

I

I ijGt_,(g(u(-,s)))ds 0

our

due to the

Further,

u".0 t-0

t=o

L[Gtuol

iu" 0 and

2

a2

t

Thus,

relation

of this

proof

the

earlier,

noted

we

by analogy.

iGtu" 0 and therefore

[Gtuo]

As

t > 0.

obviously

0, then

-12jGtuo] at

0 for

=

be made

can

If t

-2-

L[Gtuo]

indeed

that

so

trivial

solution

of this

problem

u

=-=

in the

0 from interval

=-:

C (I;

tEI, 0

X3)

of time

Let

.

Simple

I.

0')

d

Tt

j

I ux (x, t) I'dx

=

us

0

suppose

that

calculations

u(x, t)

is

show that

a

Concerning in

the

For

simplest

results

the

on

[102]. f (s)

case).

with

sP

=

NLSE

essential

an

periodic periodic =

and

u)

we

no

uniqueness

KdVE and for

of

As for

the

phenomenon

is unknown

for

the

this

[69,70]). blowing

For up

[38], f (s)

this

in

as

solutions,

NLSE) to

paper

NLSE with

=

the

sP with too.

paper

result

is p

recall

proved

that

we

1.2.10

well-

(see,

one

of the

considered

only

first

are

proved

[91] stating

Cauchy problem

of the

non-smooth

in

for

for

by

J.

problem

Bourgain in

the

solutions,

proved

for

have

(see, in

of the

one we

rigorously proved

[16] (see

[17])

the

for

to

the

where

the

(with Is IP.

usual

the

f (8)

like

it is known for

considered

vanishing

devoted

literature

is

problem

data

also

are

x

there

of the

initial

nonlinearities

Although is

with

whole

superlinear

[38].

data,

initial

well-posedness

the

review

have

it

of the

smooth

more

of Y. Tsutsumi

of the

and the

justified

2-, N

or

have

we

Theorem

with

of the blow up of we

H'

L2.

periodic

L2-solutions

KdVE. Above

and followed

subject

technique

from

possibility

the

uo

result

investigations

the

of

from

1.2.4

form is contained

investigations

lot

well-posedness

the

consideration

the

mention

and

general

in its

(we

important

data

between

Having

problem,

existence

f (u)

(with

x oo.

--+

-) (0, -N

N = I for

Theorem

1.2.7

the

initial under

difference

in

IxI

P E

equations

the

Proposition

with

a

data

of Hl-solutions

mention

some

(1.2.1),(1.2.2)

For

With

1.2.4

are

initial

with

existence

especially

We also

there

equation,

[33,37,69,70,79,88]).

one-dimensional

the

this

of Theorem

proof

the

Cauchy problem

of the

example,

for

NLSE,

the

[45].

paper

posedness

as

EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

CHAPTER1.

38

=

NLSE but

the

simplest

results

on

formal

presented

only

example,

[90] and, also,

paper

[90]

that

a

there

exist

Chapter

2

problems

Stationary As

noted

already

have

we

standing

for

sentation

AO 0

is

a

real-valued

supply

their

_

(IJ.0-1)

Usually

the

problem

N

1,

=

with

following

jxj

as

11. 1_00

QC

RN is that

Suppose function

and

problem

pairwise

By analogy,

a

oo

N

02)0

f(X, 02)

as

it

will

it

Q.

It

has

solutions.

then, as

domain

3 and in

is

=

(11.0.1)

O(x).

O(x)

equation,

of this

Rlv,

E

(11.0.2)

conditions

jxj

as

infinity

the

on

vanish

--+

oo,

i.

(11.0.3)

be reduced

assumptions

natural

KdVE under

can

for

e.

0.

(11.0.1),(11.0.3)

bounded

>

k(x) > 0 (11.0.4),(11.0.5)

different

starshaped,

now

x

some

=

the

for --+

If

is

f

proved

be shown

=

IOIP-',

in

[741,

further

P.E. Zhidkov: LNM 1756, pp. 39 - 78, 2001 © Springer-Verlag Berlin Heidelberg 2001

X

equation

to

(11.0.1)

(11.0.2),(11.0.3)

and

on

with we

E

Q,

0

with

also

(11.0.4)

O(X),

(11.0.5) a

smooth

sufficiently

k(x)jOjP-1,

=

=

0,

=

p >

that

if p E

positive

in

p >

7Nq j2,

known

solution

a

0,

=

Olau

the

the

to

problem:

WO+ f(X,

_

0,

=

solutions

problems

the

similar

AO

where

repre-

NLSE leads

generalization

a

with

that

waves

solutions

Along

too.

the

consider

solitary

finding

of

of these

behavior

RN, 0

E

consider

(11.0.2)

and

O(X) The

x

02)0

WO+ f(X,

suppose

we

0,

=

We also

function.

equations

solutions.

f(02)0

wO +

-

AO and

e"'O(x),

general

of the the

into

R,

E

w

equation

stationary

Here

u(x, t)

=

substitution

the

Introduction,

in the

waves

N-2

this

problem

(see

Example

1,

(1, N 2)

and

2

Q and w

has 11.0.1

boundary.

where

an

w

infinite

0 and no

k(x)

the

a

C'-

0, then of

sequence

domain

nontrivial

and

is

Section

fl

is

solutions. 2 of

this

chapter), critical

the

fying

condition

f(02)

lim

if

and

infinite

an

JxJ)

=

been

have

problem

(11.0.2),(11.0.3),

voted

it.

to

simple

k(.)

11.0.1

tends

rapidly

Let

to

zero

as

f (x, 0')

above,

under

similar

:N 22

p


0 and

1: I

N > 3,

x,

C, :5 k(xi)




+oo for




is

[99].

from

C'-function,

a

has

[1,77,78]

investigations

of

specific

some

radial

with

deal

that

mention

we

taken

Example

lot

a

are

f(.)

function

and

not

(11.0.1),(11.0.3)

We illustrate

example

is

there

problem

of the

lim

f (.) have been intensively

shall

we

f

problem

the

example,

interesting

many

book

present

literature,

In the

see,

nonlinearities

obtained

in the

However, lutions

(on

if

speaking,

(resp.

for

N+'

=

1951-oo

different

subject

p*

f(o 2) 0 (or f(x, 0 2) 0)

subcritical

subcritical

a

pairwise

of

this

superlinear

with

0 and

>

w

sequence

solutions

Problems there

with

N-2

1, and

p >

The exponent

called

is

OP',

=

nonlinearity

Roughly

+oo.

=

(0')

3, f

N+'.

the +oo

=

101-

(or (11.0.1),(11.0.3))

with

f(02)

Jim k61-00

N>




0,

usually the

we

O(x)

have

look

important

for

radial

results

-=

0.

solutions

by B. Gidas,

of the Ni

problem

Wei-Ming

(11.0.1),(11.0.3). and

L.

Nirenberg

In

41

[34,35]

be mentioned:

should

solution

problem

and of the is shown we

also

[9]

on

above

in the

assumptions

authors

under

the

with

assumptions

of

by

problem

a

function

of

general

similar

of

this

is not

and

P.L.

Now

so.

from

Lions

(H.0.1),(H.0.3)

to

x

As it

is radial.

type

sign

positive

independent

f

Berestycki

H.

arbitrary

an

a

alternating

with

result

that

proved

ball

a

solutions

solutions

general

very

a

have

in

proof)

a

of radial

existence of

for

papers,

(without

present the

these

(H.0.4),(H.0.5) (11.0.1),(11.0.3)

problem

of the

under

kind. 00

Theorem

11.0.2

9(-)

Let

U C((-n,

E

n); R)

be

odd,

3 and let

N :

n=1

(a)

'()

0 < lim inf

(b)

lim inf

(c)

there

'()

Jim sup 9W U

where

(0, +oo)

71 E

p*

again

-

exist

+00;


0

up

1-1-+00




=

f g(s)ds.

-2

0

Then,

problem

the

AO has

a

countable

In this

chapter,

qualitative

u(jxj),

=

of pairwise

different

we

shall

this

prove

establish

same

form

(N

2)

-

N,

solutions

radial

Theorem

011xj_.

E R

x

and

=

0

solution.

radial

positive

a

particular

in several

and

cases

study

the

of solutions.

also

we

in the

u

set

behavior

Now, taken

g(u),

=

as

Pohozaev

the

for

identity

(H.0.1),(H.0.3)

problem

the

[87]:

in

1 IV012

dx

=

-(N

1 Og(O)dx

2)

-

=

I

N

RN

RN

G(O)dx

RN

0

g(o)

where

Wo

=

f(02) 0

_

G(0)

and

-2

=

f g(s)ds.

equality

A similar

also

was

0

[74]

obtained

in

bounded

and

for

the

the

equation.

the

above

For

equality

equation

(11.0.1)

multiply

the

RN with

only

the

the

the

N 2 (N 2

independent

this

of

w

the

>

in

O(x) is

valid,

.5)

taking

into

integration

80('ax,),

sum over

i

1, 2,...,

case

the

function

1,

are

sufficient.

(11.0.6)

=

our

on

p >

of the

in

domain

one

result

may,

over

N and

first,

RN

integrate

and,

Q is of

the

f(o 2) To

in

get

multiply second, the result

by parts.

particular =-

jolp-1,

=

when the

case

Of course,

x.

account

further xi

the

assumptions

f(02)

0 and

in

of

additional

integration

identity This

3).

needs

by

solution

sign.

is

case,

with

equation use

The Pohozaev

change

f

example,

by 0

trivial

(11.0.4),(11.0

one

in

same

over

problem

function

(H.0.1),(H.0.3)

problem

p >

the

0 if

for

yields the

that

function

example,

for

(H.0-1),(H.0-3) NG(0) + (N 2)0g(O) does w > 0 and 101P-1 f(02) the

problem

-

=

has not

with

CHAPTER2.

42

Existence

2.1 first,

At

we

solitary

consider

u(x, t)

substitution

of solutions.

O(x

=

for

waves

ct),

-

-Wol Assuming

0(oo)

that

following

the

to

we come

a

NLSE with

a-

to

The

1.

=

equation

0.

=

0"(00)

and that

constants,

are

the

N

0,

=

equation: +

7(0)

0"

+

(II.1.1)

a

=

0

7(0)

and

-wa-

=

the

KdVE leads

the

into

a+ and

-wO with

KdVE and for

f M01 + 01"

+

where

a,

=

An ODE approach

the

R,

E

c

STATIONARYPROBLEMS

f f (s)ds.

is clear

It

that

substitution

the

general

of the

a-

for

representation

of

First

of this

solution

onto

(11.1.1) is bounded, 0"(-) is bounded, too,

equation

derivative

similar

a

However,

is

a

on

a

continuously

equation.

it

to

make

a

function

and

a

then

[a, b),

of the

follows

it

therefore

point

from first

the

where

this

if

equation

also

Setting

bounded.

Oo

0(a)

=

+

Oo, 0'(b) similar

Cauchy

the

00,

=

we

reasoning Let

fj(0)

problem

immediately several

=

for

get

times

f 0'(x)dx

the

0

second

solution

and

00

0'(a)

=

+

f 0"(x)dx

and

a

(II.1.1)

with

In this

statement.

this

specifying

not

can

b

equation

our

it

solution

of this

a

considering

then a

that

0'(.)

derivative

b,


2.

takes

and

real

parameter

_

following

the

g(y)

=

with

f (y2)y

0y

initial

obtain

First

we

of the

functions

0'(w,x)dx KdVE

2))]

2A 2t _

1)(v

and

+

2A2

w

as

family

one-parameter

+

vanishing

One

(v+l)(v+2)

easily

can

a

point 0
d.

In view




C,

By points

constant

=

we

obtain

i

I + 1

k

11.1.3,

(T;ml+l )2+ C2-

sequence.

a

of Lemma

statement

:5 C,

first

exists

1

the

1, 2,...,

C

1,

estimate

(11.1.9)

of extremum

lie

the

>

0 such

m=

that

1, 2,3,....

and

from the

since

the

outside

0

EXISTENCE OF SOLUTIONS. A VARIATIONALMETHOD

12.2.

the

Thus,

function

y(ro)

than

than

I roots.

the

on

At the

the

time,

same

yo and

parameter

definition

in view of the

imply

0

==

less

no

of solutions

dependence

continuous if

V has

the

theorem

the

on

that

y(ro)

V cannot

have

fact

the function

of -go that

49

=A

0

more

I roots. Let

that

prove

us

-9(r)

lim

Suppose

0.

=

this

is

the

not

Then,

case.

either

r-00

solution

the Zn

be

only

negative

for

can

number y E

first a-,

=

0) U(O, a,).

Thus,

contradiction

implies

of the m

y(r)

lim

=

y-

by

implies,

(11.1.4),(11.1.5)

0, and Theorem

H.1.3

theorem

the

than

more

on

parameter

I roots.

completely

is

domain

sufficiently

the

on

have

cannot

in the

for

negative

is the

since

case,

extrema

is

energy

problem

solutions

second

have

E(r)

energy

of the

negativeness

values

that

the

should

it

'g(r)

solution

the

of

which

asymptote

an

In the

r.

of extrema

sequence

a

has

P(r)

energy

1,

has

it

or

function

argument to

case

of solutions

large

sufficiently

this

in

But the

too.

the

of the

V is equal

solution

dependence for

that

values

r

of this

Hence,

a,.

=

large

of the

of r,

y

or

large

graph

the

case,

sufficiently

values

continuous yo,

y

of roots

(a-,,

large

the

In

sufficiently

for

V is monotone

+00-

-4

This

proved.0

r-oo

Existence

2.2 In this

section

the

to

we

shall

of radial

existence

f(02)

case

of solutions. consider

of the

jolp-1,

p > 1.

=

application

an

solutions

AO

A variational

wO

=

consider

we

101P-10,

-

Our result

the

on

Theorem

solution

We note

ul

that

right-hand

a

side

similar

sense

to

Theorem

tinuously

r(.)

ul(r),

'(rV)lr=r(,,)

=

point

a

N > 3 be

0,

where

r

=

takes

of the

equation

of

Loo

-

we

present

H be

Let

real-valued

Then, S,

jxj,

with

place

problem N

E R

(11.2.1)

,

(11.2.2)

a

solution

and,

precisely

1 roots

for

1

any on

g(o)

kind

(1, N+2). N-2

Then,

half-line

the

for

the

1,2,3,...,

=

(11.2.1),(11.2.2)

problem

the

general

more

and p E

integer,

for

then

if

a

below

real

are

Hilbert

functional differentiable

continuously

0.

vo E

attention

a r

> 0.

with

functions

g(O)

Theorem

11.2.1.

the in

a

101P`0.

which

a

the

proving

our

0.

radial

positive

result

11.2.2

be

>

similar

differentiable > 0

Jr

critical

Lo

has

=

Two results

Let

Let

to

We restrict

following.

is the

(11.2.1),(11.2.2)

problem radial

H.2.1

the x

=

existence

methods

(11.0.1),(11.0.3).

problem So,

of variational

method

the

Y(h)

space

in

j(v) -0.

when

proving

with

a

H, and S

function

functional

Ih=r(vo)vo

used

=

on

J(r(v)v)

norm =

f

S such

11

-

11,

h E H that

considered

be

J :

for

I IhI I any on

a

con-

1}.

=

v

E

S has

S a

Proof

is clear.

By conditions

Further, such

consider

-y(s)

that

E S for

d

0

arbitrary

an

(-1,

E

s

of all

consisting

Thus,

E L.

w

Remark

by

sults

S.I.

J(r(vo)vo),

H.2.3

The second

Theorem

of H1 with

Lq

is

E

the

compact

for

In

0.

=

is

a

Let

product

;

N-2

function

continuous

For any g E

1)

into

H

r(vo)-j'(0)

+

space

H

>=

>

subspace

the
=0

I

any

E H. 0

w

simplification,

sufficient

for

goals,

our

of

re-

fh

=

for

jxj

as

at

Clearly,

point

the

set

Thus,

0

x

C07,

=

Then,

h(jx 1)}

the

h E

we

H,'

0 and

there

of all

vanishing

exists

that

-->

except element

oo.

C000

from

into

unique

a

each

Ix I

as

functions

radial

subspace

of H,1

everywhere

accept

can

be the

embedding

RN and continuous

in

any

h

:

arbitrary

an

oo.

---+

H'

E

of H'.

norm

everywhere

0

--->

following.

H,'

also,

h almost

Proof.

any

a

and the

4(jxj)

of the

is

is the

2N

2 < q
3 and

addition,

Sketth

H,1.

need

we

scalar

H,' coZnciding

of H,1 in

H.2.4

J.,=0vo

[r(-y(s))]

therefore

(-1,

Then,

0.

[75,76].

from

result

H.2.2

0

obviously

is

,=0

to

>= 0

w

Theorem

Pohozaev

d

1,

7'(s)

kind

-y'(0)

and

vo

=

0.

map -Y from

X(r (vo) vo), -y'(0)




0 there

C

exists

00

h (r)

(s)

lim n-oo

and is

for

any

continuous

a, b on

:

0


r

r

(+oo)

hence

Also,

0.

=

obviously

we

have 2

00

00

1 (r)j

h(s)

ds

I


Rj)

I

r

N-1

(R)jjh,,jjj2,r'

:5 DNCq-2rq-2

Ih.,,(r)lldr

R

Since

here

for

large

ciently

Remark

Sobolev

spaces

Now,

we

functions:

u(jxj)

> 0 and a] I

of functions

from

Clearly

u


C 1

C,

n ak >

-

-

(n

> 0

-

by analogy.

that

J(U)

inf

b

n

a1

R.

set

the set

on




0

--+

are

n

+00

exists =

chosen

a

sequence

1,2,3,...). for

the

Then functions

bn

--+

+0 such

JO.jp+j vn to

:5

C2)

satisfy

EXISTENCE OF SOLUTIONS. A VARIATIONAL METHOD

2.2.

(11.2.8)

condition

1 1 VnJJ21

with

a

b

0 and

=

C3, C4

some

0

>

H,' satisfying

E

theorems,

embedding

using

we

get

for

all

C3 IVVnl2,2 therefore


Cap n- 10njP+1 4 P 2-

Hence,

n.

since

10,,Ip+l

earlier

as

! C5

> 0

have

we

2

C6b,,P,-'

0 < an
0 be

fUn}n=1,2,3....

sequence

arbitrary

its

set

we

H',

E

u

jV such that

C

fixed;

0 is

>

following.

is the

Let p E

A

JE(u)/

Inf

=

prove

fYn}n=1,2,3,...

sequence

compact

to

A where

=

the

of

fUn(*

sequence

point

is

+

(11.3.1) Yn)}n=1,2,3....

the

of

solution

a

Then L >

problem

the

and

-00

there

exists

relatively

is

problem

minimization

(11.3.1). Remark and any

A

>

If p

11.3.2 0

has

one

+

-I-,N

then

0.

To

see

> I

IA




Remark

providing

the

M N

1 +

e

T

1

(p+l)+N

2

07

P.L.

+0 when p

-4

sufficiently Lions

example,

he

0 and

f (x, u)

11.3.4

As P.L.

relative

is

large

[57,58]

in

c(x)u

p + 1

I I-I--

something Lions

compactness

the

=

like

>

'

x

and,

N

1 u (1, x) jP+1

dx.

for

E

p

(1,

1 +

N

0.

problems

considered

f (x, u),

=

1 +

> cr

investigated

U

c(x)

(1,

RN

-Au +

where

For P E

function

2

N

RN

Hence,

A > 0.

any

the

=A and

(o,, x)) =20r2

E (u

consider

VA

u(c, X) We have

for

-oo

=

of the

essentially

problem N

E R

0,

k(x) ju IP-'u his

with

publications,

noted

in

of any

minimizing

sequence

k(x) the

>

0.

principal

up to

relation

translations

as

58

CHAPTER2.

in Theorem

11.3.1

I,\

is

I,, + I,\-,,


0

c

there

for which

R> 0

U2k(X +Yk)dx>A-c, BR(O) fx (ii) (vanishing) =

k=1,2,3,...

BR +(0)

yl,

(here

satisfying

properties:

(i) (compactness) exists

there

A,

=

RN:

E

jxj

R});




there

exist

[V and satisfying

in

I

u""

(UA;

-

the

U2)1

+

k

(0, A)

fUk1}k=1,2,3....

and sequences

and

JU2k}k=1,2,3,...

following: --+0

k--+oo

as

for

q

I (Uk1)2dX

lim k-oo

G

a

-

a

I (Uk2)2dX

lim

=

k-00

RN

2N

2

oo;

dX > 0.

RN

With from

Part

the I of

Proof

[57].

of Lemma 11.3.5

We introduce

Qn(0

we,

the

=

actually,

concentration

SUP YERN

f

the

repeat functions

proof of

of Lemma 111. 1

measures

U2 (x)dx.

y+Bt (0)

jQn(t)Jn=1,2,3....

Then, functions quence

that

on

R+

is

and

k-oo

Qn, (t)

=

sequence

liM t-

f Qnk } k=1,2,3.... lim

a

+00

and

Q(t) for

a

Qn(t) function

nondecreasing,

of

A.

=

By

the

nonnegative, classical

Q(t) nonnegative

any t > 0.

result, and

uniformly there

nondecreasing

bounded

exist on

a

subse-

R+ such

OF RLIONS METHOD CONCENTRATION-COMPACTNESS

2.3THE Let

a

+00

t

for

the

Let

us

place

Q(t).

lim

=

Obviously

[0, A].

E

a

fQnk(t)}k=1,2,3,.-

sequence

If

If

a

a

0, then

=

(ii)

vanishing

the

clearly

A, then

=

59

the

takes

(i)

compactness

occurs.

briefly

arbitrary.

be

these

prove

Then,

claims.

two

let

First,

a

0 and

=

let

c

0 and

>

R

0

>

have

we

Qn,(R)

I

sup yERN

=

u',, (x)

dx




exist

sequences

Ck

integers,

such

positive

that

I

claim

+01 Rk

--+

dx

0,

have for

n

+oo

-->

Qn- (Rk)

that

2, (x)

U

proved.

is

k

as

>

Q,,,, (Rk),

=

Second,

k

--+

=

1, 2, 3,

a

all

Then,

A.

=

fMklk=1,2,3

and

oo

for

6k

let

>_

....

...

yj,+BRk(0)

Then, taking

arbitrary

an

>

c

we

I

n_(x Un

k such

all

y,,,)dx

+

A

>

that


mk-:

c.

-

BR, (0)

Now,

to

R > Rk

that

so

the

Consider

place k

in

this

the

case

for

m

all =

k

0,

1,

p(x)

0( ) .

C

>

=-

to

prove

take

to

sequences

Rk

the

that

=

W be

a,

=

A

...

=

(infinitely and O(x)

respectively.

=

exists

Rk,

=

=

k

'Ek,

a

=

k,,,

number

lQnk(4m)

and

...

Rk_+j

get required

< 1

there

m-1

R,

set

we

cut-off

=

m

Q(m)1:5

-

we

Jxj

0 for

and W,,

(iii)

dichotomy 01

Ck >

17

-

El

=

Ck

takes -+

0

as

> 0

Q(4m)l ...

=

such

'Ek,

=

=

that

m-'




k-oo

yk/

+

show that

us

I(a):5

a

ak

=

E (Unk)

lim

=

contradicts

prove

Let

ak

k-oo

Let

a

+

Let

lim

+ E (pkU2)k

IA

A

I(a)

funk I k=1,2,3.... (i) (compactness).

Then

a.

-

E(akUlk)

==

a

property

occur.

=

subsequence

a

the

AI(a)


and, again by is a ball B12 (X2) : B, (X2) C B, (xi) such that p(x) > 2 for any x E B12 (X2) Continue this process. We get a sequence of balls such that f B,. (X,,,)}n=1,2,3.... Bn,, (x +,) C 0 as we can BIn (xn) and p(x) > n for any x E B, (xn); in addition, accept that rn there is > the oo. construction n a n for Then, unique xo E nB,,,(xn). By p(xo) C

-

--+

---+

n>1

any

integer

> 0.

easily

It functional we

n

This

follows

contradiction

from

Indeed,

lemma.0

that

Lemma 11.4.6

in X is continuous.

the

proves

let

admissible

an

Then,

xo E X.

since

lower

p(x)

sernicontinuous

:5

p(x

-

xo) +p(xo),

have

p(x) On the

other

hand,

-

P(xo)

Xx

:5-

-

x0)

M1Ix




0 there

exists

that

P(xo) for

all

x:

jjx

-

xoll




c

that

arbitrary.

p from

show

to

We take

the

number

a

that

P(XO) Choose

It

65

0 such

>

continuity

P(XO)

-

Corollary

PN(Xo) -PN(X)l

that

pN).

of

P(X)

Then,

< PN

11.4.7

PN(XO)

-

(XO)

for

2

x

x

6

+

If.(X)}n=1,2,3....

Let

be

a

xoI I

-

II

xo

-

(X)

SUP Pn

-

2

2

IIx

for

< -

any


0

by

the

of the

convergence

series

en n=O

Clearly,

continuity.

case.

uniformly

M+P

E




0.

STATIONARYPROBLEMS

CHAPTER2.

66

g'= (a,,)-'h'=

Then,

E bkek

n

+ en

0 in H

--*

I

as

But

oo.

--+

E bke-k

then

H as 1

g'

clearly

and

oo

--+

g'

--+

in

kon

kon

bA;ek for

bk,

coefficients

real

some

hence

k96n

1:

+

en

bk ek

0

=

H,

in

kon

i.

get

we

e.

Thus

contradiction.

a

indeed

coefficients

an

linear

continuous

are

func-

in H.

tionals

00

Let

E an(en

F

hn)

-

and

F

Uf.

=

The operator

U is

linear

and

is

it

n=O

everywhere

determined

BR(O)

If

=

E H

11f1l

:

H.

in

R}


0 such

exists

E

that

a

2

M211fJ12




e

exists

number

a

N> 0

large

so

F,

that

I I en

hn

-




a

E

x

solution

as

A, x))

and for

exists

for

[u'(a,

=

all

0 for

increases

2

0 and

>

A > 0 there

any

values

these

whole

>

F(u'(a,

-

(11.4.6),(11.4.7)

n

u(a, A, x)

for

A > 0 satisfies

a

In

integer

time,

A, x)

problem

of the

that

G(A, A)

Au'(a,

+

arbitrary

an

theorem

comparison A.

solution fix

x)]'

all x

[0, 1].

E

x

be

can

con-

Ju(a, A, x) I

satisfies

the

0

--+

equa-

tion

where

the

A > 0 is of the

problem

values

A

A, be the

most

> n

has

0

A

than

more

above

arguments

by

above

arguments.

the in

(0, 1)

as

dependence

a

> 0

has

to the

roots

continuous

of values

set

(11.4.6),(11.4.7)

problem

According

A,(a)

E

(0, 1),

with respect to large uniformly theorem the solution by the comparison

Hence,

(11.4.6),(11.4.7)

x

arbitrary

> 0 is

large.

0,

n

(0, 1)

in

roots

for

E

x

[0, 1]

if

u(a, A, x)

sufficiently

large

> 0.

Let of the

c(A, x)

function

sufficiently

=

function

theorem

such that

at

least

the

set

The

(n

for each of them the +

1)

roots

as

A,, is nonempty.

corresponding

of the

argument u' X (a,

a

A,, (a)

Let

x

of =

x

because

u(a, A, x) E (0, 1).

A,,.

inf

u(a, A.,'(a),

solution

A, xo) :

solution

function

x)

otherwise,

Then has

at

due to the

0, there (a, A, xo) values must exist A < Xn (a) belonging to A,. as a By analogy u (a, A,, (a), x) regarded function of the argument at least in (0, 1) and u(a, A,'(a), x has n roots 0 because 1) in the opposite the solutions, case to A E An sufficiently close to An(a), corresponding must have at most n roots in (0, 1). Let us prove the uniqueness of the above value A A,,(a), for which the solution u(a, An(a), x) of the problem has precisely in the interval n roots (11.4.6),(11.4.7) the condition x E (0, 0. that there exists 1) and satisfies Suppose u(a, An(a), 1) A' 54 A,,(a) these conditions. satisfying Using the autonomy of equation (11.4-6) and and

since

0 if

u

=

=

=

=

its can

invariance

easily

with prove

that

respect

to

the

changes

of variables

x

---+

c

-

x

and

u

--+

-u,

one

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

(a) for odd with

respect

(b)

the

point

2.

1);

is

the

xo

is

strictly

on

any

(c) u(a, A, x) minimal

ul(x)

exists

the

minimal

value

of the

Ul(X)=h U2(X) byU2(X), the identities

for

x

all

from

2(n+1)

U2(X)

right

a

E

x

(0,

such

x

that

x,

each

that

also

for

point

1

x

f Ul(X)U2(X)

[f(U2(X))

=

where

x,

result

U2(X)

if

2(n+1)

ul(x),

for

the

obtained

[0, Y],

segment

be the

1

=

written

subtracting the

over

Y

let

clear

is

T > 0

or

a

and

it

Let

and

achieves

An(a)

>

0.

(11.4-6),

.

the

and

(11.4.8),

of

view

) Multiplying equation for U2(X), by ul(x),

integrating

to

Section

[0, 1]

E

A'

ul(x)=

that

such

2(n+1)

and

solution;

u(a, An(a), x)

definiteness

the

of the

( 0,

+ 2xi

x

of

respect

also

see

is

u(x)

this

with

increasing

half-neighborhood E

[0, 1];

C

of

2

of two solutions

monotonically

is

'+ 2(n+1) written

equation, another

C

in

1-

"+'2

x0

b]

solution

[XO, X2] and even [0, 1] (on this subject,

Then,

x

+

x0

solution

on

u(a,A',x).

=

b,

this

u.

(a)-(c) (11.4.6),(11.4.7), Let

-

arbitrary

an

extremum

and

any

function

the

argument

same

one

of

and

=

that

2xi) for

properties

point

u(a, An(a), x) ul(x)> U2(X) in

-

+

x

root

problem

of the

[xo

u(a, A,

=

of

point

on

of

< X2

x,

[xi, xo] b, xo + b]

[xo

segment

roots

(11.4.6),

of equation

arbitrary

unique

a

segment

solution

an

nearest

monotone

from the

maximum at

xo

on

two

positive

It follows

u(a, A', x)

point

there

arbitrary

an

arbitrary

(11.4.6)

solution

of

xo

the

to

between

equation this

root

any

69

we

get:

7

0 >

f(U2(X)) 2

_

1

-

An (a) + A'] dx.

(11.4.9)

0

by

since

But

inequality

suppositionUl(X)

our

positive,

is

i.

Lemma 11.4.9

We 11.4.9.

is

keep

Let

we

e.

An(a)

The property

get

(r(n

!

U2(X)

> a

+

for

E

x

(0, Y)

the

,

right-hand

side

of this

contradiction.

1))2

follows

from the

theorem.

comparison

Thus,

proved.0 the

A An(a) for the value of the parameter x)= Un(a, x). By Lemma 11.4.9 these definitions

notation

u,,(a,An(a),

from

Lemma correct

are

1

An (a)

and

0 for

>

any

a

>

0 and

integer

n

Let

> 0.

also

an

f Un' (a, x)

(a)

dx.

0

Lemma 11.4.10 continuous

on

Proof.

contrary, the

U2(X) each

i.

e.

properties increases

of them.

a,

that

>

An (a,)

(a)-(c) on

a2

integer

any

half-line

the

Let

For


0

An(a)

is

nondecreasing

and

> 0.

a

>

n

Let

proof

[0, have

that

prove ul

(x)

=

of Lemma 1-

2(n+1)

u',(xi)

]

and >

x

un

11.4.9, =

U2(X2)

An(al) (a,, x) each

! and

An(a2)U2 (X)

of the

Suppose =

functions

Un

the

(a2 x). By ul(x) and ,

1is the point of maximum of 2(n+1) for any y > 0 for which there exist

(0,

X1) X2 E x

STATIONARYPROBLEMS

CHAPTER2.

70

-(n+jj)-

(0,

E

we

satisfying

2(n+l)

I

I

(xi)

ul

Proceeding

U2

as

(X2)

==

Therefore,

y.

(x)

ul

(11.4.9)

inequality

deriving

when

and

> U2

taking I

(X)

for

all I

-

x

=

2(n+l)

get 2

2(X))

U1(X)U2(X)[f(U

0

f(U2(X)) 2

_

1

n(aj)

-

+

An(a2)]dx,

0

which

obviously

is

proved

is

that

Let that

the

there

the

exists

0 such

>

ao

nondecreasing

is

of the

continuity that

definiteness

the

by analogy). for

each

Then,

1) d(a) 2) u, (a, x)

+0

-+

3)

Un

first

as

one

sufficiently

> ao

a

the

ao+O

as

> un

(a, d(a))

a

inequality

take

easily

verify,

close

to

(ao, x)

E

x

(ao, d(a))

Un

:--:

for

the

An(a). An(ao)

>

(the

place

is

positive.

a

Suppose

the

second

case

0 such

>

it

i.

e.

contrary,

An(ao).




An(a)

liM a-ao-O

d(a)

exists

here

half-line

or

follows

it

there

side

(11.4.8)

and

that

that

0;

+

ao

-

An(a)

can

ao

on

function

liM a

for

right-hand

the

because

An(a)

function

prove

us

contradiction

a

(0, d(a)); -A-

and

< -- - u, dx

(a, d(a))

dx un

(ao, d(a)).

of the point x d(a) Un(a, x) < Un(ao, x) in a right half-neighborhood follows from it Then, as above, equality d(a))). (because 0,xx(a, d(a)) < u",xjao, close to ao and for all that Un(a, x) < Un(ao, x) for all a > ao sufficiently (11.4.8) x E (d(a), ; '-+ ). Using the identity similar to (11.4.9) with the integral over the 2(n+l) a contradiction. we get So, the function ' n(a) is continuous, segment [d(a), 2(n+l) Therefore

=

n

n

and Lemma 11.4.10

an(a)

Lemma 11.4.11 a

+0

a

Proof.-

The

(see (11.4.6),(11.4.7)

Lemma

0

as

a

liM a

+0

(0,

all 2(n+l)

and also

addition,

x

us

in

an(a)

x

G

an

(a)

12[

A.

follows

from the

dependence

continuous a

function

continuous

on

the

half-line

+oo.

=

function

parameters

that

prove

such

2(n+l) that

our

lim a-+oo

1

f(U2)U by

an(a)

lim

and from the the

increasing

Further,

[0, 1] (see

the

as

proof

continuity

of solutions

it

proved

is

of

) n(a)

Of

of the

problem

un(a,

earlier,

10),

Lemma IIA.

x)

--->

therefore

0.

(0,

(-=

strictly

a-+oo

11.4.10)

=

a

of the

continuity

+0 uniformly

an(a)

is

0 and

=

on

--*

Let

for

an(a)

liM

0,

>

proved.0

is

-

).

Indeed,

u"

n,

xx

An(a)u supposition

if

(a, xo) is

a

=

we

> 0.

First

+oo.

f (u 2(a, xo))

all,

we

u',x (a, xo) function

on

n

-

observe

0

it

as

the

An (a)Un (a, xo)

that

then

contrary, >

nondecreasing n

the

suppose

But

of

0.

:!'

0

xo

E

n

there

was

half-line >

u",xx (a, x) exists

indicated u

Hence,

E

earlier

[0, +oo); we

get

in

that

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

Un',.,(a, all

x) E(01

x

0 for

>

all

E

x

(xo, 11,

i.

lim

a,,(a)

e.

have

we

So, un,,(a,

contradiction.

a

71

x)

2(n'+I))'

Now,

to

that

prove

+oo

a

suffices

it

+oo,

=

1

u,,(a,

show that

to

2(n+l)

it is proved as (because, above, Un(a, x) is a concave function on the +oo the following segment [0, n+1 T' ]). Suppose that for a sequence ak C Consider the < two cases: place: :5 +oo. Un(ak7 2(n+l) separately following ; '+- ) < +oo. +oo and B. f (+oo) f (+oo) +oo

as

for

< 0

a

of

+oo

--+

x

takes

-+

A.

=

f (+oo)

A. Let we

would

(11.4.8)

get the

functions

Therefore,

An(a) u,(ak, k-oo

Then

+oo.

=

from

Un

(ak x) satisfy +

n,xx

where

9k

ing

standard

to

+oo

-*

Un(ak7 X)

for

all

B.

:!

an(a)

lim +oo

It

for

all

E

x

that

Un

Then,

(r(n

equation

it

is

We also

note ==

Let

the

1))2

e.

we

that

an(a)

is

x

have

E

(0,

)

the

equations

Hence,

2(n+l)

each

of the

; '+-

(0,

in

root

a

But

+oo.

get

by

hand,

one




this

arguments

in < a2

Suppose

the

Un(a2lx)

for

interval

such

(a2 X) right xj) un,x(a27 Xl)- In An(a2), hence, in view in

7

half-

a

=


Un

An(al) e.

a,,

takes

the

on

0 < a,

:

un(ai,x)

(a,, x)

Un

(11.4.8) written for x xi, we > (11.4.6), u",x, ,(a,, xi) u",xx(a2;X1)i that u,,(al, x) > un(a21 X) in a right .

any

that

such

a2

show that

us

for

x) :5 un(a21 X)


x,

Suppose

a

suffices

To prove

n

proved

; '-+

2(n+l)

theorem,

comparison

on

+

i.

+oo,

E

contradiction).

a

e.

(0,

x

otherwise

01

-=

to

k must

i.

impossible.

equality

of

An(al)

the

X)

respect

numbers

A is

+

prove

; '-+

E (0, (a,, xi)

of

with

on

'-+ ) 2(n+l) e.

neighborhood view

+oo.

=

to

(0,

i. x




xi.

X1)

if

An(a2)X2

X2

(n+l)

when

deriving

be the if there

is

inequality

no

such

(11.4.9)

a

E

point with

(xj, in

the

such

2(n+l)

(xi,

1

2(n+l)

integration

that

Un(aj) x)

). Repeating over

the

the

=

Un(a27 X)

procedure

segment

[XI, X21

or

used ,

we

CHAPTER2.

72

STATIONARYPROBLEMS

get: 272

I

0 >

un(al,

X)[f(U2(a,,

x)u,,(a2,

x))

n

f(U2 (a2 X))

_

An (a,)

-

,

n

An (a2)] dx;

+

X1

addition,

in

An(a2).

by

Thus

increasing

the

above

get

we

function

of the

problem

the

interval

integer

any

the

inequalities

following

Proof.

Suppose x

E I

for

all

x

by

the

[X I

i

2(n+2)

21

X

we

,

as

suffices

following < X2

Un+I(X2)

Un

7

(X)) satisfying

has

precisely

+1

of

in

roots

n

function

the

Un;

It

to

Un+I(X)

strictly

be that

un+,(x)

f (u' (x))

>

An for

-

n

exist

two

with

B

increases

E

x

can

and

also

the

integral

Un(X) hence

over

that

occur.

un+,(x) Un+I(X2)

that

!

I,

E I such

xo

A and

such

Un(X2)7

point

a

cases

(11.4.10).

inequality

prove

cannot

2(n+2)

>

(11.4.10)

...

solution

(11.4.9)

deriving


- An+1 for

that

for

u'

In view

monotonically

a

place:

A2




a

the

is

it

and such that W-4.1)-(H.4.3) (0, 1) and this pair is unique

addition,

in

So,

argument

For

have

we

contradiction.

a

Lemma 11.4.12

the

arguments,

> =

the

un(x) Un(X2)

if if

segment

get the inequality X2

0 >

i

2

Un+l(X)Un(X)[f(U

n

+1

(X))

_

f(U2 (X))

An+1

-

n

A,,]dx,

+

X1

where,

as

Thus,

we

proof

in the

get

a

of Lemma

contradiction,

11.4.11,

and the

the

A is

case

inequality

strict

place

takes

if

An+,

An

==

impossible.

un+-I(x) < u,(x) for all x E I. Observe that Un+I(X) < Un(X) for some 1 have u'( we would (because otherwise 0). Further, un+,(x) :! ' Un(X) 2(n+2) ) is obvious Let x that E [0, on a then us visually (it picture). prove n+1

B. Let xo E I

for

Un+1

the

equal


Y.

k

v(x) Repeat

procedure

this

for

of the

properties

I v (x) I

lies

that

such

k

each

(a)-(c)

under

the

Jv(x)j




Un(bn7 X))

G(, n(bn),

(r(n

!

dn

large

sufficiently

all

we

would

for

all

get that

in

large

sufficiently

So,

contradiction.

X)

n

-

F(u'(bni

X)),

n

(11.4.11)

that

implies

for

1))'u'(bn7

+

the

numbers

(11.4.11)

Indeed,

n.

if

right-hand

the

numbers

< 20

n

and

for

side all

boundedness

uniform

we

suppose

is

than

greater

satisfying

x

of the

this

that

is

the

not

left-hand

20, jUn(bn,x)j jUn(bn7X)Jn=0'1,2.... =

sequence

then

so,

one

i.

e.

a

is

proved. Let

hn(X)

=

properties

us

show

sin[7r(n (a)-(c)

10

that +

from

jun(bni -)IL2(0,1) 1)x] and observe the

proof -h"

> I

that

of Lemma

=

n

/-tnhn7

for

all

u',x(bn) n

11.4.9, X

G

large

sufficiently

0)

=

u'n,,,(bn) (0, 1),

h' n (0)

1)

=

numbers

and,

hn (1). '

in

view

n.

Set

of the

We have:

CHAPTER2.

74

h,(O) where

un

(11.4.6)

(7r (n

=

for

the

+

1))'.

Therefore,

w,,(x)

functions

"+ -Wn

Wn(O) where

family

the

boundedness

x)

Wn(l)

family

C,

constant

a

Multiplying

this

[0, 1]

segment

independent

> 0

of

applying

and the

place

by

the

by parts,

integration

because

12

An I Wn L2(0,1)

_JW/'xJ2

L2 (0,I)

n

-

-

all fact

for

C2

numbers and

all

independent

0 is n,

we

f




n

-)

0, 1, 2,....

=

C3

exists

C3 and

u,,,(bn,

functions

the

for

as

show that

n

2W2(X)dx

X

n.

enIL2(0,1)

shall

for

uniformly

is

-

> I

(11.4.13)

using

Jun

equality

In view of this

n.

1

that

fUn(*)In=0,1,2....

functions

For this

of

the

over

0

JUn(bn, *)IL2(0,1)

have

Lemma IIA.11

sufficiently It

the

>

equality

get

0

where

theorem.

comparison

obtained

we

1 xw',,,(x)Wn(x)dx

2

uniform

estimate

the

1




11.4.5,

Theorem

To prove

function

( n+1 t )

u,,

let

aim,

L2(0, 1)

space

independence

linear

the

this

For the

in

STATIONARYPROBLEMS

for

us

each

inte-

Fourier

the

in

in

series:

00

Ea ne-k(') k

n+1

where

ank

coefficients.

real

are

Then

k=O 00

Un(*)

r

=

b,,,e,,,(-)

(11.4.17)

n

M=0

in the k

=

0, 1, 2,

Then,

=

holds

valid

is

each

and,

n

in view of

real

exist

(n+1,

L2

our

Also,

space

=

a

0

points

r

n+1

the

1,

==

obviously

bn0

because

the

where

n+1

(X),

e(n+l)(k+l)-l

(11.4.17)

equality

too,

where

> 0

n

the

the functions

points,

)

r+1 n+1

r

(01

of Lemma 11.4.9

precisely

are

to these

respect

spaces

proof

the

shows that

verification

for all

,

2,

also

Therefore,

n.

bnn-1 functions

0 for

=

U.

and e.

sign everywhere. that

suppose

us

which

L2(0, 1). acceptation, bnn

of the

sense

same

Let

there

the

of the

from

roots

m: (n+l)(k+l)-i place in the space L2

0 if

=

takes

(a)-(c)

to its

respect odd with

are

obviously

properties

the direct

of each

sense

in

of the

are

is odd with

bn,,

nand k

a

=

(11.4.17)

equality of the

0, 1, 2,...,

in the

bn (n+l)(k+l)-l

where

and since

n

k

,

view

un(x)

where

it

in

1, 2,

1)

Indeed,

....

since

function r

L2 (0)

space

the

coefficients

fUn}n=0,1,2,...

system

Cn,

n

=

0, 1, 2,...,

independent.

linearly

is not

equal

all

not

to

such

zero

Then,

that

00

ECnUn

(11.4.18)

0

=

n=O

in the

b1jcj

L2(0, 1). Let (11.4.18)

space

Multiply =

But

0.

contradiction, in the

L2 (07

1)

large

is

radial

solutions

a

proved.

f(02)

method,

for

Independently, existence

jolp-1 =

in

of

a

contained.

presented;

number

N

In

the

the

existence

Thus,

problem

(p

positive

space

M, =A

0 and

that

co

L2 (0) 1) we

as

of the

Theorem

IIA.5

cl-I

In

-

of

view

supposed

cl

=

:

0.

proved,

is

54

0 and cl

(11.4.17)

we

So,

0.

get:

get

we

a

fUn(X)}n=0,1,2....

of functions

system

1)

>

devoted

publications

of

3 and

[96]

the

too.0

remarks.

of the

=

in

independence

linear is

el

above

Additional

There

is

by

proved

is

and the

space

2.5

with

it

as

I > 0 be such

index

the

equality

is

(11.0.1),(11.0.3).

for

the

solution,

[97]

paper

of

based

a

a

solution

f,

function

same

on

methods

refinement with

of

an

I

[71],

a

< p

of the

of the

arbitrary

[71],[96],

papers

paper

existence

of the

questions

In the

In the

considered.

1 < p < 4 the

the

to

< 3

and

N

qualitative

technique given

a

solution =

theory of the

number

variational

proved.

is

3

of

problem

this

by using

positive

existence

proof

a

of

paper

of roots

of

ODEs,

[71] on

is the

77

ADDITIONAL REMARKS.

2.5.

half-line

r

[82]

in

[71] (N

the

the

there

in

bounded

solution

is

i.

satisfies

it

e.

also

applicable

f (0') 0

in

principal achieving

the

W. Strauss

[87]; (non-radial)

mistake

no

nontrivial

=

paper

solutions

that

stating

Till the

101 investigated has

0; sufficient

a

--+

have

reviewed,

Another recent

paper

on

root, for

conditions of roots

case

positive

the

the

half-line

solved

of

is based

a

similar

to are

nonlinearities

for

proved

first

[98];

in

of solutions

of the existence

unsolved

remained

has

aim but

for this

exploited

method.

variational

obtained

by

by

methods

of the

a

the

(11.1.4),(11.1.5)

Cauchy problem

was

theory

qualitative

of

model

is considered

[110]. w-f (0)

> 0

r

of

jxj

an

it

paper,

is

radial

solution

are

obtained

101-1,

=

11.1.2.

supposed with

a

of

results

the

case

f( 02)

One

more

the

an

by using

made

number

that Jim i(Al-00

finite

a

so.

in the

a

of

has

given is not

(02)

Theorem

exists

> 0

f

it

(11-0.1),(11.0.3)

and there a

author

arbitrary

[77,78];

proposed

was

symmetrization

a

the

example

with

In this

=

with

problem

existence

concept

Rabinowitz

the

of the

of

Unfortunately,

solution

a

of P.H. with

the

solutions

radial

positive

on

H1.

from

solutions

oo.

0

>

existence

from results

in the

unique

the

method

existence

of radial

existence

as

number

we

was

of the

r

any

[47].

function

the

directly

now,

half-line

for

two papers

3 and was

(3,5),

p E

ODEapproach

an

completely

proving

jolp-1,

of solutions

the

problem

the

=

:

11.1.2

also

[110].

from

f(02)

3 and

see

his

101P-10.

N

Theorem

latter

0,

=

possesses

A result

0.

of the

r

paper

101P`

=

>

r

for

solutions

methods

[10],

was

[59].

paper

of radial =

half-line

Methods

positive

that

result

f(o')

3 and

=

the

on

ODEapproach,

=

on

N

of this

the

are

solved

point

of the

have

we

an was

solutions

I < P < 5 any

methods

fact,

proof

completely

was

for

that

that

so

with

considered

neighborhood

a

OIP-'

f (0')

3 and

(11.0.1),(11.0.3)

problem

In

of roots

existence

in

follows

roots

N

Shekhter,

way of

nonnegative

g(o)

p

4 < p < 5 the

for

This

in

with

f(02)

paper

Another

+oo

proved

is

5 in the

for

=

problem but

paper

sign

in the

of the

value

B.L.

this

N

the

(11.0.1),(11.0.3).

exploited

problem

by

of ODEs

in

alternative

to

the

on zero

The indicated

on

are

whether 0.

=

derivative

number

we

In the

time.

r

(11.0.1),(11.0.3)

with

result

first

its

sense

a

(11.0.1),(11.0.3) long

proved

been

the

framework

In the

by

there

for

5,




0

we

the

on

is

Also,

[119]

in

way consists

that

in

exploited in

upper

of a

general

triangular

even

for

the

in

basis

a

analog

attempt

[118].

to

properties

and

all

system

11.4.5 use

the we

of the

elements of functions

of its

on

expansions note

that

where

JUn}n=0,1,2....

of

s

that

the

< so

and

on

the

is considered.

Bary

the

(11.4.17) b,',

(the

diagonal to

theorem. for

example

an

coefficients

principal

probproblem

proved

problem

based

which,

transform

Fourier

eigenvalue

However,

the

in

of the

of Theorem

proof an

is

an

is

systems

boundary-value

[1181 H'(0, 1)

IIA.5

Similar their

is

of

monograph errors

for

first

discussion

[119].

it

paper

nonlin-

in the

was

essential in

a

of the

small

of this

basis

a

[118],

In

nonlinear

a

being

In the

denumerable,

eigenfunctions presented is

property

results

interesting

of Theorem

proof

published

knowledge We only

thorough

approach

the

eigen-

completeness

the

contains

of

direction.

some

more

The first

of

page

111.

best

arising Bary theorem

the

are

that

under

A

proof

the

is considered.

which

approach

[39];

[6].

in

[115-117].

in

[6].

in

in

paper

The

corrections

presented

are

proved

contained

in that

parameter)

over

and

approach

the

results

L2

have

shows

[5]

proved.

is

These

constant.

is (bn )n,m=0,1,2,... insufficient are zero) L2(0) 1)M

from

solutions,

natural

This paper

in

However,

spectral

negative

half-line

aim.

in

of its a

proof)

corresponding

eigenfunctions system

a

is

equation,

operator

problem,

theorem

this

in

note

system

author's

[62] containing author [63] where

same

linear

be corrected.

can

and

lems

[114].

in

the

nonlinear

a

results

no

on

of the

papers

wo- 101"o,

=--

error

properties to the

nonlinearity in the

We also

principal

a

uniqueness

the

(for g(o)

exists).

problem,

almost

there

of positive

proved

is

proved

it

basis

on

for

Makhmudov

by

paper

differs

theorem

2.4

are

a

this

around

questions this

by

(if

contains

Liouville-type there

A.P.

of

of

(without

announced

to

and

and the

perturbations

ear

field

eigenelements

of

[83]

in

as

unique

from Section

Sturm-

monograph

subject

this

on

new

a

the

mention

presented

the result

nonlinear

a

quite

is

always

is

particular,

it is

the

uniqueness

solution

positive

[54]:

in

knowledge

our

concerns

the

on

In

of the

of

chapter)

results

(11.0.1),(11.0.3).

is obtained

solution

only

best

introduced

proof.

this

in

are

uniqueness

result

result

similar

Concerning this

the

-

of

there

problem

of the

solutions

considered

literature,

In the

==

(not

problem

The second

methods

to the

the

to

of the

variant

our

close

are

However,

precisely

containing

paper

no

11.2.1

Theorem

proving

our

be

this from

matrix are

complete

non-

in

Chapter

3

Stability

of solutions

it

chapter,

this

In

noted

is

Sobolev

JxJ

with

respect

the

to

distance

stability

the

study

to

of functions

spaces

the

p in

solitary

of

we

(for

definitions

named

a

solitary

see

u(x, t),

wave

Introduction

where

(x, t)

u

O(w,

=

=

x

x

mathematical

pioneer

by

paper

field.

investigations

in the

like

vanishing

solutions

respect like

the

to

O(x -Lot) can

or

such

that

the

i.

the

"forms

if

Thus, the close

two sufficient

which

we

u(x

sense

of the for

conditions

suppose

t)

-r,

graphs"

as

to

x

be

--+

solution

u

p for

O(x

-

graph

wt)

the oo

of

stability for

one-dimensional,

P.E. Zhidkov: LNM 1756, pp. 79 - 104, 2001 © Springer-Verlag Berlin Heidelberg 2001

our

of usual

as

is

of

is

an

arbitrary

perturbed

---->

with

i.

respect

e.

stable,

to

the

Simultaneously

with

N

=

1.

wave

of functions 'T

"almost

then

for

solution

distance we

close

functions

these

almost

are

In Section

coincide".

KdVE.

spaces

u(x, t)

as

and

clearly

x

soliton-

a

travelling

translation

a

functions

O(x -wt) oo JxJ

p "almost

distance

0, then

of

with

u

=

KdVE are

standard

of the

t >

t > 0 there

of the functions

vanishing

visually:

of distances

for this

and

(x, t)

some

of soliton-

form a

the

numerous

f (u)

if

point

a

waves,

stability

the

of

stability

the

a

0(+oo).

further

the

KdVE

is

xo

=

KdVE with

understood

easily

sense

however,

and of the

graph

a

in the

solution

vanishing

solutions

other

of

of the

in the

to it

stability

be

distance

of the

spaces,

graphs

of its

forms

this

and

oo

-+

each

soliton-like

a

x

sense

Sobolev

Lebesgue e.

to

of the

can

proved

has

standard

oo

p; he called

as

in the

close

be not

x

origin author

the

the

that

E R and

x

0,

=

initiated

which

d for

recall

of solitary

stability

the

to

the

was

paper,

--->

terminology

vanishing other

to each

In this as

distance

This

solution.

[7]

Benjamin

T.B.

devoted

literature

as

of the

all

x

In the

vanishing to

case

-

are or

We also

Section

or

NLSE

Lebesgue

KdVE and

of the

case

as

waves

3.1). wt) in the and u(x, t) e'wto(w, x) for the NLSE, a kink if 0' (w, x) = 0 for if there is a unique xO E R such that solution soliton-like 0' (LO, xo) and the of x function of a of extremurn 0(-oo) argument O(Lo, x) as

NLSE

of p and d

As

waves.

KdVE and

of the

waves

of standard

and it is natural

spaces oo

-4

respect

solitary

usually

distances

to

of solitary

stability

of the

questions

consider

Introduction,

the

in

with

unstable

shall

we

3.1

as

r(t)

=

G R

identical, coincide". each

t

> 0

sufficiently we

consider

p of soliton-like

study

the

NLSE

STABILITY OF SOLUTIONS

CHAPTER3.

80

As it behavior

noted

is

as

x

Section

3.2

p under

assumptions

we

x

In two

stability

cases

prove

solitary

of

stability

of

stability

a

the

I

can

=

and kinks. the

to

respect

In

distance

defining

NLSE nonvanishing

interesting

new

vanishing

solutions

stability

the

of the

waves a

type.

solutions

of soliton-like

with

N

on

type.

stability

We begin

considered. one-dimensional

NLSE with

KdVE with

of soliton-like

the

section,

In this

assumptions solutions

soliton-like

are

of the

of kinks

a

we

Stability

3.1

these

natural

and the

of solutions

x

types:

consider

we

in

two

general

of

3.3

oo.

--+

only

the

prove

In Section as

of

waves

KdVE under

the

2.1,

oo of derivatives

--+

solitary

have

Section

in

of such

as

x

solutions

for

be

will

00

-+

KdVE and

the

NLSE. Let

p(u, v)

=

I Ju(-)

inf TER

v(-

-

r)

-

H1

E

v

u,

and

d(u, v) where

H1 is the real

prove

that First.

we

in the

space

in each

case

I Ju(-)

inf yE[0,21r]

'rER,

first

greatest

the

Cauchy problem +

ut

lower

f (u)ux

remark

for

We also

u(.,t)

let

[0, a) I Ju(-, t) 111

continued

solution >

Let

C for

1), (111.

1.

=

1.

2)

C,

0 be

in

t C-

in

I,

the

(uo

> 0

then

[0,

a

interval

an a

twice

a

constant,

existing

0,

2 E H

(x);

(111.1.2)

H2-solution

from Theorem

holds

1.1.3

continuously

differentiable

differentiable

function

twice

(M.1-1),(X.1-2) if

exists

a

=

0).

that

exists >

in

If there

6 > 0 such

8) (resp. there of time [0, 6), 6

+

1.

R,

x, t C:

continuously

twice

0, if

in

in view

an

interval

case

of the

a

=

solution

the

H2 -solution

a

=

can

be

embedding

> 0

of

time

that

u(-, t)

can

be

of the problem

0). u(., t) be a H2_ [0, a) or I [0, a], considered by analogy).

function of time

0

C

and

such

interval

an

exists

a

differentiable

continuously

1.

easily

can

here.

problem

there

a

One

second.

KdVE:

the

arbitrary

an

(Ill. 1), (111. 2) < C (the IV: Jju(-,t)jjj

problem a

of a

interval

f (-) be

0, bounded >

all

the

Let

of the

[0, a],

be

uo

of

result

f(.)

Let

H'-solution

I

onto

Proof.

a

a

or




be

-

taken

uniqueness b

exists

H2-solution

a

f (.)

for

obviously,

Then

due to the

there

fl(.)

C and let




u(x, t)

solution

t > 0 and

half-line

entire

[2,4);

the

NLSE:

0,

X,t

E

Then,

0 such

half-line

and

uo

of

this

H'

E

0 and

>

formulating 111. 1. 1 if

in

of the

>

according one

a

proves

suitable

parameter

u(x, t)

solution

method

following to which or

result

E

f (.)

supposes

sense

of

We consider on

connected

2 is

v

satisfy-

the problem for all t > 0

solutions. the

v

V(x,t)

solution

u(x, t) of t

of soliton-like

However, values

any

uniqueness

Proposition

(111.1.1),(111.1.2)

this

concentration-compactness

stability When

call

for

H'-solution

entire

existence

we

that

requirement

other

(111.1.3) soliton-like

=

H'.

of the

0.

R,

f(JU12)U in equation e'wtO(x) be a U(x, t)

let

the

function.

problem

b

uo(x).

the

onto

of the

Jul',

the

for

corresponding

application

investigation f (u)

differentiable

continuously

c (the L2.4).

an

of Theorem

sumptions


0 there

corresponding

the onto

of

solution

local

unique

a

t > 0.

iUt + AU +

Definition

then

Cauchy problem

the

if for

be continued

can

all

has

soliton-like

a

stable

O(w, .))

H and

the

one

w

solution

2

(111.1.1),(111.1.2)

problem

the

wt),

-

call

we

H2

E

uo

0(w,

let

Then

if

that,

such

of

for

111.1.1

sition

with is

a

the as-

twice

the

(for example, (0, 4), then all

local in the

the

from

arguments

Theorem soliton-like

Let

O(x,t) from ftn}n=1,2,3.... as

n

111.1.4

is

(n) uo

=

Let

the

=

has

solution

a

clearly

and

boundary

> 0 and

w

E(u)

functional

is defined

I V

(the parameter

w

this

boundary-value

belongs

01

two functions

A,

we

problem n --+

n

JUn(*) tn) 122 a

02

the

P(U(n), 01t=0)

0

_

0

Cauchy

of the

Theorem

to

following

for a

A,

(11. 0 (up

family =

1

-3)

11.3.1, with

equation

is

shown

it

some

2.1,

of Section

2.1, 0 that

solution

positive

of the

values

of

y,,)

any

according minimization

our

+

for

parameter

Therefore,

wn(-

that

102 122

10, 122 :

since

different

C R such

in Section

beginning

translation). fWn}n=1,2,3....

a

sequence

fYn}n=1,2,3....

sequence

it

the

Further,

with

to

0

translations,

0.

>

=

as

As at

up to

minimizing

any

U(00)

solutions). some

A and

=

our

we

take

--+

A

as

vn n

--+

=

oo

-+

e.

in the on

the

a

(-,tn) .Since JU.(*,tn)12 and E(Un('7 0)) Aun

for

sequence

n

-+

contradiction.

111-1.5

For the

papers

by

stability

the

above

C R such as

fUn}n=1,2,3,...

of solutions

sequence

jYrjn=1,2,3.... 0 p(vn, 0)

oo7 i.

fact,

results

sequence

a

1.1.3.

According

2.3.

otherwise,

unique,

a

with

from the

sequence

--+

0,

>

E(u),

LOU,

=

because

has

(11.1.3)

and

to

Remark In

problem

exists

minimizing

Hence, n

solution

0

--+

in H'

as

oo.

any

a

nontrivial

11.3.1,

there

Return For

positive

no

A,

that

get

Theorem

to

be

has

family

the

to

jul"u

+ 1

must

problem

above

is stable.

problem

satisfies

solution

its

every

E

that

H2-solutions

are

3)

1.

conditions

U" +

the

in Section

(11.

exist

by Theorem

given

inf JU12=A>o 2

uEHI,

the

Un(Xi t)

A > 0 the

any

corresponding

the

there

minimization

IA where

that

from H2such

where

e

for

family

the

Then,

0

P(Un(*i tn)i Olt=tn) > with uo (111.1.1),(111.1.2) lo(.,t)12 2 A > 0. Consider

problem

Suppose

ju(n)}n=1,2,3....

sequence

a

Then,

1) from

1.

stable.

not

[2,4).

E

v

(Iff.

A > 0.

(11.1.3)

R+ and

C

arbitrary

an

where

Kd VE

the

hold.

below

Jul"

=

and

oo

-+

fix

family

the

f(u)

Let

O(x, t) of

us

theorem

of the

proof

solution

Proof.

is

STABILITY OF SOLUTIONS

CHAPTER3.

82

with

proofs

+

lun (-,tn)

12

time,

based

on

1,

111.1.4

Lions the

is

[23]

we

1.1.3

is clear

it

that

therefore in

converges -->

111.1.4

Theorem and P.L.

Theorem

problem,

y.,,)

A

Theorem

Thus, first

tn))j

v,,(.

that

T. Cazenave

E(un(-,

minimization

Since

oo.

according

to

=

problem

of the

get

H' to 0 P (Un

('I tn)

JUn(.,0)12

2

f vn},,=,

2,3....

there as

exists

n

oo.

--+

0

7

as

proved.0 was

and

concentration-

proved by

P.L.

in the

paper

Lions

compactness

[57,58]

[1011. the

method

83

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

for

investigated

in these

applying

this

method

Now we consider

the

"Q- criterion"

possibility

ishing

of

jxj

as

"Q-criterion"

name

often

by

P and

stability of the

KdVE

in the

the

close

used

waves.

to

conser-

of the

condition

in the

law Eo

conservation

the

van-

necessary.

the

in which

here

rename

variable,

example

solutions

stability

KdVE)

We also

a

is

solitary

of

literature

physical

of the

case

by Q.

denoted

been

has

(E0

NLSE

law P of the

vation

originates

the

an

of soliton-like

of the

conditions

spatial

with

stability

of the

stability

of the

from

on

particular,

in

the

illustrate

to

problem

to the

sufficient

gives

which

oo

--+

only

wanted

we

considered:

are

depending

coefficients

Here

papers.

kind

general

more

NLSE, admitting

multidimensional

The

essentially

NLSE of the

a

U

that,

recall

the

in

7(u)

KdVE,

the

on

case

=

f f (p)dp

and

0 U

F(u)

f 7(p)dp.

=

0

Theorem

111.1.6

plex argument

f(JU12 )u NLSE (111.

Let

for the

u

differentiable

continuously

be

a

1.

3)

with

(f (u)

N= I

of the

function

differ-

continuously

be twice

com-

'r

entiable

for

KdVE).

the

f f (s)ds

F(r)

also

Let

and let

there

wob 2

0 and F (02)

exist

wo E

R and b > 0

0

f (0)

that

such

-

0, f (b 2)

wo


L,,o

-

wo < 0, 7(b) (0, b) (resp., f (0) the KdVE). As for 0 E (0, b) for

0

G

there

tions

U(x, t) 0(w,

exists

O(wo,

x

E

-

wot) for

it

-

0, F(b)

>

proved

is

U(x, t)

solution

soliton-like

a

wob

-

-

F(b 2)

0,

2

!Ib

-

2

e'woto(wo,

=

as

x) for

these

0




In

'.

A

N

[15].

in

v(x, t)

the

in the

NLSE if p

We

(111. 1.3).

prove

the

if p E

proved

of variables

that

0 if

By analogy,

NLSE.

NLSE

2,

>

from

results

0 is satisfied

>

place

the

can

and

condition.

dw

consider

of the

12 ]dx.

this

takes

the

1)

=

dp(o)

for

> 4

making

12 +wolh(x)

under

instability

solution

a

N

condition

the

We first

wo > 0

u(x, t)

where

is stable

when p

111.1.6.

(for

111.1.6

the

instability

of Theorem

11hl 121

=

show that

and, respectively,

0

=

if necessary, that

we

U(x, t) I u IP

f (u)

KdVE with

follows,

similar

f (0)

According 1) the solution

Theorem

to

Chapter dw

RN

0 < p < -1. N

solu-

=

also

accept

lower

bound

we

greatest

that accept 6- f (O)tu(x, t)

-00

in the

expression

remark u

that

in the

real).

d( U, u)

for

generally

is achieved

and

-r

e-'(-/W+w0t)u(-

form

Differentiating

7

t)

r(t),

+

the

expression

T(t)

7(t)

(we a perturbed solution unique). 0 + h(x, t) where h(x, t) v + iw (v, w are with to and T d(U, u) respect 7, we get at

some

T

=

E R and

E R

We represent

not

are

=

for

=

00

I

V[f(02)

+

202f/(02)]

O'dx

=

X

(111.1.5)

0,

00

00

j

Wof(02 )dx

0.

=

(111.1-6)

00

Further, AE+

WO

dx!

AP

a(s)

where d2

2

=

o(s) f(02).

=

+ W0

_

as

Lemma 111. 1.8

s

--

There

2

[P(O+h)-P(O)]

+0 and

C

exists

+

dX2

0 such

1f(L+v,v)+(L_w,w)}+a(jjhj

12)

2 d2

L+

>

>

that

W 0

(L-

-

w,

[f(02)

w)

+202f/(02)]

CIIWI12 for

all

WE

(111.1.6).

satisfying Proof. of the

Wo

E(u)-E(O)+

operator

Let

w

L_

=

ao

+ wi-

where

corresponding

(L-w,w)

to

=

(0, wi-) the

=

0.

eigenvalue

Then, A,

=

since

0

0,

have

(L_w.L,w_j -) ! A2 JU,_L12 21

we

is the

eigenfunction

H'

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

because,

and,

hence,

is

[28]).

see

0

since

is

I

(on

L-

operator

0 is minimal

=

this

subject,

+IW_L0f(02)dX

02f(02 )dx

0.

=

(012

W002 )dx

+

0,

>

get:

we

-00

-00

Jal IW12

C11WI12,

!5

and of

independent

C2

>

sup

I f(02) 1)

0

A,

00

00

f 02f(02 )dx=f

hence

1 2 is positive

L-;

eigenvalue

of the

spectrum

of

spectrum

by (111.1.6):

Then,

a

Since

positive

corresponding

the

of the

point

isolated

an

of the

function,

positive

a

bound

lower

2 is the greatest

where

85

(111-1.8)

therefore

H'.

E

w

CIW112i

:5

(L-w,w)

implies

k

For

independent

0

>

C21 W2 12 with

! of

some

(M

have

w we

X

(L-

I

w)

w,

(1W 1122 )k -+1 + 1

1

W01W122)

+

2f(02 )dx

W

00

Thus,

(L-w, w) ! for k

k >

>

sufficiently

0

0 the

I

00

1

small

independent

and

k+1

2

i-T-1 OW/12

+

W01W12). 2

IIWI12 1

k + 1

of w, because

f W2f(02 )dx

(IW112+WOIW12)_ 2 2

1

expression

k

k

+

for

is not

a

sufficiently

smaller

small

than

-00

00

C2 k + 1

IW12

I

k

2 -

k + I

W2f(02 )dx

(C2 k +

>

k _

k + 1

1

M) IW12

>

2

0.

-

-00

Lemma 111.1.8

In what

is

proved.0

follows,

condition

the

we use

(V, 0) Lemma 111.1.9

(111.1.5)

satisfying

Since

Proof. value

there this

There

2

=

0 and

exists

eigenvalue

an

(111.1.9).

clearly

0'

0'

has

is

an

precisely

eigenfunction is

C

exists

and

minimal.

Lot

gi

that

eigenfunction > >

root,

of

92

(L+v, v) >CIIVI12

L+

A2 is the

0 with

0,

(111.1.9)

0.

such

> 0

one

g1(x)

=

a =

with

second

corresponding

mo'

be

the

for

all

corresponding of L+

eigenvalue

eigenvalue

eigenfunctions

A,

of the

E

v

H'

eigenso

< 0

that

and

operator

CHAPTER3.

86

L+

o,',

normalized =

W

ag,

L2 and let

in

b92

+

01,

+

v

192

+

+

(L+v, v) It

follows

from

the

spectral

C,

of

v)

0

Using

now

Schwartz

the

using

0


v-L,

0-

this

obtain: 00

JIMI

j

00

(of 12 f(02) X

+

202f/(02)

}dx

j

11m I

=

00

[(0/1

XX

WO(of )2]dX X

-00

CC)

1 0/ [f(02) X

+

202f/(02)]

(kgl

+ vi

-)dx




where

VI

=

(L+v where

subspace

L_L be the

kgl

=

STABILITY OF SOLUTIONS

with

C5

>

0.

(111.

1.

14) implies:

v-L)

=

(111.1.14) equality,

we

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

Proceeding we

further

proof of

end of the

at the

as

and Lemma 111.1.9

CIIvII1,

::

Lemma III.1.10

I lh(.,

Proof.

for

We have

IIU(',tl)

t) Ill

is

e

-

i('Y(t2)+WOt2)

I I h(.,t2)

0(.

Now h

Theorem

prove

Then,

=

by (111.1.7)

111.1.6.

P(O

+

aj(0)I

-

lim

0. 8

+0

AE +

WO

2

all functions condition

Also,

I I,,

JIU(*itl)

h

Let

P(0)

-

-

ao

=

=

-(t2))Ill

T


C2(lllmhj-+a20l, 2 where

,

-

-u(*,tl) :5

_

of t; hence

lai(t) Further,

ei(Y(t2)+WOt2)0(.

7-(t2))Ill

functional

the

AP

independent

1

+

proved.El

is

we can

0.

7

1 IU(*,t2)

:5

_

U(* t2)1 11

-

Ilh(-,t2)III

-

t.

e'(^f(t2)+WOt2)0(.

-

JIU(')t2)

-

I IU(i t1) I 1,

of

function

JIU(*)t2)

-

7(t2))Ill

-

III -I Ih(-,tj)

and Lemma III.1.10

0,

>

IIh(-,t2)II1

-

7(tl))Ill

-

I Ilh(.,tl)lll

the

inequality

and t2:

tj


O,

jail

:

For

large

I IIrn

+

h

6 > 0 let

+

,

t)

Suppose

+

IlRe h_LIIj

Let

us

prove

a20(',

+

h(.,O)EO6

06 be the

a20111

constant.

h

STABILITY OF SOLUTIONS

t)II1

+

neighbor-

open


0 and with u.,&, t) problem (111-1.3),(111.1.4) sequence hn(', 0) E 06,,, either +0 as n 8,n c or I IIrn hi -n( -, tn)+a2n(tn)O(*7tn oo, such that aln(tn) )112+ 1 2 > h c2for First of some n > IlRe In tn tn) 112 0, 1, 2,3, all, (111. 1.15) implies that for all sufficiently < euntil large n wehave laln(t)l IIImhj_n(*it)+a2n(t)0 t) Ill 2+ E and t IlRe hj-n(., t)112 I < _C2f2 because if lain(t)l + a2n(t)O ( IlIm hj-n(.,t) 1112 + -6 V < h IlRe E2, then by (111.1-15) n(', t) 112I ! 2 +C562 which is a contradiction lain(t)l as

+0.

-->

this

right.

is not

of solutions

a

-->

Then,

there

exist

a

of the

-+

....

=

_

-

1

-

because

tn

>

0,

n

Then,


0

c

large

all

(111. 1.16),

to

independent

sufficiently

[AE

of n,

large Wo

+

Thus,

we

arrive

which

the

solution

2 at

taking

we

numbers

+

to

the

place

for

all

exists.

soliton-like

for those

t > 0 for

solutions solution

which

exist

sufficiently

a

Loo'2 'P] Ltn At the

[AE

+

I I Re h_Ln ('; tn) 112

+

IlRe hj-n(*,tn)lll];

large

2

=

02,E2.

>

m-'

and

sufficiently

a

solutions

AP] I

t=0

:5

also

c8i

t >

exist.

t

C7,8,n2

=

yield

1.

0 in the

Hence,

0

-+

as

a

n

--+

proved

are

priori

get

we

oo.

for

all

t for

estimates

0)

(111.

problem

point

(111.1.17)

from

(111-1-18)

relations


0

time,

relations

of the

the

C6

>

same

wo

and

These

u(x, t) V(-, t) at

these

+

n:

-

contradiction,

u(., t)

there

get

n.

API I t*=tn a

assumption,

our

+a2n(tn)0(',tn)II1

JIU(*it)lll taking

112

numbers

m[IIIm hln(*,tn)

[AE for

to

that

sufficiently

all




where the

0

s

-->

NLSE,

of the

case

as

+0 and L we

=

-4 -

-

dX2

+

h E H1

all

[P(u(.,

t))

-

flo(wo,

.))]

!

Loo

f (O(wo, -)).

-

Proceeding

further

as

in

get the estimate

(111.1.19)

satisfying

2

7(1 IhI 11),

+

(Lh,h) for

Wo

El (O(wo, .)) +

-

>

the

and

C911hl 121 condition

00

j

h (x)

0 (Loo, x)

dx

0.

=

00

The end of the

111.1.6

is

proof

Under

III.1.11

NLSE and

KdVE the

d

In

< 0.

the

only these

the

our

of the Here

restrictions,

N,

2

(to

then

we we

2, the problem

< P
a

the

0 for

radial

f(JU12) w

>

N

IUI,'.

0 has

positive

for

a

1, 2);

cases

takes

[15];

paper

=

=

in both

distances

only

questions

case

(11.0.1),(11.0.3)

111.1.6

above

for

the

in

the

to

solution

addition,

solution

place

NLSE

According positive

0.

of the

if

NLSE,

and,

to

results if and

if p satisfies

According

to

CHAPTER3.

90

111.1.7,

Remark

the a

Let w

wo > 0 and

positive

a

the

prove

)

4

N-2

positive

radial

and

b such

constants

all

x

a

e"OtO(wo, x) a

complex

Section

in

as

of the

(p

4

>

N

for

with

respect

to

We present

N=

only

solution

h

=

ih2

+

for

given

a

(11-0-1),(11.0.3)

problem

of the

IVO(x)l

+

Further,

h,

0

all,

of

First

1.3.

number

there

exist

that

z(t)

=-

0 for

can

easily

the

jY0+h(0)-Y95(0)j

arbitrary

not

and

functions

Cauchy problem h

is

function

these

of the

function

a

(1.2.19)

of

decreasing

t)

+ h (x,

choose

can

of the

values

are

2. 1:

Conditions for

a

kink

O(w,

x

providing -

the

wt) satisfying

existence

of kinks

equation

(11.

1.

1)

are

and

STABILITY OF KINKS FOR THE KDVE

3.2.

0(oo)

conditions

the

ing conditions

0

=

it

sufficient

is

and

f f(s)ds,

=

that

necessary

.0

(hereAo)

satisfied

are

exist,

to

91

fj(0)

7(0)

=

-

the

follow-

wO + wo-

and

00

f fi(s)ds):

F1(0)

0-

A:

fi (0-)

B:

F, (0-)

C: F, We also

fl (o+)

0;

F, (o+)

(0)

for

< 0

0;

all

(0-, 0+).

G

require -W

Clearly,

(111.2.1)

condition

provides

I O(W, X) Without

(111.2. 1)

and a

suitable

the we

result

loss

shall

of

(111. 1. 1),(Ill. following.

Theorem

Then,

.

there

of

these

with

Let

the

such

solutions

the

that

of the

of

a

0-

> x

-

0-

Under

wt).

solution

u(00,

infinity

the

>

i

O(w,

kink

on

C1 C2

0+

that

uniqueness

conditions

For this

u(x, t) t)

aim

A-C we

of the

This

need

Cauchy result

(111.2. 1) be valid, f (-) be a twice and a function uo(-) be such that u0(-) O(w, -) E and solution the a (0, a) unique u(x, t) of problem For any O(w, -) E CQ0, a); H2) n C1([0, a); H-'). A-C and

assumptions

-

half-interval

a

1XI)

!5 C1 e-C2

accept

conditions

function

exist

(111.1.2)

stability and

(111.2.1)

< 0.

estimates

we

existence

differentiable

continuously H2

generality

1.2)

111.2.1

the

A0)

I Ox'(W, X) 1

+

show the the

on

problem is the

0 I

-

+

u(-, t)

-

quantity

2IU (X, t) 2

IM-1 0)

=

F, (u (x, t))

-

x

dx

00

does the

that

of

not

above

depend

I Ju(-, t) this

-

solution

and

exists

0(w, -)111

The Proof 1.1.3

i.

t,

on

solution

onto

a

of this

Proposition




there >

law.

0, and there exists

a

In

exists

(unique)

addition, continuation

0.

by analogy

with

the

proof

if

C > 0 such

of Theorem

STABILITY OF SOLUTIONS

CHAPTER3.

92

Remark

111.2.1,

111.2.2

suffices

it

analogous

nition

Remark

t))

get

write

to

I.I.I.

careful

a

the

Since

111.2.3

I(u(-,

quantity

To

to

equation

by

IF 1 (U)I

construction

is well-defined.

difference

the

for

A formal

u(x, t)

of solutions

definition

u

C(U


0 such

that

if uo(.)

corresponding can

Pq(U(',

t), O(W, -))

We first

111.2.4.

_


0

place.

takes

E

-

the

to

respect

u(., t) of

solution

be continued

that

the

stability

from

prove

the

following

estimate:

CP2(U, 0)

I(0)

q

+0 and C

-*

T

=

T(t)

>

Theorem

111.2.4

(P2(U' 0)),

-a

is

chosen

(111.2.2)

.

independent

0 is

C- R is

for

of

Let

u.

Pq(U, 0)

to

u(x, t)

=

be minimal

O(W, (one

x

-

can

T). Then,

of such

AI 00

=

I(U)

-

1(0)

00

Ih,2

2

kink

form.

J(U) where

corresponding

qlu(x)

+

111.2.1

see

conditions

Let also

set

exists

the

easily

can

be the

T)122

_

Then,

0 there

then

inequality

any t > 0 the

1}.

+

>

c


0

technical; 0- < 0+.

is that

minimizing

number

have

a

with

X_+00

definiteness Let

c

=

(a)

assumption the

for

of

0-]e-"

-

+

-

1, conditions

(111.1.3)

NLSE

existence

U02 -i

_

STABILITY OF SOLUTIONS

-

X_+00

follows.

H1.

-

We also

We denote

(since 0(0(-) E Hl).

-

0'(x)e'

lim

=

T)

-

suppose

by

0(-)

-

Of course,

0.

=

a

To

real

E

H',

as

earlier,

we

non-unique.

X'-solution

u(x, t) of the NLSE (111.1.3) be such that lu(-, t) I 0(.) E H' for some t > 0. We set v(x, t) for g(x) u(x To, t), where To is taken lu(x, t) 1, and a(x,t) As if is lv(x,t)l then O(x). earlier, Ila(.,t)lll sufficiently small, 0 < cl value :5 O(x) + a(x, t) :5 C2 < oo for this t and respectively there exists function a real-valued continuous in finite inan absolutely w(x,t), arbitrary terval and unique the term 27rm, m to it, such that up to adding 1, 2,..., Since v(-, t) E X' and vx'(x, t) v(x, t) (O(x) +a(x, t))e'1'_0t+'0(x,t)1. [0'(x) +a'(x, t) + we have i(O(x) + a(x,t))w.,(x,t)]e'l'-t+w(x,t)), E L2 if Ila(.,t)lll is sufficiently small. if u(x, t) : 0 for some t > 0 and all x E R, where By analogy, u(x, t) is a Let

a

-

=

=

-

=

-

=

=

=

X

X'-solution

of the

Theorem

of

tion

the

111.3.7

complex

corresponding for

if

X1, luo(-)l

uo G

any

ing X1 -solution onto

the




let

the

of

the

0(-) E H1, Ila(., O)l 11 u(x, t) of the problem (Iff. 0)

t >




the

following

0 such

that

ILO,1(*) 0) 12 < 6, then the correspondis global 3), (111.1.4) (it can be continued t > 0 one has I U(-, t) fixed 0(-) E H1,

6 and




(R (T))

1

'r='ro

of minimum

of the

function

"0

2

I

00

a(x,t)o'(x)[q

opera-

C1 Ig 1 22 for

proved.0

Therefore,

2

=

-w"

202 fl(02

+

spectral

the

(111.3.6),

t) 0

Oa)2f/((o

Mi(g)

that

Ao is the smallest

sign, from

=

=

operator

L with

operator

it follows

number

-r,

the

of

H' satisfying

The -

0

f(02 )

+

spectrum

is of constant

Hence,

tor

>

LW

operator

_Zj

=

of the

0'(x)

since

the

Consider

continuous

+

condition

the

(91 01) Proof.

2(0

-

(0, 1).

E

Lemma 111-3.9 g E

202fl(02)

ay +

f(02)

+

202f/(02

)]dx.

R('r)

STABILITY OF SOLUTIONS

CHAPTER3.

100

We take

jZj

sup

f (0'(x))

-

20'(x)f'(O'(x))

-

I

+ 1 and

xER

10112 1 01(V

K

f(02)

+

+

202ff(02))

12

00

f

+

f(02)

202fl(02))dX

+

-00

g E

H' sat,sfying

C2Ig 12,2

MI(g)

Lemma 111.3.10 the

C2

where

Cj(1

=

K) -2, for

+

all

real-valued

condition

W

I

g(x)o'(x)(q

Zj

-

+

(02(x))

f

+

202(X)f1(02(X)))dX

(111.3.7)

0.

=

00

Proof. g

=

ao'+

Represent

p)

(0',

o where

arbitrary

an

0.

=

function

Then M,

(g)

CIJW12.2

M, (W)

=

(111.3.7)

H1 satisfying

9 E

in the

form

from condition

We get

(111.3.7): 00

a

j

0/2rq

_

f(02)

Zj +

202f/(02

+

+j

)]dx

00

W01[q

_

ZU +

f(02)

+

202f/(02

)Idx

=

0,

00

-00

hence, co

f 001[-q I 1101 1 2 a

=--

101 12

f(02)

+

+

202f1(02)]dX
0

n

We define

IV.1.6.

measures)

Gaussian

dimensional

kind

Definition

from

measure

arbitrary

an

H of the

space

.Clearly,

0

constant.

centered

of Borel

sequence

follows.

115

Borel

sets

setting

and,

n

n

wn(M)

(27r)-'T

=

][I

n

L

Ai

2

i=1

obviously

we

Mn.

get

procedure

this

Repeating

Gaussian

dimensional Now

we

sigma-algebra

Borel

Hn

that

subset

I el,

span

=

each

Min .

-

-)

A n Hn

sigma-algebra Borel

M1 is

sigma-algebra M, obviously

subsets

because

by

all

and

open

Now,

separable

closed

let

the

and

only

SO,

(n

=

open

1, 27 3,...)

IVn}n=1,2,3....

C

:

and closed

be

weakly

real-valued

arbitrary

Lemma IV.1.10

the

Then,

c

--+

>

as

Borel

with

get

measures

Borel =

easily

can

we

the

1,

the

to

is

a

M,

set

M. But

then,

contradiction

containing

in H.

in

complete

a

We recall

1, 2,3,....

=

that

that

a

measures n

Borel

verify

sigma-algebra

converging

I W(x)vn(dx)

measure

v

in

Mif

W(x)v(dx)

M

n

H,

of

minimal

vn(M)

=

one

a

M}

A E

some

Consider

coinciding

subsets

nonnegative

v(M)

that

Then,

whole

A E Msuch

exists

supposition.

is the

A n Hn is

show that

A n Hn for

the

onto

if

lim

an

of finite-

A E M(we recall

Hn),

n

there

in Mand not

is called

n-oo

for

=

the

be considered

Wn can

Msuch

space

sequence

Then,

sigma-algebra

Borel

sets.

v, vn

metric

that

the

definition

sigma-algebra

sequence

our

extended

wn(A to

A n Hn E M1. n

that

all

=

suffices

it

Mn by

n

contains

get

we

the

on

naturally

wn(A)

C H

Mn7 M1 :

in H contained

a

0,

>

be

opposite.

fC

=

A of H such

of all

.3ince

the

n

and M1 C n

dx,,,

...

defined

wn

n

can

this,

To prove

M'

Clearly,

Mn.

Wn

rule:

the

by

H

Suppose

of Hn if A E M.

integer

measure

1).

en

dxj

jWn}n=1,2,3,.-

measures

show that

e

measure

all

for

1

2

F

additive

countably

a

Y>'-1.?

-1

Let

of

sequence

M

bounded

the

functional

continuous

measure

w

from Definition

jWn} weakly

measures

0 in M.

be

IV.1.6

converges

to

the

countably

measure

additive. w

in

H

as

oo.

Proof.

First

0 there

exists

of a

all,

one

compact

can

set

prove

If,

as

in the

C H such

proof

that

of Theorem

w(If,)

> I

-

e

IV.1.8 and

that

1,Vn(K,)

for > 1

any -

c

CLIAPTER 4.

116

integer

all

for

functional

let

Further,

0.

>

n

W defined

take

us

j

lim

W(x)w(dx).

W(x)w,,(dx)

H

Take also

arbitrary

an

>

E

one

easily

can

that

verify

there

6

exists

=

8(e)

>

0

that

such

IWW W(Y) I any

ifn

Let

=

if,

H,,,

n

Ix

satisfying

and y E H

If,

E

x

n

f

p(x)w,(dx)

so

that

f

-

I W(x) 1,

M= sup


0

n

0, such

show the

of C > 0,

existence

I

n-oo

W(x)w(dx)

in view of the

(IV. 1.4),

i.

Y E H:

Y

Then,

K,

C

the

e.

=

Kn,

arbitrariness

of

statement

of the

E

Hn7

sufficiently

all

I all

the

by

w,,

Hilbert the

large

sufficiently

product

rule:

0

I w

on




equation tor

Borel

arbitrary

an

Borel

the

Then,

[to

-

class

it

is

of

(IV.2.5)

+

T]).

C(I;X).

eigenfunctions

Pn be the orthogonal

projector

Xn & Xn.

Consider

also

Xn

=

trans-

small

sufficiently T,to

invariant

an

val-

Therefore, Further, of the in the

the

let opera-

space

following

120

CHAPTER4.

problem

approximating 1

2

Un

+ Un,xx + Pn[f(X,

t

2

Un

_U

t

U

n

Let

pn

in the

space

Then,

the

the

Pn

0

0

Pn

)

1 n

,

X-P n[f(X,

gi

eigenelements

of the

(U2)2)U2]

n

0,

t E

R,

(IV.2.6)

(Ul)2

+

(U2)2)Ul]

n

0,

t E

R,

(IV.2.7)

n

(eO, 0),

=

is

(0, eO),

=

Clearly,

that

the

Xn is

the

onto

-

-

orthonormal

S.

nU2(X). 0

n

has a unique local solution (IV.2.6)-(IV.2.8) (as it is well known, in a finite-dimensional and

n

projector

92

an

operator

n

U2(X,to)=p

orthogonal

f9n}n=1,2,3,...

system

+

PnUI(X), 0

=

be the

(Ul)2 n

X

(X, to)

Let also

X.

(IV.2.1)-(IV.2.4):

problem

the

INVARIANT MEASURES

92n+1

7

-

basis

for

linear

=

(en) 0), the

in

X

space

integer

n

with

Xn 92n+2

(U1(X't)'U2(X'

=

any two

space

equipped

subspace

positive

any

un(x, t)

(IV.2.8) Xn (D Xn

=

(0, en)

n, the

n

norms

7

consisting

....

of

problem

t))

E C (I;

are

equivalent,

Xn)

X). In for 0 these solutions. 112 t) X for any n and for any uo Therefore, X the problem 1, 2, 3, (Ul0 (.), U20 has a unique global solution (IV.2.6)-(IV.2.8) Un(*) t) E C(R; Xn). it is clear the above solutions that Further, Un(*,t) of the problem (IV.2.6)the equations satisfy (IV.2.8) 1, 2, 3, (n we

mean

addition,

direct

the

space

shows

verification =

the

-dt-JlUn(*,

that

of the

norm

space

=

...

=

Un(', t)

=

A(t

_

to)pnUo

f

+

t

B(t

-

S)pn V(.' JUn(*, S)12 )Un(*) s)]ds.

(IV.2.9)

to

Hence,

u(., t)

(IV.2.5)

from

(IV.2.9)

and

one

has for

those

of t for

values

which

the

solution

exists:

JU(', t)

Un

-

(', t) I IX

:5

C1JJUo

pnUoJJX

_

+

I

C2

t

I JUn(*, S)

-

u(., s) I lxds+

to t

J

+C3

I JU(., 3)

pnU(.,

_

s) I Jxds.

(IV.2:10)

to

the

Here the

constants

solution

right-hand respect

u(-, t) side

to t E

C1, C2, C3 do

of this

[to,

for

exist

to +

t E

[to,

to +

T]

obviously

inequality

TI,

depend

not

therefore

we

on

where tends

get from

the

initial

value

T > 0.

Then,

to

as

zero

n

the -4

(IV.2.10)

inequality

to and t.

uo,

third

+00

term

in the

uniformly

with

by the Gronwell's

lemma that

lim n-oo

By analogy,

if the

max

tE[to,to+Tl

u(-, t)

solution lim n-oo

exists

max

tE[to-T,to]

JJU(',t)-Un(',t)JJX=Oon

a

segment

JJU(',t)-Un(',t)JJX=O-

[to

-

T, to],

Let

T >

0, then

ANINVARIANT MEASURE FOR THE NLSE

4.2.

121

Hence, lim for

all

fact

I

segments

implies,

tEI

of

the

it is easy

then

for

for

verify

to

any fixed

T2]

+

(c)

that,

if

u(-, t).

solution

and, hence,

IV.2.2

This

global

the

X.

u(., t)

function

a

of the

existence

function

t E R this

(IV.2.11)

0

=

of Theorem

uo E

any

t) I Ix

u,,(.,

-

of the

statement

(IV.2-5)

equation

Further,

(IV-2.5),

T1, to

-

particular,

in

solvability

[to

=

I Ju(-, t)

max

n-oo

is

G

C(R; X) of the

solution

a

satisfies

equation

following

equation:

T

u(.,,r)

A(T

=

t)u(-,

-

+IB(,r

t)

_

S) [f (.' I U(.' 5) 12) U(.' s)]ds,

R,

E

r

t

which, any

the

as

fixed

earlier,

for

the

map uo

t

transformation

(t

uo

fixed

any

has

t

u(., t) u(., t) as

is

--+

--+

Therefore,

X-solution.

from

map from

a

global

unique

a

one-to-one

X.

X into

X follows

X into

for

The

continuity

from

the

of

estimate

to)

>

t

U

('i t)

Vt)

-

X

C1

!

U

('; to)

V

-

(*) to)

X

+

C2

)rI

lu(.,,.s)

v(-, s) I Jxds,

-

to

u(., t)

where estimate

v(., t)

and

for t


0

c

all

numbers

n

=

T,to+T]

1, 2, 3,

(IV.2.6)-(IV.2.8),

problem

for

and

...

taken

I JUn(*, to)

(here

u,,(.,

to)

Proof

=

pnuo

follows

v,,(.,

and

from the

to)

=

estimate

-

same

(c)

of Theorem

8

exists

value

>

and

IV.2.2

0 such

a

similar

proved.

are

that




I JUn(i t)

max

tE[to

for

solutions

(a),(b)

the statements

u,,

(., t)

satisfying

n,

t) of

and Vn (*) the

the

condition


to) t

I jUn(',

t)

-

V.(-, t) I IX

!5 C1

I JU.(',

to)

-

Vn(*i to) I IX

+

C2II

JUn(', 8)

-

Vn(-, s) I Jxds,

to

that

results

from

By hn(UO) t)

Un(')

t +

to)

where

equation we

(IV.2.9),

denote

Un('i t)

the

is the

and

function solution

an

analogous mapping

of the

for

estimate

any

problem

uo

E

t
0

phase

the

with

system

space

Since

result

the

to

Borel

in the

S.

in X.

following

dynamical

a

consider

us

operator

in Xn

the

hn is

function

1, 2, 3,

=

INVARIANT MEASURES

-VaEn(a,

=

bi(to)

(t)),

b(t)

b),

(IV.2.13)

(U2 (-,to),ej)L2(0,A)i

=

(i==1,2,...,n),

n

(bo (t),

=

(IV.2.12)

bn (t)),

...'

un

(IV.2.14) b)

(Ul,

=

U2n ) and En (a,

n

A

f I! [(UI,X)2

E(Un)

2

(U2,J2]

+

n

F(x, (Ul)2

-

n

+

n

(U2)2))

Then,

dx.

n

according

The-

to

0

the

IV.1.3,

orem

dynamical

(W.2.12)-(W.2.14)

system

with

system possesses

Borel

a

phase

the

invariant

n

y' (A)

ii

-(n+l)

(27r)

=

n

A,

'=O

A C R2(n+l)

where

IV.1.1,

there

space

R2(n+l)

is

and

A C R2(n+l)

is

an

natural

a

Borel

arbitrary

Borel

Q of the

subsets to

Borel

a

e

set

ji':

measure

En(a,b)

n

db,

da

Further,

set.

space

according

between Xn defined

0 C Xn if

an

by

A of the

subsets

the

element

Proposition

to

Borel

rule:

to

9 when and

only

when

(a, b)

E A where

ul

Un

(Ul,

=

(ao,

...,a

correspond measure

n);

,

to

it,.

b

each

These

=

(bo,...,

other

bn).

In

in this

arguments

addition,

sense,

easily

then

aiej,

imply

if two sets

jUn(Q) the

=

A C

y' (A) by

statement

n

set

U2 ) ben

n

n

U2

E biei

and

i=O

'=0

a

Borel

a

n

longs

by the

generated

A

correspondence

one-to-one

corresponds

f

R2(n+l)

space

R2(n+l) the

and

definition

of Lemma IV.2.4.0

1

C X"

of the

ANINVARIANT MEASURE FOR THE NLSE

4.2.

According the

to

verges

measure

w as

n

Lemma IV.2.5

p(Q)

Proof.

arbitrary

the

IV.1.10,

Lemma

to

of Borel

sequence

measures

weakly

Wn

con-

oo.

--*

t)) for

tt(h(fl,

=

123

any

bounded

open

Q C X and

set

for

any

t E R.

Fix

according

Then,

set.

is open

4) is bounded

4.1, the

proved

too,

and

Q,

c

=

For any A C

Lemma

ly

IV.2.3,

for

B6(x)

x

and for

n

K,

of the

x).

y

-

B6,(xi)

be

the

since

Further,

< e.

h(K, t)

=

h(Q, t)

functional

from

of results

obviously,

Section

according

is

>

a

that

for

to

compact

a

set,

11hn(u,t)

covering

of the

According

0.

any

has

one

finite

a

Then,

aQ,)}

Then,

1,2,3,...

=

IV.2.1,

A and let

set

6 > 0 such

exists

any

B6, (xi),...,

Let

IV.2.2,

a.Q); dist(Ki,

yEB

E K there

x

61




bounded

open

of Theorem

X and in view

space

of Theorem

aA be the

X, let

dist(A,

again

of the

e

arbitrary

an

(a),(b),(c)

arbitrary

an

K C Q such

(a) and (b) h(f2, t).

a

(where

fix

us

set

Q C X be

statements

subsets

compact

a

statements

K,

Let

bounded

on

exists

proved

to(...

let

E R and

t

the

to

bounded,

and

there

an

u,

E

v

to

B6(x)

-h,,,(v,t)llx compact

0'


no.

n

y

of

:5 p (B) +

inf yn

< lim

n-oo

(because Hence,

p(Q)

yn(B) due

>

=

to

p(Qj).

open

an

Remark be

unbounded,

the

lim

=

R

arbitrary

proximation

p(Q)

Xn) =

proved

(B)

+

E

=

yn(hn(B of

tt(Qj),

c

n

Xn' t))

0,

>

(hn (B, t))

(c)

u

E B and

we

+

c

(f2j)

< y

+

c

hn(B n Xn' t) C hn (B, t)). By analogy ft(Q) < jz(,Qj).

and

have

and Lemma IV.2.5

statement

2

all

and IV.2.4

lim inf yn


2

n

>

2, A > 0, T

Hpner(A) to),

u(., t)

where

Eo,

...'

E,,,-,

>

Hpn ,JA)

into

by Theorem L1.5,

functionals

dynamical Let

the

IV.2.2

of

view

in

As

0.

>

of such

any

KdVE

u(x

this

on

of the

is dense

points

ut+uux+uxxx=O,

the

r

sense

f (x, s)

and

1+s

series

An infinite

tion

the

of Theorem "

X and

c

phase

new

a

By analogy,

R > 0.

of the

set

for

bounded

4D is

any

any

set

balls

case.

4.3

the

0


there

0

R

exists

and

d,


2 be

n

of equation

Lmk -solution

the

+oo

--

the

is

obviously

is

by

-dt-Ej(um)

=

u'(x,

solution

generated

(IV.3.4),(IV-3.5)

problem

the

is

1Um(*,tO)jL2(0,A)

=

of the

local

in L,,,

-dt-Eo(u')

that

verified

onto

(IV-3-5)

classical

unique

a

finite-dimensional

a

solution

Proposition be such

solution

be continued

can

addition,

the

(x).

uo

(the topology

1UM(*7t)1L2(0,A) for

P"'

=

INVARIANT MEASURES

the

(n

law

conservation

>

2):

A

2

Eo(u)

I(DnU)2 2

+ En(U)

X

1

+

2

u

2+ cnu(D'-'

X

U)2

-

Dn-2U)

q,,,(u,...'

X

dx >

0

1

211UG)11n where

n

mates

for

Ej(u)

(s)

is

have

1U1L2(0,A))1'_1

2

p

function

!Eo(u) 2

,

in view where

of the

-

and

continuous

functionals

the

we

a

2

+

?7n(JjU(-)jjn-1)) increasing

!Eo(u) 2

E2(u),...'

known

on

inequality

+

[0, +00). En-1 (u).

lUlLp(o,A)

Repeat For

1+1

'EO(U)

2

2

IJUI12 1

-

?71(1U1L2(0,A))(jJUJj12*

+

I)-

esti-

functional

JU12L2 (0,A) (jDxujL2(0,A) P

p > 2:

El (u) +

the

these

+

ANINFINITE

4.3.

We get

SERIES OF INVARIANT MEASURESFOR THE KDVE 127

by step

step

from

the

obtained

I I u 111 for

all

There

Lemma IV.3.5

(R, s)

on

[0, +oo)

E

satisfying

0

(d),...,

:5 C,

and Lemma IV.3.4

R,

t E

estimates:

proved.

is

n

functions

exist

[0, +oo),

C,,, (d)

u

(R, s),

-y,,

such

that

(R, 0)

-y,,,

nondecreasing

monotonically

in

=-

and

8

defined

0,

continuous

following:

the

d

Tt En(umk(*7t)),


En (um (-, to))

-

On(IIUM(*)t)lln-1)

-

R

(t

estimate

n

on

the

half-line

Then

< R.

,

t

aking

by step:

get step

C,,,(R),

:5

0

any

E

uo

Hp', ,,(A),

T > 0 and

integer

n

> 2

the

following

place

11 UMk(.' t) -U(',t)lln-1--->O

max

tE[to-T,to+T]

Proof.

IV.3.1,

I

arbitrary

an

from

1 E,, (um (-, t))

2

I um (-,

is

> 2

to) IL2(0,A)

Um(.'

earlier

as

and Lemma IV.3.6

takes

-

Indeed,

that

account

IDx

n

L

where

into

t)122(

follows

(-, t) I In-1)

u'

an

tEI

an(s) [0, + oo) (n

of Lemma IV.3-5

in view

we

First,

let

Using

H'.

E

uo

as

Lemmas IV.3.4

k--+oo. and

IV.3.6

and

Theorem

get: A

I d

I D'-'(u",

2 dt

(-, t)

X

_

U

(.' t)) 12 dx

=

0

A

21 Dx

'-'

(UMk (.' t)

-

U

(.' t))

n xDX [(Umk

(.' t))2

_

(U(., t))2 ]dx+

0 A

+

2

I

P,, L, [ Dn- (U (.' t))]D'[(umk(.,

t))2 ]dx

1

X

X




that

such

jn(P u)-En(hm(u,t))

(Q, t))

Proposition

to

right-hand

integer

respect

pm (hm

-

0 and

can

be

obtain

the

that

uniformly a

with

compact

proved

as

set

in the

IV.3.12, -

tt,,

(h,,, (K

n

Q, t))]

=

0,

M-00

hence,

by Proposition

IV.3.11,

we

lim sup

get the relation

[ftm (Q)

-

ym (hm

(Q, t))]




bounded

I tt,, (Q)

-

0, yields

open

p,,,

(h

the

set

statement

0 c

(Q, t)) I

Hpn,-'(A) =

0.

of Lemma IV.3.13.0

and

for

any

t E R

134

CHAPTER4. Lemma IV.3.15

,,n(Q)

,n

=

By Theorem

IV.3.1

set,

B)

Clearly, E

v

that

,n

inf

=

and

(0 \ K) < c. f2j. (Q, t) JJU Vjjn-j

these

u

-

balls.

any

also

Let

P

=

aA is

and

B1,

E

v

of

Proposition

r'(A),

for

sufficiently

all

,,n(f2)

,n




Corollary

(B, t))

+

IV.3.14 e




c

uEA, vEB > 0.

a

for all

bounded

a

Lemma IV.3.4

arbitrary

an

and K, C h n-1

too,

dist(A,

Take

too.

K C Q such

be

(Q, t)).

n-1

Hpn,,-,'(A),

in

set

Q C

(h

Proof.

Hpn,,-,'(A)

Let

INVARIANT MEASURES

=

,n(Qj)'

n

IV.3.2.

First,

let

Q C

Hpn,,-,'(A)

be

an

open

(generally

We set

f2k

=fuEQ:

11h

n-1

(U, t) I 1 _j

+

jjujj.,,-j




0.

Then

U Qk

Q

,

and

each

set

f2k

is

open

and

k=1

bounded;

in

addi-

00

tion,

hn-1

(n, t)

U

h n-1

(Qk, t)

and

/,,n(gk)

.

,n

(h

n-1

k=1

(Qk' t)) by

Lemma IV.3.15.

Therefore,

,,n (h

n-1

(n, t))

=

liM k-00

Un (h

n-1

(Qk' t))

=

liM k-oo

ttn

(f2k)

=

,n

(f2).

135

ADDITIONAL REMARKS

4.4.

Let

hn-1

(A, t)

now

can

last

is

a

Eo,...,E,,_1

all,

of

trajectories based

not

[16,20,53,66]).

Here

Concerning Some of them

[105]

in

an

with

the

sociated

invariance In

Poincar6

consider

there

equation.

is

nonlinear are

papers

for

with

quite

different.

In

the

and A

for

finite

for

any

f(X, S) :5 C(l p E (0, 2) for N

=

result

this

the 1 in

our

ball

The obtained

B C X.

+

S

d2)

A > 0.

for

is obtained

problem

all

for

by an

x,

>

J.

=

paper

initial-boundary with

Bourgain

arbitrary

>

A.

initial

[16,17] This

seem

to

data

this

allowed

AJuJP

for

in these

exploited

two

explicitly

more

this

a

author

AluIPu

and

to

space

be

p >

question

sufficient

and

nonzero

I_L(B) < -C(l+s di.)

00

:

0 if A < 0 and remains

(IV.2.1)-(IV.2.4)

open

with

L2_ The required

well-posedness of this

0

p >

0