Mathematical structures of integrable systems, their developments and applications

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RIMS Kôkyûroku Bessatsu B94

Mathematical structures of integrable systems, their developments and applications

August 8㹼10, 2022 edited by Keisuke Matsuya

November, 2023 Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed. ‫ף‬2023 by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. All rights reserved. Printed in Japan.

Preface This volume is the proceedings of the workshop “Mathematical structures of integrable systems, their developments and applications” held from August 8 to August 10, 2022. This workshop was supported by the joint research program of the Research Institute for Mathematical Sciences (RIMS), an International Joint Usage/Research Center located in Kyoto University. The workshop was held at RIMS. Since the COVID-19 pandemic was not yet over, Zoom video meeting system was also used to hold the workshop. The aim of the workshop was to connect researchers working on classical or quantum integrable systems with professions working on various mathematical objects such as applied mathematics, discrete differential geometry, blow-up of solutions for partial differential equations, etc. There were 13 lectures and 88 participants during the workshop. This volume collects 6 contributions from the speakers. All the papers have been refereed and are in their final forms. I would like to express my gratitude to all participants, invited speakers, and RIMS staff for their kind cooperation under these difficult circumstances. I also appreciate the helps of anonymous referees. September, 2023 Keisuke Matsuya

i

Program RIMS Workshop 2022 Mathematical structures of integrable systems, their developments and applications 2022 年度 RIMS 共同研究（公開型） 「可積分系数理の発展とその応用」 August 8 (Mon) 13:40–14:30 大久保 直人 非正規団代数によるワイル群の実現とタウ関数 14:40–15:30 川上 拓志 高次元の離散 Painlev´e 型方程式について 15:50–16:40 岩崎 雅史 シフト付き LR 変換から見た離散可積分系と箱玉系の広がり August 9 (Tue) 東 康平 9:30–10:20 特異積分核をもつ非線形可積分系について

10:30–11:20 佐々木 多希子 異なる伝播速度をもつ半線形波動方程式系の爆発曲線の特異性について 11:40–12:30 川原田 茜 特異関数によるセル・オートマトンの軌道の特徴付け 13:40–14:30 大森 祥輔 低次元トロピカル差分方程式の力学的性質 14:40–15:30 松永 秀章 線形差分方程式の漸近安定性における時間遅れの影響 15:50–16:40 梶原 健司 可積分系による幾何学的形状生成：弾性曲線・対数型美的曲線から「美的曲面」へ August 10 (Wed) 10:30–11:20 延東 和茂 確率セルオートマトンの漸近解と超幾何関数 11:30–12:20 礒島 伸 遷移図による符号付き超離散系の解析 13:40–14:30 丸野 健一 ホドグラフ変換が関わる非線形微分方程式の可積分離散化 ii

14:40–15:30 辻本 諭 一般化固有値問題に付随する力学系と直交関数系について

iii

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

A q-analogue of the matrix fifth Painlev´ e system . . . . . . . . . . . . . . . . . . . . . . . Hiroshi Kawakami

1

Box and ball system with numbered boxes and balls . . . . . . . . . . . . . . . . . . . 21 Yusaku Yamamoto, Akiko Fukuda, Emiko Ishiwata and Masashi Iwasaki The blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Takiko Sasaki On dynamical connection between continuous and tropical discretized dynamical systems in one-dimensional . . . . . . . . . . . . . Shousuke Ohmori and Yoshihiro Yamazaki

55

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shin Isojima and Seiichiro Suzuki

65

Integrable discretizations of the SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuta Tanaka and Ken-ichi Maruno

iv

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RIMS Kôkyûroku Bessatsu 㹀㸵㸷 Stochastic Analysis on Large Scale Interacting Systems 㹀㸶㸮 Regularity, singularity and long time behavior for partial differential equations with conservation law 㹀㸶㸯 Study of the History of Mathematics 2019 㹀㸶㸰 Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations 㹀㸶㸱 Algebraic Number Theory and Related Topics 2017 㹀㸶㸲 Inter-universal Teichmüller Theory Summit 2016 㹀㸶㸳 Study of the History of Mathematics 2020 㹀㸶㸴 Algebraic Number Theory and Related Topics 2018 㹀㸶㸵 Mathematical structures of integrable systems, its deepening and expansion 㹀㸶㸶 Harmonic Analysis and Nonlinear Partial Differential Equations 㹀㸶㸷 Study of the History of Mathematics 2021 㹀㸷㸮 Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties 㹀㸷㸯 Various aspects of integrable systems 㹀㸷㸰 Study of the History of Mathematics 2022 㹀㸷㸱 Research on preserver problems on Banach algebras and related topics

RIMS Kôkyûroku Bessatsu Vol. B94 䠎䠌䠎䠏ᖺ㻝㻝᭶Ⓨ⾜ 䚷Ⓨ⾜ᡤ䚷䚷ி㒔኱Ꮫᩘ⌮ゎᯒ◊✲ᡤ 䚷༳ๅᡤ䚷㻌㻌᫂ᩥ⯋༳ๅᰴᘧ఍♫

RIMS Kˆ okyˆ uroku Bessatsu B94 (2023), 001–019

A q-analogue of the matrix fifth Painlev´ e system By

Hiroshi KAWAKAMI∗

Abstract We consider a degeneration of the q-matrix sixth Painlev´e system. As a result, we obtain a system of non-linear q-difference equations, which describes a deformation of a certain “nonFuchsian” linear q-difference system. We define the spectral type for non-Fuchsian q-difference systems and characterize the associated linear problem in terms of the spectral type. We also consider a continuous limit of the non-linear q-difference system and show that the resulting system of non-linear differential equations coincides with the matrix fifth Painlev´e system.

§ 1.

Introduction

The Painlev´e equations are non-linear second order ordinary differential equations that define novel transcendental functions. Historically, the Painlev´e equations were classified into six equations. We refer to them as PI , PII , . . . , PVI . The sixth Painlev´e equation PVI serves as the “source” from which all the other Painlev´e equations can be derived through degeneration processes. Since the 1990s, various generalizations of the Painlev´e equations have been proposed in the literature, such as discretizations, higher-dimensional analogues, quantizations, and so on. Recently, Painlev´e-type differential equations with four-dimensional phase space have been classified from the perspective of isomonodromic deformations of linear differential equations [4, 9, 10, 11]. This series of studies shows that, in the four-dimensional case, there exist four “sources” as extensions of the sixth Painlev´e equation. Namely, they are Received January 30, 2023. Revised August 3, 2023. 2020 Mathematics Subject Classification(s): 34M55, 34M56, 33E17, 39A13 Key Words: q-difference equation, connection-preserving deformation, isomonodromic deformation, Painlev´ e-type equation, integrable system. Supported by JSPS KAKENHI Grant Number JP20K03705 ∗ School of Social Informatics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamiharashi, Kanagawa 252-5258, Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

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Hiroshi Kawakami

• the Garnier system [3], which is a classically known multivariate extension of PVI , • the Fuji-Suzuki-Tsuda system [2, 22], which is an extension of PVI with the affine Weyl group symmetry of type A, • the Sasano system [19], which is an extension of PVI with the affine Weyl group symmetry of type D, • the matrix sixth Painlev´e system [1, 4], which is a non-abelian extension of PVI . Note that each of the four equations has its extensions defined in arbitrary even dimensions. These four families are expected to have an impact on fields such as integrable systems, special functions, and so on. On the other hand, Sakai [16] established an algebro-geometric theory which provides a comprehensive understanding of two-dimensional (or second order) Painlev´e equations. According to Sakai’s theory, when the phase spaces are two-dimensional, the discrete Painlev´e equations are more fundamental. Roughly speaking, by classifying a certain kind of rational surfaces, 22 different surfaces are obtained. From the discrete symmetry of each surface, a discrete dynamical system (a system of difference equations) is generated. The theory classifies these discrete Painlev´e equations into three types: additive difference, multiplicative difference (q-difference), and elliptic difference equations. The Painlev´e (differential) equations are understood through the continuous limit of these discrete Painlev´e equations. In this sense, we can say that the discrete Painlev´e equations are more fundamental than the Painlev´e differential equations. Our aim is, inspired by the two-dimensional case, to construct a unified framework for discrete Painlev´e-type equations in higher dimensions. However, it is difficult to classify algebraic varieties when the phase spaces have four or more dimensions. Instead, from the standpoint of the classification theory (by Katz [7] and Oshima [15]) of linear differential equations and the isomonodromic/connection-preserving deformation theory, we would like to develop a framework for higher-dimensional Painlev´e-type equations that involves discrete Painlev´e-type equations. In [13], we have defined an equivalence relation between spectral types of linear differential equations (that is, those can be transformed into each other by M¨obius transformations, the Harnad dual, or certain scalar gauge transformations are equivalent) and shown that there is a tree structure among the equivalence classes including differential equations without continuous deformations. The tree structure of equivalence classes corresponds to the additive surfaces in Sakai’s list. As a next step, we investigate multiplicative difference Painlev´e-type equations in higher dimensions. Among the four families mentioned above, q-analogues of the Garnier systems, the Fuji-Suzuki-Tsuda systems, and the Sasano systems have been obtained and studied

´ system A q-analogue of the matrix fifth Painleve

3

by several authors [17, 20, 21, 14]. Recently, a q-analogue of the matrix sixth Painlev´e system, which we call the q-matrix PVI , has been obtained [12]. In this paper we investigate a degeneration of the q-matrix PVI with the aim of constructing a degeneration scheme for higher dimensional q-difference Painlev´e-type equations. As a result, a system of non-linear q-difference equations is obtained, which corresponds to the matrix fifth Painlev´e system in the continuous limit. We tentatively denote the non-linear system by the q-matrix PV . The q-matrix PV is expressed as a deformation equation of a non-Fuchsian linear q-difference equation. This paper is organized as follows. In Section 2 we describe how to construct formal solutions to linear q-difference systems. We also give the definition of spectral types for non-Fuchsian linear q-difference systems. In Section 3 we review the q-matrix PVI . In Section 4 we consider a degeneration of the q-matrix PVI . In Section 5 we consider a continuous limit of the q-matrix PV obtained in Section 4. The appendix is devoted to a brief description of the matrix fifth Painlev´e system. Acknowledgements This work was supported by JSPS KAKENHI Grant Number JP20K03705 and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. § 2.

Linear q-difference systems

In this section, we collect some facts about linear q-difference systems. The formal normal form is used to define the spectral type for non-Fuchsian systems. § 2.1.

Formal normal form: Fuchsian case

Let q be a complex number satisfying 0 < |q| < 1. Consider a linear q-difference system with polynomial coefficients (2.1)

Y (qx) = A(x)Y (x),

A(x) = A0 + A1 x + · · · + AN xN

where Aj ∈ Mm (C) and A0 , AN ̸= O. If A0 and AN are both invertible, then the system (2.1) is said to be Fuchsian. For simplicity we assume that A0 and AN are diagonalizable. In this subsection we outline the procedure to transform the given Fuchsian system into its formal normal forms at x = 0 and x = ∞. We use the following well-known fact from linear algebra. We denote the set of all eigenvalues of a matrix A by Sp(A). Proposition 2.1. (2.2)

Let A ∈ Mm (C) and B ∈ Mn (C). Then the linear map

φ : Mm,n (C) → Mm,n (C),

φ(X) = AX − XB

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Hiroshi Kawakami

is an isomorphism of vector spaces if and only if Sp(A) ∩ Sp(B) = ∅. The linear q-difference systems that will be treated mainly in this paper are those with polynomial coefficients, while the process of constructing the formal normal form involves systems with infinite series coefficients. Therefore, in Section 2.1 and 2.2, we will discuss the linear q-difference systems of infinite series coefficients. First we explain the formal normal form at x = 0. For the convenience of later discussion, we consider a system of the following form: (2.3)

Y (qx) = A(x)Y (x),

A(x) = xr (A0 + A1 x + A2 x2 + · · · )

where A(x) is an m × m formal power series. Here A0 is invertible and diagonalizable, and r is a non-negative integer. Let the eigenvalues of A0 be θ1 , . . . , θm . We also assume that the system is non-resonant, that is, for any i, j (2.4)

θj /θi ∈ / q Z≥1 = {q n | n ∈ Z≥1 }.

P∞ n Let P (x) = n=0 Pn x be an m × m formal power series with P0 = Im . The substitution Y (x) = P (x)Z(x) yields (2.5)

Z(qx) = P (qx)−1 A(x)P (x)Z(x).

We can choose the matrix P (x) so that (2.6)

P (qx)−1 A(x)P (x) = xr A0 .

The matrix P (x) can be constructed as follows. The equation (2.6) can be written as (2.7) (A0 + A1 x + A2 x2 + · · · )(Im + P1 x + P2 x2 + · · · ) = (Im + qP1 x + q 2 P2 x2 + · · · )A0 . Equating the coefficients of xn (n ≥ 1) on both sides, we obtain (2.8)

A0 Pn − Pn (q A0 ) = − n

n−1 X

An−k Pk .

k=0

If the coefficient matrices P1 , . . . , Pn−1 are determined, then the equation (2.8) uniquely determines Pn by Proposition 2.1 and non-resonant condition. In this way, the matrix P (x) is constructed inductively. Then the matrix xr A0 is the formal normal form of (2.3) in this case. The construction at x = ∞ is almost the same. Assume that (2.9)

A(x) = xr (A0 + A1 x−1 + A2 x−2 + · · · )

where A0 is invertible, r ∈ Z≥0 . We can construct the transformation matrix P (x) = P∞ −n at x = ∞ such that P (qx)−1 A(x)P (x) = xr A0 holds in a similar way. n=0 Pn x

´ system A q-analogue of the matrix fifth Painleve

§ 2.2.

5

Formal normal form: non-Fuchsian case

In the case that A0 or AN of (2.1) is not invertible, the construction of the formal normal form can be modified as follows. Consider at x = 0 (2.10)

Y (qx) = A(x)Y (x),

A(x) = xr (A0 + A1 x + A2 x2 + · · · ).

Here we assume that A0 = diag(θ1 , . . . , θs , 0, . . . , 0) = Θ ⊕ Om−s ,

(2.11) where for any i, j

θi ̸= 0,

(2.12)

θj /θi ∈ / q Z≥1 .

P∞ n Let P (x) = n=0 Pn x be an m × m formal power series with P0 = Im . We have Z(qx) = P (qx)−1 A(x)P (x)Z(x) by the substitution Y (x) = P (x)Z(x). Set (2.13)

B(x) := P (qx)

−1

A(x)P (x) = x

r

∞ X

Bn xn .

n=0

From the coefficients of x0 in A(x)P (x) = P (qx)B(x), we have B0 = A0 . Then the coefficients of xn (n ≥ 1) gives the following relation: (2.14)

B n = An +

n−1 X

(An−j Pj − q j Pj Bn−j ) + A0 Pn − Pn (q n A0 ).

j=1

Set (n)

(2.15)

Cn =

(n)

C11 C12 (n) (n) C21 C22 (n)

! := An +

n−1 X

(An−j Pj − q j Pj Bn−j )

j=1 (n)

(n)

(n)

for simplicity. Here C11 is s × s, C12 is s × (m − s), C21 is (m − s) × s, and C22 is (m − s) × (m − s). Then we have (2.16)

A0 Pn − Pn (q n A0 ) = Bn − Cn .

Unlike the Fuchsian case, Sp(A0 )∩Sp(q n A0 ) ̸= ∅. Thus the equation (2.16) with respect to Pn does not necessarily have a solution. Instead, we partition Pn conformably with Cn ! (n) (n) P11 P12 (2.17) Pn = (n) (n) P21 P22

6

Hiroshi Kawakami (n)

(n)

and choose Pn so that P11 = O, P22 = O, and (n)

(2.18)

Bn = Cn +

O ΘP12 (n) n −P21 (q Θ) O

!

to be block-diagonal. More specifically, we set (2.19)

P12 = −Θ−1 C12 , (n)

(n)

P21 = q −n C21 Θ−1 . (n)

(n)

Thus we have the following proposition. Proposition 2.2. For any system (2.10) with (2.11) and (2.12) there exists a P∞ (n) formal power series with matrix coefficients P (x) = n=0 Pn xn where P11 = O and (n) P22 = O such that the gauge transformation by P (x) is block-diagonal: ! B1 (x) O (2.20) Z(qx) = B(x)Z(x), B(x) = P (qx)−1 A(x)P (x) = . O B2 (x) Now we apply the above construction to a polynomial coefficient system (2.21)

Y (qx) = A(x)Y (x),

A(x) = A0 + A1 x + · · · + AN xN .

If A0 is of the form (2.11), then the constant term of B1 (x) of (2.20) is Θ. Thus the formal normal form of B1 (x) at x = 0 is Θ. On the other hand, B2 (x) is of the following form: (2.22)

B2 (x) = xr2 B0′ + xr2 +1 B1′ + · · ·

where r2 is a positive integer. If B0′ is similar to Θ′ ⊕ O where Θ′ is diagonal, invertible, and non-resonant (in particular we assume that B0′ is diagonalizable), then B2 (x) can be block-diagonalized into the following form ! xr2 Θ ′ O (2.23) . O xr3 C2 (x) To summarize the above, a linear q-difference system (2.21) satisfying diagonalizability (of the first term of each direct summand) and the non-resonant condition can be transformed into the following block diagonal form:   x r1 B 1   ..  (2.24) Z(qx) = B(x)Z(x), B(x) = P (qx)−1 A(x)P (x) =  .   rk x Bk

´ system A q-analogue of the matrix fifth Painleve

7

where Bj ∈ GLmj (C). ri ’s are non-negative integers satisfying r1 = 0 < r2 < · · · < rk . Here the numbers ri ’s and mj ’s are uniquely determined only by the original system (2.21). Moreover, if we require that any eigenvalue λ of Bj satisfies |q| < |λ| ≤ 1, then the conjugacy class of Bj is uniquely determined (for example see [5]). Then (2.24) is the formal normal form of (2.21) at x = 0. Similarly, the formal normal form of (2.21) at x = ∞ has the following form:   xN −s1 B1′   ..   (2.25) .   N −sℓ ′ Bℓ x where Bj′ ∈ GLm′j (C) and s1 = 0 < s2 < · · · < sℓ . The formal normal form at x = ∞ is also unique in the same sense as above. § 2.3.

Spectral types of linear q-difference systems

First we recall the notion of spectral type of Fuchsian linear q-difference systems [18]. Consider the following Fuchsian linear q-difference system of rank m: (2.26)

Y (qx) = A(x)Y (x),

A(x) = A0 + A1 x + · · · + AN xN

where A0 and AN are invertible. We assume that, for any a ∈ C, A(a) ̸= O. In addition, we assume that A0 and AN are diagonalizable for simplicity. Let the eigenvalues of A0 be θj (j = 1, . . . , k), and let their multiplicities be mj (j = 1, . . . , k). Also, let the eigenvalues of AN be κj (j = 1, . . . , ℓ), and let their multiplicities be nj (j = 1, . . . , ℓ): (2.27)

A0 ∼ θ 1 I m 1 ⊕ · · · ⊕ θ k I m k ,

A N ∼ κ1 I n1 ⊕ · · · ⊕ κℓ I nℓ .

Then we define partitions S0 and S∞ of m as (2.28)

S0 = m1 , . . . , mk ,

S ∞ = n1 , . . . , n ℓ .

Let ZA be the set of the zeros of det A(x): (2.29)

ZA = {a ∈ C | det A(a) = 0} = {α1 , . . . , αp }.

We denote by di (i = 1, . . . , m) the elementary divisors of A(x). Here we assume that di+1 |di . For any αi ∈ ZA , we denote by n ˜ ik the order of αi in dk . For each i, let {nij }j be the partition conjugate to {˜ nik }k . Then we define Sdiv as (2.30)

Sdiv = n11 . . . n1k1 , . . . , np1 . . . npkp .

We call the triple [S0 ; S∞ ; Sdiv ] the spectral type of the Fuchsian system (2.26).

8

Hiroshi Kawakami

Example 2.3.

Consider a linear q-difference system Y (qx) = A(x)Y (x) with A(x) = A0 + A1 x + A2 x2 ,

where A0 is similar to diag(θ1 , θ1 , θ1 , θ2 ), A2 is similar to diag(κ1 , κ1 , κ2 , κ2 ), and the Smith normal form of A(x) is   (x − α1 )(x − α2 )(x − α3 )2 (x − α4 )(x − α5 )  (x − α1 )(x − α2 )     .  1  1 Then the spectral type of the system is [3, 1; 2, 2; 2, 2, 11, 1, 1]. Spectral types can also be defined for non-Fuchsian systems. Taking the formal r1 -tuple

rk -tuple

z }| { z }| { normal form (2.24) into account, we can define S0 as S0 = (· · · (λ1 ) · · · ), . . . , (· · · (λk ) · · · ) where λj is the partition of mj determined by the multiplicities of the eigenvalues of Bj . For example, if the normal form of A(x) around x = 0 is (B1 ) ⊕ (xB2 ) ⊕ (x3 B3 ) and the partitions corresponding to Bj ’s are (2.31)

B1 : 3, 1

B2 : 2, 1

B3 : 2, 2, 2

then S0 = 3, 1, (2, 1), (((2, 2, 2))). s1 -tuple

sℓ -tuple

z }| { z }| { = (· · · (λ′1 ) · · · ), . . . , (· · · (λ′ℓ ) · · · )

Similarly, taking (2.25) into account, we define S∞ as S∞ where λ′j is the partition of m′j corresponding to Bj′ . Sdiv is the same as in the Fuchsian case. Then the triple [S0 ; S∞ ; Sdiv ] is the spectral type. § 3.

q-matrix PVI

In this section we review the q-matrix PVI [12], which describes a connectionpreserving deformation of the Fuchsian linear q-difference system of spectral type [m, m; m, m− 1, 1; m, m, m, m] (see [6, 12] for the connection-preserving deformation). Consider a linear q-difference system of the following form: (3.1)

Y (qx) = A(x)Y (x),

A(x) = A0 + A1 x + A2 x2 ,

Aj ∈ M2m (C),

where (3.2)

A2 =

κ1 I m O O K

!

m−1

,

z }| { K = diag(κ2 , . . . , κ2 , κ3 ),

A0 ∼

θ1 tIm O O θ2 tIm

! .

´ system A q-analogue of the matrix fifth Painleve

9

Since Sdiv = m, m, m, m, the Smith normal form of the polynomial matrix A(x) is of the following form: ! Im O Q4 (3.3) . O i=1 (x − αi )Im Q4 That is, d1 = · · · = dm = i=1 (x − αi ), dm+1 = · · · = d2m = 1, so we have n ˜ ik = 1 (i = 1, 2, 3, 4, k = 1, . . . , m). We assume that αj ’s depend on t as follows: ( aj t (j = 1, 2), (3.4) αj = aj (j = 3, 4). We also assume qαi ̸= αj (i ̸= j). The linear q-difference systems satisfying the above conditions can be parametrized as follows: (3.5) A(x) =

W K{κ1 (xIm − F )(xIm − α) + κ1 G1 }K −1 W −1 W K(xIm − F ) −1 κ1 (γx + δ)W K(xIm − β)(xIm − F ) + KG2

where (3.6)  α = (κ1 − K)−1 (θ1 + θ2 )tF −1 − κ1 F −1 G1 − KG2 F −1 + K(F + G−1 1 F G 1 + β1 ) , (3.7)  β = (κ1 − K)−1 −(θ1 + θ2 )tF −1 + κ1 F −1 G1 + KG2 F −1 − κ1 (F + G−1 1 F G 1 + β1 ) , (3.8) 2 −1 γ = K{G1 + G2 + F α + βF + βα − G−1 , 1 (F + β1 F + β2 )G1 }K (3.9) 2 −1 δ = κ−1 − κ1 K(G2 + βF )F −1 (G1 + F α)}K −1 . 1 {t θ1 θ2 F Here the auxiliary parameters βj ’s are defined by (3.10)

4 X

j

β4−j z :=

j=0

4 Y

(z − αj ).

j=1

Also, the matrices G1 and G2 satisfy (3.11)

G1 G2 = (F − α1 Im )(F − α2 Im )(F − α3 Im )(F − α4 Im ).

The relation (3.11) allows us to introduce a new variable G by (3.12)

−1 G1 = q −1 κ−1 , 1 (F − α1 )(F − α2 )G

G2 = qκ1 G(F − α3 )(F − α4 ).

!

10

Hiroshi Kawakami

Then, from the assumption about the Smith normal form (3.3), F and G must satisfy the following commutation relation: (3.13)

F −1 GF G−1 = ρK,

ρ=

a 1 a 2 a 3 a 4 κ1 . θ1 θ2

Since (3.14)

m−1 det A(x) = κm κ3 1 κ2

4 Y

(x − αi )m ,

i=1

we have (3.15)

κ1 m κ2 m−1 κ3

4 Y

ai m = θ1 m θ2 m .

i=1

Let us consider the connection-preserving deformation of the system (3.1). We choose t as a deformation parameter. The parameters θj , κj , and aj ’s are independent of t. In the following we write A(x, t) instead of A(x) when it is necessary to emphasize that A(x) depends on t. The connection-preserving deformation of (3.1) is given by (3.16)

Y (x, qt) = B(x, t)Y (x, t)

where (3.17)

x(xI2m + B0 ) B(x, t) = , (x − qa1 t)(x − qa2 t)

B0 =

B11 B12 B21 B22

! .

Here Bij ’s are m × m matrices and given as follows: h i −1 (3.18) B11 = qW K(Im − GK)−1 GK K −1 G {F − (a1 + a2 )t} + β K −1 W −1 , (3.19)

(3.20)

(3.21) (3.22)

B12 = qW K(Im − GK)−1 G, o n −1 B21 = qκ1 q −1 κ−1 (F − qa t)G − qa t + α (Im − qκ1 G)−1 2 1 1 n o −1 −1 × GK K G (F − a2 t) − a1 t + β K −1 W −1 o n −1 = qκ1 q −1 κ−1 (F − qa t)G − qa t + α (Im − qκ1 G)−1 1 2 1 n o −1 −1 × GK K G (F − a1 t) − a2 t + β K −1 W −1 , h i −1 B22 = q −1 κ−1 − q(a + a )t}G + α qκ1 G(Im − qκ1 G)−1 . {F 1 2 1

Here the overline denotes the q-shift with respect to t: f = f (qt) for f = f (t).

´ system A q-analogue of the matrix fifth Painleve

11

Now we have the pair of linear q-difference systems: ( Y (qx, t) = A(x, t)Y (x, t), (3.23) Y (x, qt) = B(x, t)Y (x, t). Then the compatibility condition of (3.23) (3.24)

A(x, qt)B(x, t) = B(qx, t)A(x, t)

reduces to a system of non-linear q-difference equations satisfied by F , G, and W . Theorem 3.1 ([12]). is equivalent to (3.25) (3.26) (3.27)

The compatibility condition A(x, qt)B(x, t) = B(qx, t)A(x, t)

1 (F − a1 t)(F − a2 t)(F − a3 )−1 (F − a4 )−1 , qκ1    −1
−1 θ1 θ2 a1 a2 a1 a2 1 F KF = G−t G−t G− G−ρ , κ1 a 1 a 2 θ1 θ2 qκ1   1 −1 −1 −1 G− W W = qκ1 (G − K ) K −1 . qκ1 GKG =

We call the system (3.25) and (3.26) (with (3.13)) the q-matrix PVI . Although this system appears to have eight parameters (θi ’s, κi ’s, and ai ’s with a single relation (3.15)), the number of parameters can be reduced to five by rescaling F , G, and t. § 4.

Degeneration of q-matrix PVI

Now we consider a degeneration of the q-matrix PVI which corresponds to the limit κ1 to 0. § 4.1.

From q-matrix PVI to q-matrix PV

Consider the following transformation: (4.1)

t = εt˜,

F = εF˜ ,

˜ G = εG,

a4 = −ε−1 κ ˜1 ,

a3 = −ε˜ a3 ,

˜, W = εW κ1 = ε,

κ2 = ε−1 κ ˜2 ,

κ3 = ε−1 κ ˜3 .

m−1

z }| { ˜ = diag(κ ˜ The other parameters We set K ˜2 , . . . , κ ˜2 , κ ˜ 3 ) so that we have K = ε−1 K. a1 , a2 and θ1 , θ2 are not changed. This transformation is compatible with the commu˜ F˜ G ˜ −1 = ρ˜K ˜ holds where ρ˜ = a1 a2 a˜3 κ˜ 1 . From the tation relation (3.13), that is, F˜ −1 G θ1 θ2

relation (3.15), we have (4.2)

κ ˜ 1m κ ˜ 2m−1 κ ˜ 3 a1 m a2 m a ˜3m = θ1 m θ2 m .

12

Hiroshi Kawakami

Substituting (4.1) into (3.25), we have ˜K ˜G ˜ = ε (F˜ − a1 t˜)(F˜ − a2 t˜)(F˜ + a (4.3) εG ˜3 )−1 (ε2 F˜ + κ ˜ 1 )−1 . q Letting ε → 0, we obtain ˜K ˜G ˜ = 1 (F˜ − a1 t˜)(F˜ − a2 t˜)(F˜ + a G ˜3 )−1 . q˜ κ1

(4.4)

Similarly, from the equation (3.26) we have    −1  −1 θ θ a a a a 1 1 2 1 2 1 2 2 ˜ F˜ = ε ˜ − t˜ ˜ − t˜ ˜− ˜ − ρ˜ G G G . (4.5) εF˜ K ε G a1 a2 θ1 θ2 q Letting ε → 0, we obtain ˜ F˜ = −q θ1 θ2 F˜ K a1 a2

(4.6)



˜ − t˜a1 a2 G θ1



˜ − t˜a1 a2 G θ2



−1 ˜ G − ρ˜ .

From the equation (3.27) we have

  1 ˜ −1 −1 ˜ −1 −1 2˜ ˜ ˜ ˜ ε G− W W = q(G − K ) K . q

(4.7)

Letting ε → 0, we obtain ˜ −1 W ˜ = −(K ˜G ˜ − Im )−1 . W

(4.8)

Omitting the tilde, we obtain the following system of non-linear q-difference equations 1 (4.9) (F − a1 t)(F − a2 t)(F + a3 )−1 , GKG = qκ1   
−1 θ1 θ2 a1 a2 a1 a2 (4.10) F KF = −q G−t G−t G−ρ , a1 a2 θ1 θ2 W −1 W = (Im − KG)−1 .

(4.11)

The associated linear system (3.1) can also be degenerated in the same manner as above. Set x = ε˜ x. Notice that (4.12) α = −˜ κ1 ε−1 + O(1), β = O(ε), γ = O(1), δ = O(ε), G1 = O(1), G2 = O(ε2 ). We set ˜ := lim β , γ ˜ := lim γ, β ε→0 ε→0 ε Then it is easy to see that (4.13)

δ δ˜ := lim , ε→0 ε

˜ 1 := lim G1 , G ε→0

˜ 2 := lim G2 . G ε→0 ε2

(4.14) ˜ x) := lim ε−1 A(x) = A(˜ ε→0

˜ K{˜ ˜ κ1 (˜ ˜ 1 }K ˜ −1 W ˜ −1 ˜ K(˜ ˜ xIm − F˜ ) W xIm − F˜ ) + G W ˜ W ˜ xIm − F˜ ) + K ˜ −1 ˜ xIm − β)(˜ ˜G ˜2 (˜ γx ˜ + δ) K(˜

! .

´ system A q-analogue of the matrix fifth Painleve

13

Remark. The multiplication of A(x) by ε−1 can be realized by a simple gauge transformation of the linear system. For example, consider the transformation Y = xlog ε/ log q Y˜ or use the ratio of theta functions (5.12) instead of xlog ε/ log q . Then we have Y˜ (qx) = ε−1 A(x)Y˜ (x). Thus we obtain (by omitting the tilde) A(x) = (4.15)

W K{κ1 (xIm − F ) + G1 }K (γx + δ)W −1

−1

W

−1

W K(xIm − F ) K(xIm − β)(xIm − F ) + KG2

!

=: A0 + A1 x + A2 x2 ,

where (4.16)

β = K −1 {(θ1 + θ2 )tF −1 − F −1 G1 − KG2 F −1 + κ1 },

(4.17)

γ = K{G1 − κ1 (F + β + GF G−1 − (a1 + a2 )t + a3 )}K −1 ,

(4.18)

δ = F −1 (G1 − κ1 F − θ1 t)(G1 − κ1 F − θ2 t)K −1 .

From the determinant of (4.15) we have

(4.19)

m

κ1 κ 2

m−1

κ3

3 Y

ai m = θ1 m θ2 m .

i=1

The matrices G1 and G2 are given by (4.20)

G1 = q −1 (F − a1 t)(F − a2 t)G−1 ,

G2 = qκ1 G(F + a3 )

and satisfy (4.21)

G1 G2 = κ1 (F − a1 t)(F − a2 t)(F + a3 ).

The matrices F and G satisfy the following commutation relation: (4.22)

F −1 GF G−1 = ρK,

ρ=

a 1 a 2 a 3 κ1 . θ1 θ2

The system in t-direction (3.16) can also be degenerated in the same manner. As a result, we have (by omitting the tilde) (4.23)

x(xI2m + B0 ) B(x, t) = , (x − qa1 t)(x − qa2 t)

B0 =

B11 B12 B21 B22

!

14

Hiroshi Kawakami

where Bij ’s are m × m matrices given by (4.24)

 B11 = qW K(Im − GK)−1 F − (a1 + a2 )t + GKβ K −1 W −1 ,

(4.27)

B12 = qW K(Im − GK)−1 G, n o
−1 B21 = (F − qa2 t)G − qκ1 F − a2 t − a1 tGK + GKβ K −1 W −1 n o
−1 = (F − qa1 t)G − qκ1 F − a1 t − a2 tGK + GKβ K −1 W −1 ,

(4.28)

B22 = F − q(a1 + a2 )t − qκ1 G.

(4.25) (4.26)

We obtain the following theorem by a direct calculation. Theorem 4.1. The compatibility condition A(x, qt)B(x, t) = B(qx, t)A(x, t) with (4.15) and (4.23) is equivalent to (4.29) (4.30) (4.31)

1 (F − a1 t)(F − a2 t)(F + a3 )−1 , qκ1   
−1 θ1 θ2 a1 a2 a1 a2 F KF = −q G−t G−t G−ρ , a1 a2 θ1 θ2

GKG =

W −1 W = −(KG − Im )−1 .

We call the system (4.29) and (4.30) (with (4.22)) the q-matrix fifth Painlev´e system (q-matrix PV ). Although this system appears to have seven parameters (θi ’s, κi ’s, and ai ’s with a single relation (4.19)), the number of parameters can be reduced to four by rescaling F , G, and t. § 4.2.

Characterization of the linear system

The matrix (4.15) satisfies (C1): A0 is similar to θ1 tIm ⊕ θ2 tIm . (C2): The formal normal form of A(x) at x = ∞ is ! x2 K O (4.32) . O x(κ1 Im ) (C3): The Smith normal form of A(x) is ! Im O Q3 (α1 = a1 t, α2 = a2 t, α3 = −a3 ). (4.33) O j=1 (x − αj )Im Conversely, it can be shown that a polynomial matrix A(x) satisfying the above three conditions can be written (generically) in the form (4.15). Thus the linear system associated with the q-matrix PV is characterized by the conditions (C1), (C2), and (C3).

´ system A q-analogue of the matrix fifth Painleve

15

From the definition given in Section 2.3, the spectral type of the system is written as [m, m; m − 1, 1, (m); m, m, m]. § 5.

Continuous limit of q-matrix PV

The system (4.29) and (4.30) can be viewed as a q-analogue of the matrix PV (Appendix A.6). That is, taking the limit q → 1, one can obtain (Appendix A.6) from (4.29) and (4.30). In fact, let us define the parameter ε by q = 1 − ε. We set (5.1)

θi = 1 − σi ε (i = 1, 2),

κ1 = −1 − µ1 ε,

ai = 1 + ζi ε (i = 1, 2),

a3 = ε−1 ,

κi = ε(1 + µi ε) (i = 2, 3),

m−1

z }| { and M = diag(µ2 , . . . , µ2 , µ3 ). Moreover, we introduce new dependent variables Q and P which are related to F and G by (5.2) (5.3) (5.4) (5.5)

˜ −1 Q, ˜ F = −(P˜ + ε−1 t)(ϕ1 − ϕ2 Q)

˜ −1 , G = (P˜ + ε−1 t)(ϕ1 − ϕ2 Q)

σ1 + σ2 ζ1 + ζ2 ϕ1 = ε−1 − 1 − ζ1 − ζ2 − , ϕ2 = ε−1 − , 2  2  ζ − ζ + σ − σ ζ − ζ 2 1 1 2 1 2 −1 ˜ = Im − Q ˆ , P˜ = t (Q ˆ − Im )Pˆ Q ˆ+ ˆ+ Q Q , 2 2 ˆ = g −1 Qg, Pˆ = g −1 P g. Q

−1 = 1t M . Here g = tM , which is a solution to dg dt g Then, taking the limit ε → 0, we find that Q and P satisfy the following equations:

(5.6) dQ t = Q(Q − 1)(P + t) + P (Q − 1)Q − (ζ1 − ζ2 )(Q − 1) + (σ1 − σ2 )Q, dt (5.7) dP t = −(Q − 1)P (P + t) − (P + t)P Q − (ζ2 − ζ1 + σ1 − σ2 )P − (ζ2 + ζ4 + σ1 )t. dt These equations coincide with (Appendix A.6) by the following correspondence of the parameters: (5.8)

σ1 − σ2 = θ 0 ,

ζ 1 − ζ2 = θ 1 ,

µi + ζ2 + σ2 = θi∞ (i = 1, 2, 3).

Expanding (4.22) with respect to the small parameter ε and taking the coefficient of ε , we have the commutation relation between P and Q: 1

(5.9)

P Q − QP = (µ1 + ζ1 + ζ2 + σ1 + σ2 )Im + M.

16

Hiroshi Kawakami

The linear system (4.15) also admits the continuous limit in a similar way. To see this, we first change the dependent variable Y to Z: Y (x) = f (x)Z(x), where f (x) is a solution of the following q-difference equation f (qx) = −(x − t)f (x).

(5.10) For example, we can take (5.11)

f (x) =

ϑq (x/t) , (x/t; q)∞ ϑq (x)

where (5.12)

(a; q)∞ =

∞ Y

(1 − aq ), n

ϑq (x) =

n=0

∞ Y

(1 − q n+1 )(1 + xq n )(1 + x−1 q n+1 ).

n=0

Then we have (5.13)

Z(x) − Z(qx) 1 = (1 − q)x εx



 1 I2m − A(x) Z(x). −(x − t)

Set W = tU −1 . Define matrices A0 , A1 , and A∞ by   1 1 A0 A1 (5.14) lim I2m − A(x) = + + A∞ . ε→0 εx −(x − t) x x−t It can be shown that the matrices A0 , A1 , and A∞ (almost) coincide with (Appendix A.2). More precisely, performing suitable scalar gauge transformations (in other words, adding suitable scalar matrices to A0 , A1 , and A∞ ), performing the gauge transformation by (5.15)

h=

O Im Im O

! ,

and setting x = t˜ x, we have (5.16) h−1 (A0 − σ2 I2m )h = A0 ,

h−1 (A1 − ζ2 I2m )h = A1 ,

h−1 (tA∞ − tI2m )h = A∞ .

Thus the resulting system of linear differential equations (5.17)

dZ˜ = d˜ x



 A0 A1 + + A∞ Z˜ x ˜ x ˜−1

coincides with the x-direction of (Appendix A.1).

´ system A q-analogue of the matrix fifth Painleve

§ Appendix A.

17

The matrix fifth Painlev´ e system

In this appendix, we review the matrix fifth Painlev´e system (matrix PV ) [8, 11]. The matrix PV is derived from the isomonodromic deformation of a certain linear differential system. There are several Lax pairs for the matrix PV , one of them is the following:    A0 A1 ∂Y   = + + A∞ Y,  ∂x x x−1 (Appendix A.1)    ∂Y = (−E2 ⊗ Im x + B1 )Y, ∂t where (Appendix A.2) Aξ = (Im ⊕ U )−1 Aˆξ (Im ⊕ U ) (ξ = 0, 1), !  0 ∞ 1 QP + θ + θ 1 ∞ 0 ∞ ˆ Im − Q, {(Q − Im )QP + (θ + θ1 )Q − θ1 } , A0 = t tIm !  1 0 ∞ 1 )Q + θ (Q − I )P Q + (θ + θ m 1 0 ∞ ˆ Im , {(Im − Q)P − θ − θ1 } , A1 = t tQ ! ! ! Om Om 00 θ2∞ Im−1 A∞ = , E2 = , Θ= . Om −tIm 01 θ3∞ Furthermore, the matrix B1 is given by (Appendix A.3)

B1 = (Im ⊕ U )

−1

Om ˆ0 +A ˆ1 ]21 [A t

ˆ0 +A ˆ1 ]12 [A t

Om

! (Im ⊕ U ),

where [Aˆ0 + Aˆ1 ]ij is the (i, j)-block of the matrix Aˆ0 + Aˆ1 . The Fuchs-Hukuhara relation is written as m(θ0 + θ1 + θ1∞ ) + (m − 1)θ2∞ + θ3∞ = 0. P and Q satisfy [P, Q] = (θ0 + θ1 + θ1∞ )Im + Θ. The system in x-direction of (Appendix A.1) is characterized by the spectral type (m)(m − 1 1), mm, mm. The compatibility condition (in other words, isomonodromic deformation equation) for (Appendix A.1) has two descriptions, which are mutually equivalent. One is the Hamiltonian form and the other is the “non-abelian” form. The Hamiltonian is given by  0  −θ − θ1 − θ1∞ , θ0 − θ1 Mat,m tHV (Appendix A.4) ; t; Q, P θ1 , θ0 + θ1 + θ1∞ + θ2∞ = tr[P (P + t)Q(Q − 1) + (θ0 − θ1 )P Q + θ1 P + (θ0 + θ1∞ )tQ]. Then the compatibility condition can be written as follows: (Appendix A.5)

Mat,m ∂HV dqij = , dt ∂pji

Mat,m ∂HV dpij =− . dt ∂qji

18

Hiroshi Kawakami

On the other hand, the non-abelian description is given as follows [11]: (Appendix A.6)  dQ  t = Q(Q − 1)(P + t) + P Q(Q − 1) + (θ0 − θ1 )Q + θ1 , dt  t dP = −(Q − 1)P (P + t) − P (P + t)Q − (θ0 − θ1 )P − (θ0 + θ∞ )t. 1 dt

References ´ [1] P. Boalch, Simply-laced isomonodromy systems, Publ. Math. Inst. Hautes Etudes Sci. 116, No. 1 (2012), 1–68. [2] K. Fuji and T. Suzuki, Drinfeld-Sokolov hierarchies of type A and fourth order Painlev´e systems, Funkcial. Ekvac. 53 (2010), 143–167. [3] R. Garnier, Sur des ´equations diff´erentielles du troisi`eme ordre dont l’int´egrale g´en´erale est uniforme et sur une classe d’´equations nouvelles d’ordre sup´erieur dont l’int´egrale g´en´erale ´ Norm. Sup´er. 29 (1912), 1–126. a ses points critiques fixes, Ann. Sci. Ec. [4] K. Hiroe, H. Kawakami, A. Nakamura, and H. Sakai, 4-dimensional Painlev´e-type equations, MSJ Memoirs 37 (2018). [5] C. Hardouin, J. Sauloy, and M. F. Singer, Galois theories of linear difference equations: an introduction, Mathematical Surveys and Monographs Volume 211, American Mathematical Society (2016). [6] M. Jimbo and H. Sakai, A q-analog of the sixth Painlev´e equation, Lett. Math. Phys. 38 (1996), 145–154. [7] N. M. Katz, Rigid local systems, Annals of Mathematics Studies 139, Princeton University Press (1995). [8] H. Kawakami, Matrix Painlev´e systems, J. Math. Phys. 56 (2015), doi.org/10.1063/1.4914369. [9] H. Kawakami, Four-dimensional Painlev´e-type equations associated with ramified linear equations III: Garnier systems and FS systems, SIGMA 13 (2017), 096, 50 pages. [10] H. Kawakami, Four-dimensional Painlev´e-type equations associated with ramified linear equations II: Sasano systems, Journal of Integrable Systems, Volume 3, Issue 1 (2018), xyy013. [11] H. Kawakami, Four-dimensional Painlev´e-type equations associated with ramified linear equations I: Matrix Painlev´e systems, Funkcial. Ekvac. 63 (2020), 97–132. [12] H. Kawakami, A q-analogue of the matrix sixth Painlev´e system, J. Phys. A: Math. Theor. 53 (2020). [13] H. Kawakami, Four-dimensional Painlev´e-type difference equations, arXiv:1802.00116. [14] T. Masuda, A q-analogue of the higher order Painlev´e type equations with the affine Weyl group symmetry of type D, Funkcial. Ekvac. 58 (2015), 405–430. [15] T. Oshima, Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs 28 (2012). [16] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlev´e equations, Comm. Math. Phys. 220 (2001), 165–229. [17] H. Sakai, A q-analog of the Garnier system, Funkcial. Ekvac. 48 (2005), 273–297. [18] H. Sakai and M. Yamaguchi, Spectral types of linear q-difference equations and q-analog of middle convolution, Int. Math. Res. Not., Volume 2017, Issue 7 (2017), 1975–2013.

´ system A q-analogue of the matrix fifth Painleve

19

[19] Y. Sasano, Coupled Painleve VI systems in dimension four with affine Weyl group sym(1) okyˆ uroku Bessatsu B5 (2008), 137–152. metry of type D6 . II, RIMS Kˆ [20] T. Suzuki, A q-analogue of the Drinfeld-Sokolov hierarchy of type A and q-Painlev´e system, AMS Contemp. Math. 651 (2015), 25–38. [21] T. Tsuda, On an Integrable System of q-Difference Equations Satisfied by the Universal Characters: Its Lax Formalism and an Application to q-Painlev´e Equations, Comm. Math. Phys. 293 (2010), 347–359. [22] T. Tsuda, UC hierarchy and monodromy preserving deformation, J. Reine Angew. Math. 690 (2014), 1–34.

RIMS Kˆ okyˆ uroku Bessatsu B94 (2023), 021–036

Box and ball system with numbered boxes and balls By

Yusaku Yamamoto∗, Akiko Fukuda∗∗ , Emiko Ishiwata∗∗∗ and Masashi Iwasaki†

Abstract Box and ball systems (BBSs) are known as discrete dynamical systems in which motions of balls among successive infinite boxes are governed by an ultradiscrete integrable system. The equation of motion in the simplest BBS is the ultradiscrete version of the discrete Toda equation, which is one of famous discrete integrable systems. The discrete Toda equation is extended to two types of discrete hungry Toda (dhToda) equations, and their ultradiscretizations are shown to be the equations of motion in the BBSs in which either boxes or balls are numbered. In this paper, we propose a new box-ball system in which both boxes and balls are numbered, and show that its equation of motion is the ultradiscretization of a variant of the dhToda equations. With the help of a combinatorial technique, we describe conserved quantities of our new numbered BBS (nBBS). We also clarify its relationship to the hungry ε-BBS, which is derived from the ultradiscretization of another extension of the discrete Toda equation.

§ 1.

Introduction

The box and ball system (BBS) that was first proposed by Takahashi and Satsuma [8] is a cellular automaton in which each ball, in order from left, moves to the nearest Received February 27, 2023. Revised June 5, 2023. 2020 Mathematics Subject Classification(s): 37B15, 37C79, 65F15 Key Words: Box and ball system, Numbered boxes and balls, Discrete hungry Toda equation, Ultradiscretization, Conserved quantity. This work was partially supported by a joint project of Kyoto University and Toyota Motor Corporation, titled “Advanced Mathematical Science for Mobility Society”. ∗ Department of Computer and Network Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan. e-mail: [email protected] ∗∗ Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan. e-mail: [email protected] ∗∗∗ Department of Applied Mathematics, Tokyo University of Science, Tokyo 162-8601, Japan. e-mail: [email protected] † Faculty of Life and Environmental Sciences, Kyoto Prefectural University, Kyoto 606-8522, Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

22

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki

empty box on the right among an infinite number of boxes arranged in a straight line. The interactions of groups of successive balls can be regarded as those of the simplified solitons. Under discrete time evolution from n to n + 1, the motion of m ball groups is described using the equation:  ( k ) k−1 ∑ ∑   (n) (n+1) (n) (n+1)  Qi − Qi , Ek Q = min ,    k (1.1)

i=1

i=1

 (n+1) (n+1) (n) (n)  Qk + Ek = Qk+1 + Ek ,     (n) (n) E0 ≡ +∞, Em ≡ +∞, (n)

k = 1, 2, . . . , m,

k = 1, 2, . . . , m − 1,

(n)

where Qk and Ek respectively correspond to the number of successive balls in the kth ball group from the left and of successive boxes in the box array between the kth and (k + 1)th ball groups at discrete time n. Equation (1.1) is derived from the ultradiscretization of the famous discrete Toda (dToda) equation:  (n+1) (n+1) (n) (n)  q + ek−1 = qk + ek ,   k (1.2)

(n+1) (n+1) (n) (n) qk ek = qk+1 ek ,    (n) (0) e0 ≡ 0, em ≡ 0.

k = 1, 2, . . . , m,

k = 1, 2, . . . , m − 1,

Thus, (1.1) is called the ultradiscrete Toda (udToda) equation. By the way, the dToda equation (1.2) can generate LR transformations of tridiagonal matrices where one step first decomposes a tridiagonal matrix into the product of lower and upper bidiagonal matrices, and next inverse the product. In fact, the dToda equation (1.2) is just equal to the recursion formula of the well-known quotient-difference (qd) algorithm for computing tridiagonal eigenvalues [7]. As an extension of the simple BBS, Tokihiro et al. [9] proposed a numbered BBS (nBBS) in which each ball is given one of the numbers 1, 2, . . . , M where M is a positive integer. Concerning the order of ball motion, the assigned number has priority over the ball position. In this case, the equation of motion is given by an extension of the udToda equation (1.1) as:  ) ( k ∑ (n) k−1 ∑ (n+M ) (n)   (n+M )  Qi , Ek , Q = min Qi −    k (1.3)

i=1

k = 1, 2, . . . , m,

i=1

 (n+M ) (n+1) (n) (n)  Qk + Ek = Qk+1 + Ek ,     (n) (n) E0 ≡ +∞, Em ≡ +∞.

k = 1, 2, . . . , m − 1,

The inverse ultradiscretization of (1.3) leads to the discrete hungry Toda (dhToda)

Box and ball system with numbered boxes and balls

23

equation:

(1.4)

 (n+M ) (n+1) (n) (n)  + ek−1 = qk + ek , k = 1, 2, . . . , m,  qk (n+M ) (n+1) (n) (n) qk ek = qk+1 ek , k = 1, 2, . . . , m − 1,   (n) (n) e0 ≡ 0, em ≡ 0.

In [3], we related the dhToda equation (1.4) to LR transformations of totally nonnegative (TN) Hessenberg matrices, and proposed its application to computing the eigenvalues of TN Hessenberg matrices. Moreover, we developed similar studies on a variant of the dhToda equation given as:

(1.5)

 (n+1) (n+N ) (n) (n)  + ek−1 = qk + ek , k = 1, 2, . . . , m,  qk (n+1) (n+N ) (n) (n) qk ek = qk+1 ek , k = 1, 2, . . . , m − 1,   (n) (n) e0 ≡ 0, em ≡ 0,

and then designed another nBBS in which boxes are numbered, where the value of N corresponds to the number of box types [4]. For simplicity, in this paper, we call the nBBSs with numbered balls and with numbered boxes nBBS-I and nBBS-II, respectively. We also distinguish (1.4) and (1.5) by referring to them as the dhToda-I and dhToda-II equations, respectively. The motion of nBBS-II is expressed by using the ultradiscretization of the dhToda-II (1.5) given as:  ) ( k ∑ (n+1) (n) ∑ (n) k−1   (n+1)  Qi , Ek , Q = min Qi −    k (1.6)

i=1

k = 1, 2, . . . , m,

i=1

 (n+1) (n+N ) (n) (n)  Qk + Ek = Qk+1 + Ek ,    (n)  (n) E0 ≡ +∞, Em ≡ +∞.

k = 1, 2, . . . , m − 1,

The dhToda-I equation (1.4) and the dhToda-II equation (1.5) can be unified as the following equation [1].

(1.7)

 (n,j+1) (n,i+1) (n,j) (n,i)  + ej,k−1 = qi,k + ej,k , k = 1, 2, . . . , m,  qi,k (n,j+1) (n,i+1) (n,j) (n,i) qi,k ej,k = qi,k+1 ej,k , k = 1, 2, . . . , m − 1,   (n,i) (n,i) ej,0 ≡ 0, ej,m ≡ 0.

Here, there are M sets of q variables, {qi,k }m k=1 for i = 0, 1, . . . , M − 1 and N sets of e m−1 variables, {ej,k }k=1 for j = 0, 1, . . . , N − 1. The discrete time index of qi,k (resp. ej,k ) consists of the main index n and the sub index j (resp. i), which takes the value between 0 and N − 1 (resp. M − 1). At the beginning of step n, j (resp. i) is set to zero and (n,j) (n,i) (n,i) (n,j) it is incremented by one when qi,k (resp. ej,k ) “interacts” with ej,k (resp. qi,k ) (n,i)

(n,j)

through (1.7). After it interacts with eN −1,k (resp. qM −1,k ), n is incremented by one

24

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki

and j (resp. i) is set to zero. Thus, we have { (n,N ) (n+1,0) qi,k = qi,k , i = 0, 1, . . . , M − 1, (1.8) (n,M ) (n+1,0) ej,k = ej,k , j = 0, 1, . . . , N − 1,

k = 1, 2, . . . , m, k = 1, 2, . . . , m − 1. (n,j)

To see that (1.7) with (1.8) is a generalization of (1.4) and (1.5), let us rewrite qi,k (n,i)

and ej,k

as: {

(1.9)

(n,j)

(M N n+jM +i)

qi,k = qk , (n,i) (M N n+iN +j) ej,k = ek .

Note that this is consistent with (1.8). Now, consider the case of N = 1. Substituting (1.9) into (1.7) and letting N = 1 and j = 0 gives  (M n+M +i) (M n+i+1) (M n+i) (M n+i)  + ek−1 = qk + ek , k = 1, 2, . . . , m,  qk (M n+M +i) (M n+i+1) (M n+i) (M n+i) (1.10) qk ek = qk+1 ek , k = 1, 2, . . . , m − 1,   (M n+i) (M n+i) ≡ 0, em ≡ 0. e0 By rewriting M n + i as n, we recover (1.4). Equation (1.5) can also be obtained by letting M = 1. Note that (1.7) (written in the so-called differential form) is referred to as the multiple dqd algorithm in [10]. It can also be derived from a reduction of the two-dimensional discrete Toda equation [5]. Equation (1.7) is similar to the two dhToda equations (1.4) and (1.5), however (1.7) differs from them in that it involves two types of arbitrary parameters instead of one. Obviously, (1.7) with N = 1 and with M = 1 are respectively equal to the dhToda-I equation (1.4) and the dhToda-II equation (1.5). Thus, we can regard (1.7) as a generalization of the two dhToda equations (1.4) and (1.5). We hereinafter call (1.7) the dhToda-III equation to distinguish it from the two dhToda equations (1.4) and (1.5). Note here that the first equation of the dhToda-III equation (1.7) can be rewritten using ∏k (n+M ) (n) ∏k−1 (n+M ) (n) the second equation repeatedly as qk = ( j=1 qj / j=1 qj ) + ek . Thus, by (n)

(n)

(n)

(n)

replacing qk and ek with exp(−Qk /ε) and exp(−Ek /ε), respectively, taking the logarithm of both sides, multiplying them by ε, and taking the limit ε → +0, we obtain the ultra-discretization of the dhToda-III (udhToda-III) equation:  ( k ) k−1 ∑ ∑   (n,j+1) (n,j) (n,j+1) (n,i)  = min Qi,k′ − Qi,k′ , Ej,k , k = 1, 2, . . . , m,   Qi,k ′ ′ k =1 k =1 (1.11) (n,i+1) (n,j) (n,i) (n,j+1)  E = Q + E − Q , k = 1, 2, . . . , m − 1,  j,k i,k+1 j,k i,k    (n,i) (n,i) Ej,0 := +∞, Ej,m := +∞. In this paper, we propose a new nBBS associated with the udhToda-III equation (1.11), and then derive the conserved quantities of the resulting nBBS. Moreover, we clarify

Box and ball system with numbered boxes and balls

25

the relationship of the resulting nBBS to the hungry ε-BBS presented in Kobayashi and Tsujimoto [6]. The remainder of this paper is organized as follows. In Section 2, we first design a new nBBS with numbered both boxes and balls, and then associate it with the udhTodaIII equation (1.11). In Section 3, based on the relationship to eigenvalue problem and the correspondence to combinatorial representation, we next derive conserved quantities of the resulting nBBS. In Section 4, we show that discrete-time evolutions and conserved quantities of the hungry ε-BBS can be grasped from the viewpoint of the resulting nBBS. Finally, we give concluding remarks. § 2.

Box and ball system with numbered both boxes and balls

In this section, we propose a box-ball system with numbered boxes and balls that has the udhToda-III equation (1.11) as its equation of motion. As in the basic BBS, we assume that an infinite number of boxes are arranged in a straight line. We assume that only one ball can be put in one box, and the number of balls is finite. The types of balls and boxes are distinguished by identification numbers, which take a value between 0 and M − 1 for balls and between 0 and N − 1 for boxes. We here emphasize that M and N are arbitrary parameters corresponding to those in the udhToda-III equation (1.11). In the following, we refer to a set of balls in consecutive boxes as a ball group and a set of consecutive empty boxes between two ball groups as a box array. At discrete time n, we assign identification numbers to boxes and balls so that the following four conditions hold: (a) Every ball has an identification number, which is one of 0, 1, . . . , M − 1. (b) For each i, 0 ≤ i ≤ M − 1, each ball group contains one or more balls with identification number i, and the balls in each ball group are lined up so that their identification numbers are in ascending order from the left. (c) Every box between ball groups has an identification number, which is one of 0, 1, . . . , N −1. Only boxes between the leftmost and rightmost ball groups are numbered. (d) For each j, 0 ≤ j ≤ N − 1, each box array between ball groups contains one or more boxes with identification number j, and the boxes in each box array are lined up so that their numbers are in ascending order from the left. We hereinafter omit “from the left” in describing the order of each box array and each ball group from the left. Now, we define the rules for discrete time evolution of our new nBBS with numbered balls and boxes. The time evolution of the nBBS from n to n + 1 consists of M N substeps. At the (jM + i)th substep (0 ≤ i ≤ M − 1, 0 ≤ j ≤ N − 1), we move the boxes and balls as follows:

26

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki n=0

…

…

0 0 0 1 1 2 2 0 0 1 1 1 1 1 0 0 1 1 2 0 0 0 0 0 1 1 0 1 2

(i)

(i, j)

1 1 2 2 0 0 1 1 1 1 1

1 2 0

…

…

1 1 2 2 0 0 1 1 1 1 1 0 0 1 1 2 0 0 0 0 0 1 1 0 1 2 0

…

=

…

(0,0)

1 1 2 0 0 0 0 0 1 1

(ii) (iii) …

…

1 1 2 2 0 0 0 0 1 1 1 1 1 1 1 2 0 0 0 0 0 0 1 1 1 2 0

(i, j)

…

2 2 0 0 1 1 1 1 1 1 1

2 0 1

…

=

…

2 2 0 0 1 1 1 1 1 1 1 0 0 2 0 0 0 1 1 0 1 1 0 2 0 1

…

…

2 2 0 0 1 1 0 0 1 1 1 1 1 2 0 0 0 1 1 0 0 1 1 2 0 1

…

(1,0)

2 0 0 0 1 1 0 1 1

(i, j)

…

0 0 1 1 2 2 1 1 1 1 1

0 1 2

…

=

…

0 0 1 1 2 2 1 1 1 1 1 0 0 0 0 1 1 2 0 1 1 0 0 1 2

…

…

0 0 1 1 2 2 1 1 1 1 1 0 0 0 0 1 1 2 1 1 0 0 0 1 2

…

(2,0)

0 0 0 1 1 2 0 1 1

(i, j)

…

1 1 2 2 0 0 1 1 1 0

1 2 0 0

…

=

…

1 1 2 2 0 0 1 1 1 0 1 1 1 1 1 2 0 0 0 0 1 1 2 0 0

…

…

1 1 2 2 0 0 1 1 1 1 1 1 0 1 1 2 0 0 1 0 0 1 2 0 0

…

(0,1)

1 1 2 0 0 0 0

(i, j)

…

2 2 0 0 1 1 1 1 1 1 0

2 0 0 1 1

…

=

…

2 2 0 0 1 1 1 1 1 1 0 1 1 2 0 0 1 0 0 1 2 0 0 1 1

…

…

2 2 0 0 1 1 1 1 1 1 1 1 0 2 0 0 1 1 0 0 2 0 0 1 1

…

(1,1)

2 0 0 1 0 0

(i, j)

…

0 0 1 1 2 2 1 1 1 1 0

0 0 1 1 2

…

=

…

0 0 1 1 2 2 1 1 1 1 0 1 0 0 1 2 0 0 1 0 0 1 1 2

…

…

0 0 1 1 2 2 0 1 1 1 1 1 0 0 1 2 0 0 1 0 0 1 1 2

…

(2,1)

0 0 1 2 0 0

n=1

Figure 1. An example of discrete time evolution from n = 0 to n = 1 in the case where M = 3 and N = 2. (i) Starting from the leftmost ball with number i, move each ball with number i, one by one, to the nearest right empty box numbered j or without an identification number. (ii) After moving the balls numbered i, delete identification number j from boxes newly filled with a ball. Moreover, assign the identification number j to boxes that become empty, except for boxes to the left of the leftmost ball group. (Boxes to the left of the leftmost ball group do not have a box identification number.) (iii) If i < M − 1, in each box array, gather all boxes with number j to the left end of the box array. If i = M − 1, in each box array, gather all boxes with number j to the right end of the box array. We refer to the nBBS defined by the conditions (a)–(d) and the rules (i)–(iii) as nBBSIII. Figure 1 shows an example of discrete time evolution from n = 0 to n = 1 of nBBS-III. Now, we show a lemma concerning discrete time evolution of nBBS-III. To this end, we define new conditions (b)i and (d)j , which are slight generalizations of the conditions (b) and (d), respectively.

Box and ball system with numbered boxes and balls

27

(b)i For each p, 0 ≤ p ≤ M − 1, each ball group contains one or more balls with identification number p, and the balls in each ball group are lined up so that their identification numbers are in the order of i, i + 1, . . . , M − 1, 0, 1, . . . , i − 1. (d)j For each q, 0 ≤ q ≤ N − 1, each box array between ball groups contains one or more boxes with identification number q, and the boxes in each box array are lined up so that their numbers are in the order of j, j + 1, . . . , N − 1, 0, 1, . . . , j − 1. Lemma 2.1. Consider the (jM + i)th substep during discrete time evolution from n to n + 1, where 0 ≤ i ≤ M − 1 and 0 ≤ j ≤ N − 1. Let i′ = mod(i + 1, M ) and j ′ = ⌊(jM + i + 1)/M ⌋ where ⌊·⌋ denotes the greatest integer part of a real number. If the conditions (a), (b)i , (c) and (d)j hold and the number of ball groups is m at the beginning of the substep, then the conditions (a), (b)i′ , (c) and (d)j ′ hold at the end of the substep and the number of ball groups remains unchanged. Proof. It is clear that (a) holds because the identification number of each ball does not change throughout discrete time evolution. It is also clear that (c) holds because the boxes that become empty are given number j unless it lies to the left of the leftmost ball group, the boxes newly filled with a ball are deprived of the numbers, and the numbers of other boxes are unchanged. Now, we show that m remains unchanged by induction. By assumption, the balls to be moved, those with number i, are at the left end of each group. Among them, the leftmost one is moved first, the second leftmost one next, and so on. The boxes that become empty due to these movements are not filled with another ball at this substep. Also, balls with numbers other than j exist in each group and they are not moved. From these facts, it is clear that no ball groups vanish nor split into two or more groups due to the removal of the balls. On the other hand, the removed balls are then attached at the right end of some ball group, because the boxes with number j lie there. Thus, the number of ball groups does not increase. Also, since there are boxes with numbers other than j between any two ball groups and they remain empty at this substep, it does not occur that two ball groups merge due to this attachment. From these facts, we can conclude that the number of ball groups remains unchanged after the substep. Next, we show that (b)i′ holds at the end of the substep. Consider the kth ball group. The empty box with number j just at the right of this ball group is filled with a ball with number i, which comes either from this ball group or from one of the preceding ball groups. Since the balls with number other than i do not move, the kth ball group still has one or more balls with number p for 0 ≤ p ≤ M − 1. Also, it is clear that the balls are lined up so that their identification numbers are in the order of i + 1, i + 2, . . . , M − 1, 0, 1, . . . , i. This shows that (b)i′ holds.

28

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki

Finally, we show (d)j ′ . Let 2 ≤ k ≤ m. When the balls with number i belonging to the kth ball group are moved, the boxes that stored them become empty, are given number j, and become part of the (k − 1)th box array. Since the boxes with numbers other than j remain unchanged, the (k − 1)th box array still has one or more boxes with number q for 0 ≤ q ≤ N − 1. Also, due to rule (iii), the boxes are lined up so that their identification numbers are in the order of j, j + 1, . . . , N − 1, 0, 1, . . . , j − 1 when 0 ≤ i ≤ M − 2 and in the order of j + 1, j + 2, . . . , N − 1, 0, 1, . . . , j when i = M − 1. Thus, (d)j ′ holds. By using Lemma 2.1 repeatedly, we obtain the following theorem. Theorem 2.2. Suppose that the conditions (a)–(d) hold at discrete time n and the number of ball groups is m. Then, these conditions hold also at discrete time n + 1 and the number of ball groups remains unchanged. Since the number of ball groups (and therefore that of box arrays) stays constant, we can describe the state of nBBS-III by specifying the number of balls of each identification number in each ball group and the number of boxes of each identification number in each box array. Noting that the number of balls with identification number i changes only at the ith, (M + i)th, . . ., ((N − 1)M + i)th substeps, we denote the number of balls with number i in the kth ball group at the beginning of the (jM + i)th substep (n,j) the transition under discrete time evolution from n to n + 1 by Qi,k . Similarly, noting that the number of boxes with identification number j changes only at the (M j)th, (M j + 1)th, . . ., (M j + M − 1)th substeps, we denote the number of boxes with number j in the kth box array at the beginning of the (jM + i)th substep of the transition under (n,i) discrete time evolution from n to n + 1 by Ej,k . Now, let us consider the (jM + i)th substep. In this substep, only balls with number i and boxes with number j are used. If we focus on these balls and boxes, their movements are exactly the same as those of balls and boxes in the standard BBS in which neither balls nor boxes are numbered. Thus, the numbers of the balls and (n,j) (n,i) m−1 boxes before and after the (jM + i)th substep, that is, {Qi,k }m k=1 , {Ej,k }k=1 and (n,j+1)

(n,i+1)

{Qi,k }m }m−1 k=1 , {Ej,k k=1 , should satisfy the same equation of motion as that of the standard BBS. The equation is exactly (1.11). Therefore, we arrive at the following theorem. Theorem 2.3. (1.11).

The equation of motion of nBBS-III is the udhToda-III equation

Box and ball system with numbered boxes and balls

§ 3.

29

Conserved quantity

Fukuda [2] derived a conserved quantity of nBBS-I with the help of a combinatorial technique. In this section, along the same line, we derive combinatorial representation of a conserved quantity of nBBS-III. § 3.1.

Conserved quantity of nBBS-I

We begin by reviewing the main result of [2]. Let us consider nBBS-I in which each ball has an identification number (color) between 0 and M − 1. We assign index 0 to one of the boxes and then assign indices 1, 2, . . . to the boxes to the right of it, starting from the nearest box. Also, assign indices −1, −2, . . . to the boxes to the left of it, staring from the nearest box. These indices denote the position of each box and do not change over time. They should not be confused with the identification numbers of the boxes defined in nBBS-III, which change during time evolution. ∑M −1 Now, consider the status of this nBBS-I at discrete time n. Let L = i=0 Q(n+i) be the total number of balls and let the indices of boxes storing a ball be denoted by i1 , i2 , . . . , iL . Also, let the color (identification number) of the ball stored in a box with index ij be aj . Then, we can uniquely represent the status of the nBBS-I using the so-called bi-word: ( ) i1 i2 · · · iL w= . a1 a2 · · · aL Using the Robinson-Schensted-Knuth correspondence, we can construct one-to-one correspondence between w and a pair (P, Q) of the semi-standard Young tableaux. To construct the P symbol, we start from an empty Young tableaux and repeat, for j = 1, 2, . . . , L, adding a new box with element aj at the right end of the top row and reconstructing the whole Young tableaux according to an algorithm called row bumping. To construct the Q symbol, we prepare another empty Young tableau and repeat adding a new box at the same position where a new box of the P symbol appeared as a result of row bumping and inserting ij into the box. Thus, P and Q become Young tableaux of the same shape. It is important to note that the P symbol can be constructed solely from the bottom row of w, whereas construction of the Q symbol requires both the top and bottom rows of w. The P and Q symbols are defined for each discrete time n. Here, we consider discrete time evolution from n to n + M , because this is a period during which all the balls are moved exactly once. Then, the following theorem holds. Theorem 3.1 (Fukuda [2]). Consider discrete time evolution of P and Q constructed from that of nBBS-I. Then, the following two holds:

30

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki

(i) The P symbol remains unchanged. (ii) The Q symbol evolves independently of the P symbol. Thus, the P symbol constructed from the bi-word w gives a conserved quantity (n) of nBBS-I. It is to be noted that the bi-word w is determined uniquely if {Qk }m k=1 , (n+1) m (n+M −1) m (n) m−1 {Qk }k=1 , . . ., {Qk }k=1 , {Ek }k=1 and the index i1 of the box containing (n) the leftmost ball at discrete time n are given. But, {Ek }m−1 k=1 and i1 are used only to determine the top row of w. Thus, the bottom row of w, and hence the P symbol, are (n) (n+1) m (n+M −1) m determined solely by {Qk }m }k=1 , . . ., {Qk }k=1 . k=1 , {Qk § 3.2.

Conserved quantity of nBBS-III

We now turn to the case of the nBBS-III. Consider the (jM )th, (jM + 1)th, . . ., (jM + M − 1)th substeps of the transition under discrete time evolution from n to n + 1, where 0 ≤ j ≤ N − 1. At these substeps, balls with identification numbers 0, 1, . . . , M − 1 are moved using only boxes with identification number j. Thus, if we focus only on these balls and boxes, the dynamics of the system is exactly the same as that of nBBS-I with M kinds of balls. Then, it follows from Theorem 3.1 that the P (n,j) (n,j) m (n,j) m symbol determined by {Q1,k }m k=1 , {Q2,k }k=1 , . . ., {QM,k }k=1 and that determined (n,j+1)

(n,j+1)

(n,j+1)

by {Q1,k }m }m }m k=1 , {Q2,k k=1 , . . ., {QM,k k=1 are the same. Since this holds for j = 0, 1, . . . , N − 1 and for any n, we obtain the following theorem concerning the P symbol in nBBS-III. Theorem 3.2.

(n,j)

(n,j)

m The P symbol determined by the variables {Q1,k }m k=1 , {Q2,k }k=1 ,

(n,j)

. . . , {QM,k }m k=1 of nBBS-III remains the same regardless of the value of j and n. This gives a combinatorial conserved quantity of nBBS-III. In the example shown in Figure 1, the bi-words at discrete time n = 0 and at discrete time n = 1 are respectively given as: ( ) 1 2 3 4 5 6 7 15 16 17 18 19 27 28 29 , 0 0 0 1 1 2 2 0 0 1 1 2 0 1 2 ( ) 14 15 16 17 18 19 26 27 28 29 33 34 35 36 37 . 0 0 1 1 2 2 0 0 1 2 0 0 1 1 2 Thus, we can easily check that the P symbols at discrete time n = 0 and at discrete time n = 1 are the same, and are both expressed as: 0

0

0

0

0

1

1

1

2

2

2

0

1

1

2

Box and ball system with numbered boxes and balls

§ 4.

31

Relationship to the hungry ϵ-BBS

In this section, we relate the hungry ε-BBS to the nBBS-III, and then derive a conserved quantity of the hungry ε-BBS from the viewpoint of the nBBS-III. According to Kobayashi-Tsujimoto [6], the hungry ε-BBS is designed based on ultra-discretization of a discrete integrable system which can be represented in matrix form as: R(n+M ) = R(n) (L1 )−1 L2 , ) L2     1 1  (n)   (n)    e¯ε,1 1   −eε,1 1     (n) (n)     (n) (n) 1 −eε,2 1 e ¯ := , L L1 :=    , ε,2 2     .. .. .. ..     . . . .     (n) (n) −eε,m−1 1 e¯ε,m−1 1   (n) 1 q1   (n)   1 q2   (n)   q3 . . . R(n) :=  ,   .  .. 1    (n) qm (n+1) −1

(4.1)

(L1

(4.2)

(4.3)

(n)

(n)

(n)

(n)

(n)

(n+1)

(n)

where eε,ℓ := εℓ eℓ , e¯ε,ℓ := (1 − εℓ )eℓ , and εℓ is a constant whose value is 0 or 1. (n)

Let L1,ℓ be a lower bidiagonal matrix whose subdiagonal entries are 0 except for the (ℓ + 1, ℓ) entry:   1   ..   .       1 (n) . L1,ℓ =  (n)   eε,ℓ 1     .   . .   1 Then, we can decompose the inverse matrix (L1 )−1 in product form as: −1  1   (n)   −eε,1 1   (n)   (n) −eε,2 1 (L1 )−1 =     . .   .. ..   (n) −eε,m−1 1 (n)

32

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki



−1 

1

    =  

      

1 ..

. 1 (n) −eε,m−1



1  −e(n) 1  ε,1  × ··· ×    

..

      

−1

1 ..

      

. 1 (n) −eε,m−2

1

1 1

−1       

. 1 1

(n)

(n)

(n)

= L1,m−1 L1,m−2 · · · L1,1 .

(4.4)

We now assume that εℓ = 1 for ℓ = ℓK , ℓK−1 , . . . , ℓ1 , where ℓK > ℓK−1 > · · · > ℓ1 , and (n) εℓ = 0 otherwise. Then, L1,ℓ with ℓ ̸= ℓK , ℓK−1 , . . . , ℓ1 are the identity matrices. Thus, using (4.4), we can simplify (4.1) to a tractable form that does not involve the inverse as: (4.5)

(n+1)

(n+1)

(n+1)

(n+1)

L1,ℓK L1,ℓK−1 · · · L1,ℓ1 L2

(n)

(n)

(n)

(n)

R(n+M ) = R(n) L1,ℓK L1,ℓK−1 · · · L1,ℓ1 L2 .

By defining R(n,0) = R(n) and introducing intermediate variables R(n,1) , . . . , R(n,K) , we can rewrite the LR transformation (4.5) as a sequence of LR transformations of bidiagonal matrices:  (n+1) (n)  L1,ℓK R(n,1) = R(n,0) L1,ℓK ,    (n) (n+1)   L R(n,2) = R(n,1) L1,ℓK−1 ,   1,ℓK−1 .. (4.6) .    (n+1) (n,K) (n)  L1,ℓ1 R = R(n,K−1) L1,ℓ1 ,     (n+1) (n+M,0) (n) L2 R = R(n,K) L2 . Let n be a multiple of M and consider (4.6) for n, n+1, . . . , n+M −1. These constitute one period of the hungry ε-BBS, in which all M kinds of balls are moved once. Now, we write n = n′ M and apply subscript-superscript swapping as follows:  (n′ ,i) (n+i)   Lj ′ := L1,ℓK−j , i = 0, 1, . . . , M − 1, j = 0, 1, . . . , K − 1, (n ,i) (n+i) LK := L2 , i = 0, 1, . . . , M − 1,   (n′ ,j) (n+i,j) Ri := R , i = 0, 1, . . . , M − 1, j = 0, 1, . . . , K. Using these variables, (4.6) for n, n + 1, . . . , n + M − 1 can be written succinctly as: (4.7)

(n′ ,i+1)

Lj

(n′ ,j+1)

Ri

(n′ ,j)

= Ri

(n′ ,i)

Lj

,

i = 0, 1, . . . , M − 1,

j = 0, 1, . . . , K,

Box and ball system with numbered boxes and balls

33

with the conditions: { (4.8)

(n′ ,M )

Lj

(n′ +1,0)

= Lj

(n′ ,K+1) Ri

=

,

(n′ +1,0) Ri ,

j = 0, 1, . . . , K, i = 0, 1, . . . , M − 1.

By writing (4.7) and (4.8) entry-by-entry, we obtain (1.7) and (1.8) with M = K + 1 and n = n′ . Note that (4.7) (or (1.7)) constitutes a doubly nested loop over i and j, and in the original ε-BBS, the loop over j was the inner loop, as can be seen from (n′ ,i) (n′ ,j) (4.5). But the order of computation of (4.7) is arbitrary as long as Lj and Ri (n′ ,i+1)

(n′ ,j+1)

are computed before Lj and Ri for all i and j. Thus, we can exchange the loops and make the loop over i (ball identification numbers) the inner one. This allows us to regard the hungry ε-BBS, which is derived from the matrix representation (4.1), as a special case of the nBBS-III with K + 1 kinds of boxes and M kinds of balls. (n,i) (i = 0, 1, . . . , M − 1, j = Below, we rewrite n′ as n. Note that each Lj 0, 1, . . . , K − 1) has a nonzero entry only in the (ℓK−j + 1, ℓK−j ) entry on the subdiagonal. From the viewpoint of the nBBS-III, we see that, for each j = 0, 1, . . . , K − 1, there are a finite number of boxes numbered j in the ℓK−j th box array and an infinite number of boxes numbered j in other box arrays. This means that when the balls are moved using boxes numbered j, the ball groups other than the ℓK−j th and (ℓK−j + 1)th ones simply move to the empty boxes to the right of them, without changing their lengths. Thus, effectively, we only need to consider the interaction between the ℓK−j th ball group and the ℓK−j th box array numbered j, which results in changes in the lengths of the ℓK−j th and (ℓK−j + 1)th ball groups and the ℓK−j th box array. On the other hand, there are a finite number of boxes numbered K in the ℓth box array when ℓ ̸= ℓK−j , j = 0, 1, . . . , K − 1 and an infinite number of boxes numbered K in the ℓth box array otherwise. Let {ℓ¯1 , ℓ¯2 , . . . , ℓ¯m−1−K } = {1, 2, . . . , m − 1}\{ℓ1 , ℓ2 , . . . , ℓK } and ℓ¯1 < ℓ¯2 < · · · < ℓ¯m−1−K . Then, when the balls are moved using boxes numbered K, we need to care about only the interactions between the ℓth ball group and the ℓth box array for ℓ = ℓ¯1 , ℓ¯2 , . . . , ℓ¯m−1−K . Until now, we have assumed that there are K + 1 kinds of boxes. However, as is clear from the explanation above, for each j, 0 ≤ j ≤ K − 1, there is only one box array such that the number of boxes numbered j in it is finite. Hence, for 0 ≤ j ≤ K − 1, instead of saying “use boxes numbered j”, we can say “use the finite number of boxes in the ℓK−j th box array”. After this has been done for 0 ≤ j ≤ K − 1, we use the finite number of boxes in the ℓth box array for ℓ = ℓ¯1 , ℓ¯2 , . . . , ℓ¯m−1−K in this order. Thus, we can do without the box identification numbers and instead consider a variant of nBBS-I in which the order of box arrays to be used is changed from the natural order. In summary, we arrive at the following alternative discrete-time evolution rule of the hungry ε-BBS.

34

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki

Algorithm 1 Alternative discrete-time evolution rule of the hungry ε-BBS. 1: for i = 0, M − 1 do 2: for j = 0, K − 1 do 3: Move the balls numbered i according to the standard BBS rule assuming that only the ℓK−j th box array has finite length and the other box arrays have infinite lengths. 4: end for 5: Move the balls numbered i according to the standard BBS rule assuming that only the ℓ¯1 th, ℓ¯2 th, . . . , ℓ¯m−1−K th box arrays have finite length and the other box arrays have infinite lengths. 6: end for Note that the loops over i and j can be interchanged, so that when a box identification number is specified, all M kinds of balls are moved successively using those boxes. Since the hungry ε-BBS can be interpreted as a special case of nBBS-III, we immediately obtain its conserved quantity from Theorem 3.2 as follows. Theorem 4.1. Under discrete-time evolution according to Algorithm 1, the P symbol computed from ball sequences before discrete-time evolution coincides with that computed from ball sequences after discrete-time evolution. Theorem 4.1 is equivalent to Proposition 4.1 in Kobayashi and Tsujimoto [6]. Our proof above is an alternative one based on the results of [2] and an interpretation of the hungry ε-BBS as a special case of nBBS-III. Example To show that the discrete-time evolution rule given as Algorithm 1 is equivalent to the rule given in [6], we show an example of time evolution below. Here, there are three kinds of balls and four ball groups (M = 3, m = 4) and ε1 = 0, ε2 = 1, ε3 = 0. These settings, as well as the initial state of the system, are the same as those of the first example of Example 3.2 in [6]. In this example, we use a modified version of Algorithm 1 in which the loops over i and j are interchanged. Since only ε2 is nonzero, we have K = 1 and ℓ1 = 2 and therefore ℓ¯1 = 1 and ℓ¯2 = 3. In the transition from n to n + 1/2, the three kinds of balls are moved using the 2nd box array. The 1st and 3rd box arrays are treated as having infinite lengths (denoted by double line) and their original lengths do not change after the transition. In the transition from n + 1/2 to n + 1, the three kinds of balls are moved using the 1st and 3rd box arrays. The 2nd box array is treated as having an infinite length (denoted by double line) and its original length does not change after the transition. By comparing the states at n = 1, 2 and 3 with those of [6], we see that our rule gives identical results.

Box and ball system with numbered boxes and balls

n=0: n=1/2: n=1: n=1+1/2: n=2: n=2+1/2: n=3:

1111222 1111222

35

112233 133 11223 11223 1333 11223 11122 1112223 1333 11223 11122 112 11223333 11223 11122 112 11223333 11223 11122 112 11223333 11223 122 11112 1122333 112233

§ 5.

Concluding remarks

In this paper, we proposed a new numbered box and ball system (nBBS) in which both boxes and balls are numbered. We first designed rules of discrete time evolutions of the new nBBS, and showed that its dynamics is described by an extension of the ultradiscrete hungry Toda (udhToda) equations corresponding to the nBBSs in which either boxes or balls are numbered. We next focused on a pair of semi-standard Young tableaux which represent the status of our nBBS, and showed that one of the tableaux constitutes a conserved quantity of our nBBS under discrete-time evolutions, by slightly extending the approach for the simplest nBBS. We also showed that the matrix LR transformation associated with the hungry ε-BBS is a specialization of that associated with our nBBS, and thereby derived an already known conserved quantity of the hungry ε-BBS from the viewpoint of our nBBS. In fact, there is another version of nBBS with numbered boxes and balls, and we have described its elementary properties in our previous paper. Our future work is thus to enrich the study of this nBBS by, for example, deriving its conserved quantities using the Young tableaux approach, and finding its relationships to other BBSs.

References [1] Akaiwa, K., Yoshida, A. and Kondo, K., An improved algorithm for solving an inverse eigenvalue problem for band matrices, Electron. J. Linear Algebra, 38 (2022), 745–759. [2] Fukuda, K., Box-ball systems and Robinson-Schensted-Knuth correspondence, J. Algebraic Comb., 19 (2004), 67–89. [3] Fukuda, A., Ishiwata, E., Yamamoto, Y., Iwasaki, M. and Nakamura, Y., Integrable discrete hungry systems and their related matrix eigenvalues, Annal. Mat. Pura Appl., 192 (2013), 423–445. [4] Yamamoto, Y., Fukuda, A., Kakizaki, S., Ishiwata, E., Iwasaki, M. and Nakamura, Y., Box and ball system with numbered boxes, Math. Phys. Anal. Geom., 25 (2022), 13 (20pp). [5] Hirota, R., Tsujimoto, S. and Imai, T., Difference Scheme of Soliton Equations, In: Christiansen, P.L., Eilbeck, J.C. and Parmentier, R.D. (eds), Future Directions of Nonlinear Dynamics in Physical and Biological Systems. NATO ASI Series, vol. 312, Springer, Boston, MA (1993).

36

Y. Yamamoto, A. Fukuda, E. Ishiwata and M. Iwasaki

[6] Kobayashi, K. and Tsujimoto, S., Generalization of the ε-BBS and the Schensted insertion algorithm, arXiv:2202.09094v1 (2022). [7] Rutishauser, H., Lectures on Numerical Mathematics, Birkh¨ auser, Boston (1990). [8] Takahashi, D. and Satsuma, J., A soliton cellular automaton, Phys. Soc. Jpn., 59 (1990), 3514–3519. [9] Tokihiro, T., Nagai, A. and Satsuma, J., Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization, Inverse Probl., 15 (1999), 1639–1662. [10] Yamamoto, Y. and Fukaya, T., Differential qd algorithm for totally nonnegative band matrices: convergence properties and error analysis, JSIAM Letters, 1 (2009), 56–59.

RIMS Kˆ okyˆ uroku Bessatsu B94 (2023), 037–053

The blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative-type By

Takiko Sasaki

∗

Abstract In this paper, we study a blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative type in one space dimension. Employing the idea of Caffarelli and Friedman [1], we prove the blow-up curve becomes Lipschitz continuous under suitable initial conditions. Moreover, we show the blow-up rates of the solution of the wave equations.

§ 1.

Introduction

We consider the following weak coupled system of semilinear wave equations with nonlinearities of derivative type:   x ∈ R, t > 0, ∂t2 u1 − c21 ∂x2 u1 = 2p1 (∂t u2 )p1 ,     ∂ 2 u − c2 ∂ 2 u = 2p2 (∂ u )p2 , x ∈ R, t > 0, t 1 t 2 2 x 2 (1.1)  u1 (x, 0) = u1,1 (x), ∂t u1 (x, 0) = u1,2 (x), x ∈ R,     u (x, 0) = u (x), ∂ u (x, 0) = u (x), x ∈ R, 2 2,1 t 2 2,2 where c1 , c2 > 0, ui,j (i, j = 1, 2) are given smooth functions and pi > 1 is a constant such that the function spi for s ≥ 0 is of class C 4 Received Feburary 17, 2023. Revised July 2, 2023. 2020 Mathematics Subject Classification(s): Key Words: system of semilinear wave equations, one dimension, classical solution, blow-up curve. This work was supported by JSPS Grant-in-Aid for Early-Career Scientists, 18K13447 ∗ Department of Mathematical Engineering, Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo 135-8181, Japan. /Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

38

Takiko Sasaki

for i = 1, 2. Through this paper, we assume c1 ≥ c2 . We only consider (1.1) in the following time-space region. Let R∗ and T ∗ be positive constants and set KR∗ ,T ∗ ,i =

[

K−,i (x, T ∗ ) (i = 1, 2)

x∈BR∗

where Br = {x ∈ R | |x| < r} (r > 0),  K−,i (x0 , t0 ) = (x, t) ∈ R2 | |x − x0 | < ci (t0 − t), t > 0

(i = 1, 2).

We are interested T (x), the maximal existence time of classical solutions of wave equations at x ∈ BR∗ . In the case of (1.1), T (x) satisfies T (x) = sup {t ∈ (0, T ∗ ) | |∂t (u1 + u2 )(x, t)| < ∞}

for

x ∈ BR ∗ .

In this paper, we call the set Γ = {(x, T (x)) | x ∈ BR∗ } the blow-up curve. Below, we identify Γ with T itself. Before we proceed to our problem, we recall some previous results. We begin with a single equation: ∂t2 u − ∂x2 u = F (u, ∂t u),

x ∈ R,

t > 0.

The pioneering study of this topic was done by Caffarelli and Friedman[1]. They showed the blow-up curve T becomes continuously differentiable under suitable initial conditions in the case of F = |u|p . Godin[2] verified such results for F = eu . Sasaki[3] proved the blow-up curve also becomes continuously differentiable under suitable initial conditions with F = |∂t u|p . On the other hand, Uesaka[4] considered systems of semilinear wave equations with nonlinearities which are not involved in the derivatives. He showed the blow-up curve becomes a Lipschitz function under suitable initial conditions. Our result in this paper is to prove that the blow-up curve becomes Lipschitz continuous and the upper bounds of the Lipschitz constants depend on initial conditions of (1.1). Moreover, we show the blow-up rates of the solution for (1.1). The article is organized as follows. In Section 2, we state our main results. In Section 3, we show that the monotonicity and the regularity of the solution of (1.1). In Section 4, we give a proof of the Lipschitz continuity of the blow-up curve and the blow-up rates of the solution for (1.1).

The blow-up curve for system of semilinear wave equations

§ 2.

39

Main results

Let i′ = 3 − i (i = 1, 2). We rewrite (1.1) as the following:  pi  x ∈ R, t > 0,  Di,− ϕi = (ϕi′ + ψi′ ) , (2.1)

  

x ∈ R,

Di,+ ψi = (ϕi′ + ψi′ )pi , ϕi (x, 0) = fi (x),

t > 0,

ψi (x, 0) = gi (x), x ∈ R,

where ϕi and ψi are ϕ i = ∂ t ui + c i ∂ x ui ,

ψ i = ∂ t ui − c i ∂ x ui

(i = 1, 2).

where Di,± = ∂t ±ci ∂x and fi (x) = ui,2 (x)+ci ∂x ui,1 (x), gi (x) = ui,2 (x)−ci ∂x ui,1 (x), (i = 1, 2). To state our main results, we give some assumptions on initial data. We consider the following system of ordinary equations:  dϕ˜   ˜ p, = (ϕ˜ + ψ) t > 0,    dt dψ˜ ˜ p, = (ϕ˜ + ψ) t > 0,   dt   ˜ ˜ ϕ(0) = γϕ , ψ(0) = γψ , where γϕ , γψ ≥ 1 and p = min {p1 , p2 }. We can easily see that there exists a positive constant T˜ such that ˜ (ϕ˜ + ψ)(t) →∞

as

t → T˜.

For i = 1, 2, we pose the following assumptions. (A1) T˜ < T ∗ . (A2) fi ≥ γϕ , gi ≥ γψ in BR∗ +c1 T ∗ . (A3) fi , gi ∈ C 4 (BR∗ +c1 T ∗ ). (A4) There exits a positive constant ε0 such that (γϕ + γψ )pi ≥ c1 (2 + ε0 ) (|∂x fi | + |∂x gi |)

in

BR∗ +c1 T ∗ .

Our main result in this paper is as follows. Theorem 2.1. Let R∗ and T ∗ be positive constants and assume (A1)–(A4). Then, there exists a unique Lipschitz function T such that 0 < T (x) < T ∗ (x ∈ BR∗ ) and a unique (C 3,1 (Ω))4 solution (ϕ1 , ψ1 , ϕ2 , ψ2 ) of (2.1) satisfying

40

Takiko Sasaki

ϕi (x, t), ψi (x, t) → ∞ as t → T (x) (i = 1, 2)  for any x ∈ BR∗ +c1 T ∗ , where Ω = (x, t) ∈ R2 | x ∈ BR∗ , 0 < t < T (x) . Moreover, there exist positive constants C1 , C2 such that C1 (T (x) − t)−ri ≤ (ϕi + ψi )(x, t) ≤ C2 (T (x) − t)−ri

in

Ω

where ri = (pi + 1)/(pi pi′ − 1) (i = 1, 2). Remark.

We see that (2.1) is equivalent to (1.1). Let ui satisfy Z 1 t ui (x, t) = ui,1 (x) + (ϕi + ψi )(x, s)ds (i = 1, 2). 2 0

Then, ui satisfies (1.1). This fact implies that ∂t ui (x, t) → ∞ § 3.

as

t → T (x) (i = 1, 2).

Preliminaries

For i = 1, 2, let {(ϕi,n , ψi,n )}n∈N∪{0} be a sequence ϕi,0 ≡ γϕ and ψi,0 ≡ γψ and  Z t   (ϕi′ ,n + ψi′ ,n )pi (x + ci (t − s), s)ds  ϕi,n+1 (x, t) = fi (x + ci t) + Z0 t for (x, t) ∈ KR∗ ,T ∗ ,1 ,  pi  ′ ′  ψi,n+1 (x, t) = gi (x − ci t) + (ϕi ,n + ψi ,n ) (x − ci (t − s), s)ds 0

ϕi,n+1 (x, 0) = fi (x),

ψi,n+1 (x, 0) = gi (x) for

x ∈ BR∗ +c1 T ∗

for n ∈ N ∪ {0}. First we note the following fact. Lemma 3.1.

Assume (A2). Then, we have

ϕi,n+1 ≥ ϕi,n ≥ γϕ ,

and

ψi,n+1 ≥ ψi,n ≥ γψ

in

KR∗ ,T ∗ ,1

(i = 1, 2)

for n ∈ N ∪ {0}. Proof. proof.

Since the proof is quite similar to the proof of Lemma 3.2, we omit the

The following Lemma plays an important role in the proof of the Lipschitz continuity of the blow-up curve T . Lemma 3.2. (3.1)

Assume (A2)–(A4). For i = 1, 2, we have

∂t ϕi,n ≥ c1 (1 + ε0 )|∂x ϕi,n |,

for n ∈ N ∪ {0}.

∂t ψi,n ≥ c1 (1 + ε0 )|∂x ψi,n |

in

KR∗ ,T ∗ ,1

41

The blow-up curve for system of semilinear wave equations

Proof.

Let λ = c1 (1 + ε0 ) and ± Ji,n = (∂t ± λ∂x )ϕi,n ,

L± i,n = (∂t ± λ∂x )ψi,n

(i = 1, 2).

for n ∈ N. We show ± Ji,n , L± i,n ≥ 0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

for n ∈ N. First, since (A4) yields that ± Ji,1 (x, 0) = (∂t ± λ∂x )ϕi,1 (x, 0)

= (ci ± c1 (1 + ε0 ))∂x ϕi,1 (x, 0) + (ϕi′ ,0 + ψi′ ,0 )pi (x, 0) ≥ −c1 (2 + ε0 )|∂x fi (x)| + (γϕ + γψ )pi ≥ 0

in

BR∗ +c1 T ∗

(i = 1, 2).

Moreover, we see that ± Di,− Ji,1 = (∂t ± λ∂x )Di,− ϕi,1 = (∂t ± λ∂x )(ϕi′ ,0 + ψi′ ,0 )pi = 0

in

KR∗ ,T ∗ ,1

for i = 1, 2. Therefore, we have (3.2)

± Ji,1 ≥0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

L± i,1 ≥ 0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

Similarly we obtain (3.3)

Next, we assume that (3.1) holds in the case of n = k. It follows from (A4) that ± Ji,k+1 (x, 0) = (∂t ± λ∂x )ϕi,k+1 (x, 0)

= (ci ± c1 (1 + ε0 ))∂x ϕi,k+1 (x, 0) + (ϕi′ ,k + ψi′ ,k )pi (x, 0) ≥ −c1 (2 + ε0 )|∂x fi (x)| + (γϕ + γψ )pi ≥ 0

in

BR∗ +c1 T ∗

(i = 1, 2).

Moreover, we have ± Di,− Ji,k+1 = (∂t ± λ∂x )Di,− ϕi,k+1 = (∂t ± λ∂x )(ϕi′ ,k + ψi′ ,k )pi

= pi (ϕi′ ,k + ψi′ ,k )pi −1 (Ji±′ ,k + L± i′ ,k ) ≥ 0 in

KR∗ ,T ∗ ,1

Hence we obtain (3.4)

± Ji,k+1 ≥0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

By the same arguments, we also obtain that (3.5)

L± i,k+1 ≥ 0 in

KR∗ ,T ∗ ,1

(i = 1, 2).

(i = 1, 2).

42

Takiko Sasaki

It follows from (3.2), (3.3), (3.4) and (3.5) that (3.1) for all n ∈ N. We shall construct a classical solution of (2.1). Fix (x, t) ∈ KR∗ ,T ∗ ,1 . Thanks to Lemma 3.1, we can define ϕ and ψ as     ϕi (x, t) = lim ϕi,n (x, t) = sup ϕi,n (x, t) ψi (x, t) = lim ψi,n (x, t) = sup ψi,n (x, t) n→∞

n→∞

n∈N

n∈N

for i = 1, 2. Moreover, thanks to Lemma 3.2, we can define the following function: X T (x) = sup{t ∈ (0, T ∗ ) | (ϕi + ψi )(x, t) < ∞} for x ∈ BR∗ . i=1,2

Lemma 3.3. Assume (A2)–(A4). Then, (ϕ1 , ψ1 , ϕ2 , ψ2 ) is a unique (C 3,1 Ω)4  solution of (2.1). Here, Ω = (x, t) ∈ R2 | x ∈ BR∗ , 0 < t < T (x) . Through this proof, we fix (x0 , t0 ) ∈ Ω and define Bi (t) as

Proof.

Bi (t) = {x ∈ R | |x0 − x| < ci (t0 − t)} for (x0 , t0 ) ∈ Ω (i = 1, 2). First, we show that there exits a positive constant C0 such that X (3.6) ∥(ϕi + ψi )(·, t)∥L∞ (B1 (t)) ≤ C0 for 0 ≤ t ≤ t0 i=1,2

by showing a contradiction. We assume that there exists t˜ such that X (3.7) 0 ≤ t˜ ≤ t0 and ∥(ϕi′ + ψi′ )(·, t˜)∥L∞ (B1 (t)) = ∞. i=1,2

Then, it follows from Lemma 3.2 and (3.7) that X ∞= (ϕi + ψi )(x0 , t0 ) < ∞. i=1,2

This is a contradiction. Hence, we obtain (3.6). Next, we shall show that (ϕ1 , ψ2 , ϕ2 , ψ2 ) ∈
4 C 3,1 (K−,1 (x0 , t0 )) . For i = 1, 2, we notice that |(ϕi,n+1 − ϕi,n )(x, t)| Z t (ϕi′ ,n + ψi′ ,n )pi (x + ci (t − s), s)ds = f (x + ci t) + 0

Z

t

− f (x + ci t) − Z

(ϕi′ ,n−1 + ψi′ ,n−1 ) (x + ci (t − s), s)ds pi

0 t

≤

|(ϕi′ ,n + ψi′ ,n )pi − (ϕi′ ,n−1 + ψi′ ,n−1 )pi | (x + ci (t − s), s)ds 0 Z t p −1 ≤ pi ((ϕi′ ,n + ψi′ ,n + θ(ϕi′ ,n−1 + ψi′ ,n−1 )) i 0

× |(ϕi′ ,n + ψi′ ,n ) − (ϕi′ ,n−1 + ψi′ ,n−1 )| (x + ci (t − s), s)ds

43

The blow-up curve for system of semilinear wave equations

for some θ ∈ (0, 1). Hence, we have |(ϕi,n+1 − ϕi,n )(x, t)| Z t
pi −1 ≤2 pi (ϕi′ ,n + ψi′ ,n )pi −1 + (ϕi′ ,n−1 + ψi′ ,n−1 )pi −1 0

× |(ϕi′ ,n + ψi′ ,n ) − (ϕi′ ,n−1 + ψi′ ,n−1 )| (x + ci (t − s), s)ds. for i = 1, 2 In the same way, we have |(ψi,n+1 − ψi,n )(x, t)| Z t
pi −1 ≤2 pi (ϕi′ ,n + ψi′ ,n )pi −1 + (ϕi′ ,n−1 + ψi′ ,n−1 )pi −1 0

× |(ϕi′ ,n + ψi′ ,n ) − (ϕi′ ,n−1 + ψi′ ,n−1 )| (x − ci (t − s), s)ds for i = 1, 2. Since (3.6) yields that  X  ∥(ϕi,n+1 − ϕi,n )(·, t)∥L∞ (B1 (t)) + ∥(ψi,n+1 − ψi,n )(·, t)∥L∞ (B1 (t)) i=1,2

≤

X

Z t

pi

2 pi 0

i=1,2



p −1

p −1

∥(ϕi′ ,n + ψi′ ,n )(·, s1 )∥Li∞ (B1 (s1 )) + ∥(ϕi′ ,n−1 + ψi′ ,n−1 )(·, s1 )∥Li∞ (B1 (s1 ))

 × ∥(ϕi′ ,n − ϕi′ ,n−1 ) (·, s1 )∥L∞ (B1 (s1 )) + ∥(ψi′ ,n − ψi′ ,n−1 ) (·, s1 )∥L∞ (B1 (s1 )) ds1 Z t X pi −1 pi +1 ≤ 2 pi C 0 ∥(ϕi′ ,n − ϕi′ ,n−1 ) (·, s1 )∥L∞ (B1 (s1 )) 0

i=1,2

+ ∥(ψ .. . ≤

X 

pi +1

2

X i=1,2

n Z

Z

t

sn−1

··· 0

i=1,2

≤

pi C0pi −1

0



i′ ,n

−ψ

 i′ ,n−1

) (·, s1 )∥L∞ (B1 (s1 )) ds1

∥(ϕi′ ,1 − ϕi′ ,0 )(·, sn )∥L∞ (B1 (sn ))

 + ∥(ψi′ ,1 − ψi′ ,0 )(·, sn )∥L∞ (B1 (sn )) dsn · · · ds1 4C0

(2pi +1 pi C0pi −1 T ∗ )n → 0 as n!

n→∞

for t ∈ [0, t0 ]. Moreover, we see that Di,− Dθ ϕi,n+1 = Di,+ Dθ ψi,n+1 = pi′ (ϕi′ ,n + ψi′ ,n )pi′ −1 (Dθ ϕi′ ,n + Dθ ψi′ ,n ), where Dθ = cos θ∂x + sin θ∂t . Let W (t) define by     1 W (t) = 2 + 1 C0p1 +p2 exp 2(p1 + p2 )C0p1 +p2 −1 t . c1



44

Takiko Sasaki

Then W (t) satisfies  W (t) = 2

 Z t 1 p1 +p2 + 1 C0 + 2(p1 + p2 )C0p1 +p2 −1 W (s)ds. c2 0

We shall show (3.8)

∥Dθ ϕi,n (·, t)∥L∞ (B1 (t)) , ∥Dθ ψi,n (·, t)∥L∞ (B1 (t)) ≤ W (t) (i = 1, 2)

for n ∈ N ∪ {0} and t ∈ [0, t0 ]. In the case of n = 0, we see that ∥Dθ ϕi,0 (·, t)∥L∞ (B1 (t)) , ∥Dθ ψi,0 (·, t)∥L∞ (B1 (t)) = 0 for t ∈ [0, t0 ]. It follows from (A4) that |∂x fi (x)| ≤

1 ′ 2ci (fi

(i = 1, 2)

+ gi′ )pi . Moreover, we notice

Di,− Dθ ϕi,n+1 = Dθ Di,− ϕi,n+1 = Dθ (ϕi′ ,n + ψi′ ,n )pi = pi (ϕi′ ,n + ψi′ ,n )pi −1 Dθ (ϕi′ ,n + ψi′ ,n ) (i = 1, 2). Assuming that (3.8) holds for n = k, one can estimate W (t) as follows. ∥Dθ ϕi,k+1 (·, t)∥L∞ (B1 (t)) ≤ ∥(cos θ + ci sin θ)∂x fi (·) + sin θ|(fi′ + gi′ )|pi (·)∥L∞ (B1 (0)) Z t

pi (ϕi′ ,k + ψi′ ,k )pi −1 (Dθ ϕi′ ,k + Dθ ψi′ ,k )(·, s) ∞ ds + L (B1 (t)) 0   Z t 1 pi ≤2 2pi C0pi −1 W (s)ds + 1 ∥(fi′ + gi′ )(·)∥L∞ (B1 (0)) + ci 0   Z t 1 + 1 C0pi + 2pi C0pi −1 W (s)ds ≤2 ci 0   Z t 1 p1 +p2 ≤2 + 1 C0 + 2(p1 + p2 )C0p1 +p2 −1 W (s)ds = W (t) (i = 1, 2) c2 0 for t ∈ [0, t0 ]. Similarly, we can show ∥Dθ ψi,k+1 ∥L∞ (B1 (t)) ≤ W (t) (i = 1, 2) for t ∈ [0, t0 ]. Therefore we have (3.8) for n ∈ N. Let C1 = W (T ∗ ) satisfy ∥Dθ ϕi,n ∥L∞ (B1 (t)) ≤ C1 ,

∥Dθ ψi,n ∥L∞ (B1 (t)) ≤ C1

(i = 1, 2)

The blow-up curve for system of semilinear wave equations

45

for t ∈ (0, t0 ) and n ∈ N ∪ {0}. We notice that |Dθ ϕi,n+1 (x, t) − Dθ ϕi,n (x, t)| = |(cos θ + ci sin θ)∂x fi (x + ci t) + sin θ(fi′ + gi′ )pi (x + ci t) Z t + pi (ϕi′ ,n + ψi′ ,n )pi −1 (Dθ ϕi′ ,n + Dθ ψi′ ,n )(x + ci (t − s), s)ds 0

−(cos θ + ci sin θ)∂x fi (x + ci t) − sin θ(fi′ + gi′ )pi (x + ci t) Z t pi −1 − pi (ϕi′ ,n−1 + ψi′ ,n−1 ) (Dθ ϕi′ ,n−1 + Dθ ψi′ ,n−1 )(x + ci (t − s), s)ds 0

Z

t

≤

pi |ϕi′ ,n + ψi′ ,n |

pi −1

|Dθ ϕi′ ,n + Dθ ψi′ ,n − Dθ ϕi′ ,n−1 − Dθ ψi′ ,n−1 | (x + ci (t − s), s)ds

0

Z

t

pi |Dθ ϕi′ ,n−1 + Dθ ψi′ ,n−1 |

+ 0

× |(ϕi′ ,n + ψi′ ,n )pi −1 − (ϕi′ ,n−1 + ψi′ ,n−1 )pi −1 |(x + ci (t − s), s)ds

Z

t

≤

pi |ϕi′ ,n + ψi′ ,n |

pi −1

0

× (|Dθ ϕi′ ,n − Dθ ϕi′ ,n−1 | + |Dθ ψi′ ,n − Dθ ψi′ ,n−1 |) (x + ci (t − s), s)ds

Z

t

+

2pi −2 pi (pi − 1) |Dθ ϕi′ ,n−1 + Dθ ψi′ ,n−1 | (ϕi′ ,n + ψi′ ,n )pi −2 + (ϕi′ ,n−1 + ψi′ ,n−1 )pi −2




0

× (|ϕi′ ,n − ϕi′ ,n−1 | + |ψi′ ,n − ψi′ ,n−1 |) (x + ci (t − s), s)ds. In the same way, we have |Dθ ψi,n+1 (x, t) − Dθ ψi,n (x, t)| Z t p −1 ≤ pi |ϕi′ ,n + ψi′ ,n | i 0

× (|Dθ ϕi′ ,n − Dθ ϕi′ ,n−1 | + |Dθ ψi′ ,n − Dθ ψi′ ,n−1 |) (x − ci (t − s), s)ds

Z

t

+

2pi −2 pi (pi − 1) |Dθ ϕi′ ,n−1 + Dθ ψi′ ,n−1 | (ϕi′ ,n + ψi′ ,n )pi −2 + (ϕi′ ,n−1 + ψi′ ,n−1 )pi −2




0

× (|ϕi′ ,n − ϕi′ ,n−1 | + |ψi′ ,n − ψi′ ,n−1 |) (x − ci (t − s), s)ds. Then, we have  X  ∥(Dθ ϕi,n+1 − Dθ ϕi,n )(·, t)∥L∞ (B1 (t)) + ∥(Dθ ψi,n+1 − Dθ ψi,n )(·, t)∥L∞ (B1 (t)) i=1,2

≤

X

C˜1

i=1,2

+

X i=1,2

Z t 0

C˜2

 ∥(Dθ ϕi′ ,n − Dθ ϕi′ ,n−1 )(·, s)∥L∞ (B1 (s)) + ∥(Dθ ψi′ ,n − Dθ ψi′ ,n−1 )(·, s)∥L∞ (B1 (s)) ds

Z t 0

 ∥(ϕi′ ,n − ϕi′ ,n−1 )(·, s)∥L∞ (B1 (s)) + ∥(ψi′ ,n − ψi′ ,n−1 )(·, s)∥L∞ (B1 (s)) ds

46

Takiko Sasaki

Hence, we have  X  ∥(Dθ ϕi,n+1 − Dθ ϕi,n )(·, t)∥L∞ (B1 (t)) + ∥(Dθ ψi,n+1 − Dθ ψi,n )(·, t)∥L∞ (B1 (t)) i=1,2

≤

X i=1,2

Z

Z

t

sn−1

···

(C˜1 )n 0

∥Dθ ϕi,1 (·, sn ) − Dθ ϕi,0 (·, sn )∥L∞ (B1 (sn ))

0


+ ∥Dθ ψi,1 (·, sn ) − Dθ ψi,0 (·, sn )∥L∞ (B1 (sn )) dsn · · · ds1 Z t Z sk−1 X n−1 X (2pi +1 pi C0pi −1 T ∗ )n−k k ˜ ˜ 4C0 + C1 C2 ··· dsk · · · ds1 (n − k)! 0 0 i=1,2 k=0

X (C˜1 T )n 1 + 2(pi +1)n+3 C0 C˜1n C˜2 (T ∗ )n →0 ≤ 8C0 n! k!(n − k)! n−1

as

n→∞

k=0

for t ∈ [0, t0 ]. Here, C˜1 = 2(p1 + p2 )C0p1 +p2 −1 ,

C˜2 = 2p1 +p2 +1 (p1 + p2 )(p1 + p2 − 1)C1 C0p1 +p2 −2 .

Thus, there exists (ϕ1 , ψ1 , ϕ2 , ψ1 ) ∈ (L∞ (K−,1 (x0 , t0 )))4 such that (1)

(1)

(1)

(1)

∥Dθ ϕi,n − ϕ(1) ∥L∞ (B1 (t)) + ∥Dθ ψi,n − ψ (1) ∥L∞ (B1 (t)) → 0 as

n→∞

(i = 1, 2)

for t ∈ [0, t0 ]. Therefore we obtain that (ϕ1 , ψ1 , ϕ2 , ψ2 ) ∈ (W 1,∞ (K−,1 (x0 , t0 )))4 . By repeating this process, we obtain (ϕ1 , ψ1 , ϕ2 , ψ2 ) ∈ (W 4,∞ (K−,1 (x0 , t0 )))4 . This means that (ϕ1 , ψ1 , ϕ2 , ψ2 ) ∈ (C 3,1 (K− (x0 , t0 )))4 . Finally, we shall show the uniqueness of solution of (2.1). We assume that ∗∗ ∗∗ ∗∗ 4,∞ ˜ 4 (ϕ∗1 , ψ1∗ , ϕ∗2 , ψ2∗ ), (ϕ∗∗ (Ω)) 1 , ψ1 , ϕ2 , ψ2 ) ∈ (W

satisfy (2.1) where we set  ˜ = (x, t) ∈ R2 | x ∈ BR∗ , 0 < t < min {T ∗ (x), T ∗∗ (x)} Ω ∗∗ ∗∗ ∗∗ and T ∗ and T ∗∗ be the blow-up curves of (ϕ∗1 , ψ1∗ , ϕ∗2 , ψ2∗ ) and (ϕ∗∗ 1 , ψ1 , ϕ2 , ψ2 ) respectively. ˜ Then, there exits a positive constant C0 such that Let us assume that (x0 , t0 ) ∈ Ω.  ∥ϕ∗ (·, t)∥ ∞ ∗ L (B1 (t)) , ∥ψi (·, t)∥L∞ (B1 (t)) ≤ C0 i (i = 1, 2). ∥ϕ∗∗ (·, t)∥L∞ (B (t)) , ∥ψ ∗∗ (·, t)∥L∞ (B (t)) ≤ C0 i

1

i

1

The blow-up curve for system of semilinear wave equations

47

for t ∈ [0, t0 ]. It follows from Lemma 3.2 that  X  ∗ ∗∗ ∗ ∗∗ sup ∥(ϕi − ϕi )(·, t)∥L∞ (B1 (t)) + ∥(ψi − ψi )(·, t)∥L∞ (B1 (t)) t∈[0,t0 ] i=1,2

≤ sup

X Z

t∈[0,t0 ] i=1,2

≤ sup ≤2

0

X Z

t∈[0,t0 ] i=1,2 pi +1

t

∗∗ pi 2 ∥((ϕ∗i′ + ψi∗′ )pi − (ϕ∗∗ i′ + ψi′ ) ) (·, s)∥L∞ (B1 (s)) ds

t pi +1

2 0

pi C0pi −1 t

sup

pi C0pi −1 X 

t∈[0,t0 ] i=1,2

= 2pi +1 pi C0pi −1 t sup

X 

t∈[0,t0 ] i=1,2



∥(ϕ∗i′

∥(ϕ∗i′

−

−

ϕ∗∗ i′ ) (·, s)∥L∞ (B1 (s))

ϕ∗∗ i′ ) (·, t)∥L∞ (B1 (t))

+

+

∥(ψi∗′

∥(ψi∗′

−

−

ψi∗∗ ′ ) (·, s)∥ ∞ L (B1 (s))

ψi∗∗ ′ ) (·, t)∥ ∞ L (B1 (t))

∗ ∗∗ ∥(ϕ∗i − ϕ∗∗ i ) (·, t)∥L∞ (B1 (t)) + ∥(ψi − ψi ) (·, t)∥L∞ (B1 (t))





for t ∈ [0, t0 ]. Hence we obtain  X  ∗ ∗∗ (3.9) ∥(ϕ∗i − ϕ∗∗ ) (·, t)∥ + ∥(ψ − ψ ) (·, t)∥ i i i L∞ (B1 (t)) L∞ (B1 (t)) = 0 i=1,2

# 1 1 . Since the constant for t ∈ 0, 2p1 +p2 +1 (p1 + p2 )C0p1 +p2 −1 2p1 +p2 +1 (p1 + p2 )C0p1 +p2 −1 does not depend on t, we obtain that (3.9) for t ∈ [0, t0 ]. Therefore, we have that "

∗∗ ∗∗ ∗∗ (ϕ∗1 , ψ1∗ , ϕ∗2 , ψ2∗ ) = (ϕ∗∗ 1 , ψ1 , ϕ2 , ψ2 ) in

˜ Ω.

Moreover, this means that T ∗ (x) = T ∗∗ (x) for x ∈ BR∗ .

The following lemma guarantees that the solution of (2.1) blows-up in KR∗ ,T ∗ ,1 . Lemma 3.4. (3.10) Proof.

Assume (A1)–(A4). Then, we have T (x) ≤ T ∗

for

x ∈ BR ∗ .

We consider {(ϕ˜n , ψ˜n )}n∈N∪{0} such that ϕ˜0 ≡ γϕ , ψ˜0 ≡ γψ and  dϕ˜n+1   = (ϕ˜n + ψ˜n )p , t > 0,    dt dψ˜n+1 = (ϕ˜n + ψ˜n )p , t > 0,   dt   ˜ ϕn+1 (0) = γϕ , ψ˜n+1 (0) = γψ ,

for n ∈ N ∪ {0} . Thanks to (A1), it is enough to show that (3.11)

ϕi,n (x, t) ≥ ϕ˜n (t)(≥ γϕ > 1)

ψi,n (x, t) ≥ ψ˜n (t)(≥ γϕ > 1) (i = 1, 2)

 ds

48

Takiko Sasaki

for n ∈ N∪{0} . First, in the case of n = 0, we can easily confirm (3.11) holds. Assuming that (3.11) holds for n = k, we obtain that ϕi,k+1 (x, t) − ϕ˜k+1 (t) = fi (x + ci t) − γϕ Z tn o (ϕi′ ,k + ψi′ ,k )pi (x + ci (t − s), s) − (ϕ˜i′ ,k + ψ˜i′ ,k )p (s) ds + 0

≥ 0 for

(x, t) ∈ KR∗ ,T ∗ ,1

(i = 1, 2)

because of p = min{p1 , p2 }. By the same arguments, we have that ψi,k+1 (x, t) − ψ˜k+1 (t) ≥ 0

for (x, t) ∈ KR∗ ,T ∗ ,1

(i = 1, 2).

Therefore we have (3.11) for n ∈ N.

§ 4.

Lipschitz continuity and the blow-up rates

This section is devoted to prove show that the blow-up rates of ϕi , ψi (i = 1, 2) and the blow-up curve T is Lipschitz continuous in BR∗ . Proposition 4.1. Assume (A1)–(A4). Then, there exist positive constants C1 and C2 which depend only on ci , pi , fi , gi (i = 1, 2) and ε0 such that (4.1) C1 (T (x) − t)−ri ≤ (ϕi + ψi )(x, t) ≤ C2 (T (x) − t)−ri

for

(x, t) ∈ Ω

(i = 1, 2)

where ri = (pi + 1)/(pi pi′ − 1). Proof.

First, we show that there exist positive constants C1 and C2 such that

(4.2) C1 (ϕi′ ,n + ψi′ ,n )pi ≤ ∂t ϕi,n , ∂t ψi,n ≤ C2 (ϕi′ ,n + ψi′ ,n )pi in KR∗ ,T ∗ ,1

(i = 1, 2).

For n ∈ N ∪ {0}, we have that Di,− ∂t ϕi,n+1 = ∂t Di,− ϕi,n+1 = ∂t (ϕi′ ,n + ψi′ ,n )pi (4.3)

= pi (ϕi′ ,n + ψi′ ,n )pi −1 (∂t ϕi′ ,n + ∂t ψi′ ,n )

in

KR∗ ,T ∗ ,1

(i = 1, 2).

By Lemma 3.2, we obtain Di,− (ϕi′ ,n + ψi′ ,n )pi = pi (ϕi′ ,n + ψi′ ,n )pi −1 (∂t ϕi′ ,n − ci ∂x ϕi′ ,n + ∂t ψi′ ,n − ci ∂x ψi′ ,n ) (4.4)

≤ 2pi (ϕi′ ,n + ψi′ ,n )pi −1 (∂t ϕi′ ,n + ∂t ψi′ ,n ) in

KR∗ ,T ∗ ,1

(i = 1, 2)

49

The blow-up curve for system of semilinear wave equations

for n ∈ N ∪ {0}. We set Jϕi ,n+1 = 2∂t ϕi,n+1 − (ϕi′ ,n + ψi′ ,n )pi for n ∈ N ∪ {0}. By (4.3) and (4.4), we have Di,− Jϕi ,n+1 ≥ 0

(4.5)

in

KR∗ ,T ∗ ,1

(i = 1, 2)

for n ∈ N ∪ {0}. It follows from (A4) that Jϕi ,n+1 (x, 0) = 2∂t ϕi,n+1 (x, 0) − (ϕi′ ,n + ψi′ ,n )pi (x, 0)

(4.6)

= 2ci ∂x ϕi,n+1 (x, 0) + (ϕi′ ,n + ψi′ ,n )pi (x, 0)   pi ≥ −2c1 |∂x fi (x)| + (γϕ + γψ ) ≥ 0 in BR∗ +c1 T ∗

(i = 1, 2)

for n ∈ N ∪ {0}. Since (4.5) and (4.6) yields that Jϕi ,n ≥ 0 in

(4.7)

KR∗ ,T ∗ ,1

(i = 1, 2)

for n ∈ N. It follows from Lemma 3.2 that ∂t ϕi,n+1 = ci ∂x ϕi,n+1 + (ϕi′ ,n + ψi′ ,n )pi 1 ≤ ∂t ϕi,n+1 + (ϕi′ ,n + ψi′ ,n )pi 1 + ε0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

for n ∈ N. Hence we have that (4.8)

∂t ϕi,n+1 ≤

1 + ε0 (ϕi′ ,n + ψi′ ,n )pi ε0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

It follows from (4.7) and (4.8) that 1 1 + ε0 (ϕi′ ,n + ψi′ ,n )pi ≤ ∂t ϕi,n+1 ≤ (ϕi′ ,n + ψi′ ,n )pi 2 ε0

in

KR∗ ,T ∗ ,1

(i = 1, 2)

for n ∈ N ∪ {0}. Similarly, we can prove (4.9)

1 + ε0 1 (ϕi′ ,n + ψi′ ,n )pi ≤ ∂t ψi,n ≤ (ϕi′ ,n + ψi′ ,n )pi 2 ε0

in

KR∗ ,T ∗ ,1

(i = 1, 2).

Therefore we obtain (4.10)

C1 (ϕi′ + ψi′ )pi ≤ ∂t ϕi , ∂t ψi ≤ C2 (ϕi′ + ψi′ )pi

in

Ω

(i = 1, 2).

Next, we show that there exist positive constants C3 and C4 such that (4.11)

−1

C3 (ϕi + ψi )1+ri

−1

≤ ∂t ϕi , ∂t ψi ≤ C4 (ϕi + ψi )1+ri

in

Ω (i = 1, 2),

50

Takiko Sasaki

where ri = (pi + 1)/(pi pi′ − 1). By (4.10), there exist positive constants C, C ′ such that C(ϕi′ + ψi′ )pi ≤

∂(ϕi + ψi ) ≤ C ′ (ϕi′ + ψi′ )pi ∂t

in

Ω

(i = 1, 2).

Hence, there exist positive constants C ′′ such that Z

(ϕi +ψi )(x,t)

s

p i′

ds ≤ C

(ϕi +ψi )(x,0)

′′

Z

(ϕi′ +ψi′ )(x,t)

for (x, t) ∈ Ω (i = 1, 2).

spi ds

(ϕi′ +ψi′ )(x,0)

Thus, there exists a positive constant C ′′′ such that (ϕi + ψi )pi′ (x, t) ≤ C ′′′ (ϕi′ + ψi′ )(pi +1)pi′ /(pi′ +1) (x, t)  p ′ /(p ′ +1) (ϕi + ψi )pi′ +1 (x, 0) − (ϕi′ + ψi′ )pi +1 (x, 0) i i × 1+ for (x, t) ∈ Ω (ϕi′ + ψi′ )pi +1 (x, t)

(i = 1, 2).

It follows from Lemma 3.2 that there exists a positive constant C¯ such that 0≤1+

(ϕi + ψi )pi′ +1 (x, 0) − (ϕi′ + ψi′ )pi +1 (x, 0) ≤ C¯ (ϕi′ + ψi′ )pi +1 (x, t)

for

(x, t) ∈ Ω (i = 1, 2).

Therefore we have ¯ i′ + ψi′ )(pi +1)/(pi′ +1) (x, t) for (x, t) ∈ Ω (i = 1, 2). (ϕi + ψi )(x, t) ≤ (C ′′′ )1/pi′ C(ϕ Hence it follows from (4.10) that we obtain (4.11). Finally, we shall show (4.1). We notice that it follows from Lemma 3.4 that X (ϕi + ψi )(x, t) → ∞ as t → T (x) for (x ∈ BR∗ ). i=1,2

Let x0 ∈ BR∗ and assume (ϕ1 + ψ1 )(x0 , t) → ∞

as

t → T (x0 ).

Then, ∂t ϕ1 (x0 , t), ∂t ψ1 (x0 , t) → ∞

as

t → T (x0 ).

It follows from (4.10) that (ϕ2 + ψ2 )(x0 , t) → ∞

as

t → T (x0 ).

(ϕ1 + ψ1 )(x0 , t) → ∞

as

t → T (x0 )

Similarly, we can prove

51

The blow-up curve for system of semilinear wave equations

when we assume (ϕ2 + ψ2 )(x0 , t) → ∞ Hence, we see that  (ϕ + ψ )(x , t) → ∞ 1 1 0 (ϕ2 + ψ2 )(x0 , t) → ∞

as

t → T (x0 ).

as

t → T (x)

for

x ∈ BR ∗ .

It follows from (4.11) that there exist positive constants C, C ′ such that Z

(ϕi +ψi )(x0 ,T (x0 )−ε)

C

z

−1−ri−1

dz ≤ T (x0 ) − ε − t ≤ C

′

Z

(ϕi +ψi )(x0 ,t)

(ϕi +ψi )(x0 ,T (x0 )−ε)

−1

z −1−ri dz.

(ϕi +ψi )(x0 ,t)

Hence we obtain (4.1) by letting ε → 0. Lemma 3.2 yields that the blow-up curve T is Lipschitz continuous. Lemma 4.2. (4.12)

Assume (A1)–(A4). Then, we have

|T (x) − T (x′ )| ≤

1 |x − x′ | c1 (1 + ε0 )

for

x, x′ ∈ BR∗ .

Proof. This proof is based on the Implicit Function Theorem. Let ε be a sufficiently small positive constant such that T (x) − ε > 0. For Mε = supx∈BR∗ (ϕ1 + ψ1 )(x, T (x) − ε), there exists an implicit function Eε ∈ C 1 (BR∗ ) such that (ϕ1 + ψ1 )(x, Eε (x)) = Mε

for

x ∈ BR ∗ .

For x1 , x2 ∈ BR∗ , let k = Eε (x1 ) − Eε (x2 ) and consider H ∈ C 1 (0, 1) such that H(ξ) = (ϕ1 + ψ1 )(x1 + ξ(x2 − x1 ), t + ξk). We notice that H satisfies that H(0) = (ϕ1 + ψ1 )(x1 , t), H(1) = (ϕ1 + ψ1 )(x2 , t + k) = (ϕ1 + ψ1 )(x2 , t + Eε (x2 ) − Eε (x1 )). Then, we have H(0) = H(1) = Mε where we set t = Eε (x1 ). By using Rolle’s Theorem, there exists ξ ′ ∈ (0, 1) such that H ′ (ξ ′ ) =(x2 − x1 )∂x (ϕ1 + ψ1 )(x1 + ξ ′ (x2 − x1 ), Eε (x1 ) + ξ ′ k) (4.13)

+ k∂t (ϕ1 + ψ1 )(x1 + ξ ′ (x2 − x1 ), Eε (x1 ) + ξ ′ k) = 0.

52

Takiko Sasaki

Since Lemma 3.2 and (4.13) yield that −∂x (ϕ1 + ψ1 )(x1 + ξ ′ (x2 − x1 ), Eε (x1 ) + ξ ′ k) |x1 − x2 | |Eε (x1 ) − Eε (x2 )| = |k| = ∂t (ϕ1 + ψ1 )(x1 + ξ ′ (x2 − x1 ), Eε (x1 ) + ξ ′ k) 1 |x1 − x2 |. ≤ c1 (1 + ε0 ) Thus, Eε is Lipschitz continuous. Finally, we will prove Lipschitz continuity of T in BR∗ . By using the definition of Mε , we have |T (x1 ) − T (x2 )| ≤ |T (x1 ) − Eε (x1 )| + |Eε (x1 ) − Eε (x2 )| + |Eε (x2 ) − T (x2 )| 1 |x1 − x2 | for x1 , x2 ∈ BR∗ . ≤ 2ε + c1 (1 + ε0 ) Since we let ε > 0 take an arbitrary value, this completes the proof. By d(x, t), we denote the distance from a point (x, t) in Ω to Γ = {(x, T (x)) | x ∈ BR∗ }. It follows from Lemma 4.2 that we obtain the following results. 𝑡 𝑇 𝑥 −𝑥

c1 (T (x) − t) p ≤ d(x, t) ≤ T (x) − t. 1 + c21

(𝑥, 𝑇(𝑥))

𝑐! 𝑇 𝑥 − 𝑡 1 + 𝑐!" 𝑡 = 𝑇(𝑥)

(𝑥, 𝑡) (𝑥 + 𝑐! (𝑇 𝑥 − 𝑥)), 𝑡) 𝑥

By replacing T (x) − t by d(x, t) in Proposition 4.1, we obtain the following Corollary. Corollary 4.3. Assume (A1)–(A4). Then, there exist positive constants C1 and C2 which depend only on ci , pi , fi , gi (i = 1, 2) and ε0 such that  C (T (x) − t)−ri ≤ ϕ (x, t) ≤ C (T (x) − t)−ri 1 i 2 (4.14) for (x, t) ∈ Ω (i = 1, 2), C1 (T (x) − t)−ri ≤ ψi (x, t) ≤ C2 (T (x) − t)−ri where ri = (pi + 1)/(pi pi′ − 1). Proof. such that

It follows from Proposition 4.1 that there exist positive constants C, C ′

Cd−ri (x, t) ≤ (ϕi + ψi )(x, t) ≤ C ′ d−ri (x, t)

for (x, t) ∈ Ω (i = 1, 2).

The blow-up curve for system of semilinear wave equations

53

Therefore, there exists a positive constant C ′′ , C ′′′ we have ϕi (x, T (x) − ε) = fi (x + c1 (T (x) − ε)) Z T (x)−ε (ϕi′ + ψi′ )pi (x + ci (T (x) − ε − s), s)ds + Z

0 T (x)−ε

≥

(ϕi′ + ψi′ )pi (x + ci (T (x) − ε − s), s)ds T (x)−2ε

≥ C ′′ ε

inf T (x)−2ε≤s≤T (x)−ε

d(x + ci (T (x) − ε − s), s)−ri′ pi

′′′

≥ C ε · ε−ri′ pi = C ′′′ ε−ri

for (x, t) ∈ Ω (i = 1, 2).

By the same arguments, we can show that there exist positive constants C1 , C2 such that   ϕ (x, T (x) − ε) ≤ C ε−ri i 2 for (x, t) ∈ Ω (i = 1, 2).  C1 ε−ri ≤ ψi (x, T (x) − ε) ≤ C2 ε−ri

References [1] Caffarelli, F. and Friedman, A., The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223–241. [2] Godin, P., The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension.I, Calc. Var. Partial Differential Equations, 13 (2001), 69–95. [3] Sasaki, T., Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity, Advances in Differential Equations , 23 (2018), 373–408. [4] Uesaka, H., The Blow-Up Boundary for a System of Semilinear Wave Equations, Further progress in analysis, World Sci. Publ., Hackensack, NJ, (2009), 845–853.

RIMS Kˆ okyˆ uroku Bessatsu B94 (2023), 055–063

On dynamical connection between continuous and tropical discretized dynamical systems in one-dimensional By

Shousuke Ohmori∗ and Yoshihiro Yamazaki∗∗

Abstract We discuss the relationship of stability and the local bifurcations between one-dimensional differential equations and their tropically discretized ones. The discretized time interval is introduced as a bifurcation parameter in the tropical discretized equation, and emergence condition of an additional bifurcation, flip bifurcation, is revealed. By reviewing the continuous dynamical systems with saddle node and pitchfork bifurcations treated in our previous study (S. Ohmori and Y. Yamazaki, J. Math. Phys. 61 122702 (2020)), correspondence of their dynamics to their tropical discrete equations is discussed.

§ 1.

Introduction: Tropical discretization

Tropical discretization, proposed by Murata[1], is a discretizing procedure that converts a differential equation into a difference equation with only positive variables. Let us consider a differential equation of x = x(t) > 0, (1.1)

dx = F (x) = f (x) − g(x), dt

where we assume that F (x) can be divided into the two positive smooth functions f and g. Then, the following discretized form can be adopted: (1.2)

xn+1 = xn

xn + τ f (xn ) ≡ Fτ (xn ), xn + τ g(xn )

Received January 25, 2023. Revised March 17, 2023. 2020 Mathematics Subject Classification(s): 37M15 Key Words: tropical discretization, discrete dynamical systems, bifurcation, ultradiscretization ∗ National Institute of Technology, Gunma College, Maebashi-shi, Gunma 371-8530, Japan. e-mail: [email protected] ∗∗ Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

56

Shousuke Ohmori and Yoshihiro Yamazaki

where xn = x(nτ ); τ (> 0) and n show the discretized time interval and the number of iteration steps, respectively. It is noted that eq. (1.1) is reproduced from the following equation that is identical to eq. (1.2), (1.3)

xn+1 − xn f (xn ) − g(xn ) = xn τ xn + τ g(xn )

by taking τ → 0. The discretization from eq. (1.1) to eq. (1.2) is called the tropical discretization. The tropical discretization is also known as a non-standard finite difference scheme with positively-preserving system of ordinary differential equations[2, 3]. In our previous studies[4, 5, 6, 7], the ultradiscrete equations that are obtained by tropical discretization from the continuous dynamical systems with local bifurcations have been focused on. The crucial point of our previous results is that the bifurcation structures emerging in some obtained ultradiscrete equations coincide with their original differential equations. The one-dimensional nonlinear differential equations that express the normal forms of the saddle-node and transcritical bifurcations are typical examples; the ultradiscrete equations derived from them possess the piecewise linear saddle-node and transcritical bifurcations, respectively[4]. Meanwhile, we found some inconsistent cases. For instance, the ultradiscrete equation for the normal form of the supercritical pitchfork bifurcation exhibits the bifurcations of the supercritical pitchfork and the flip types. The flip bifurcation does not exist in the original one-dimensional differential equation. We consider that such emergence of the additional bifurcation is caused by the tropical discretization or ultradiscretion processes. The present article provides a brief report of general identification that how the original bifurcations retain and how additional bifurcations emerge. § 2.

General Relations

§ 2.1.

Fixed Point

First, we focus on the relation of fixed points between eqs. (1.1) and (1.2). Suppose that x ¯ is a fixed point of eq. (1.1): F (¯ x) = 0 i.e., f (¯ x) = g(¯ x). Then, x ¯ is also a fixed x ¯+τ f (¯ x) point of eq. (1.2), since Fτ (¯ x) = x ¯ x¯+τ g(¯x) = x ¯. On the other hands, if eq. (1.2) has a fixed point x ¯, we have x ¯(f (¯ x) − g(¯ x)) = 0. Thus, the following property is obtained. Prop. 1 (Fixed Point Condition) A fixed point of eq. (1.1) is identical to a fixed point of eq. (1.2). § 2.2.

Linear Stability

Next, we consider the relation for stability of fixed points. For eq. (1.1), the following linearized equation at the fixed point x ¯ (> 0) provides its stability: dx dt =

Dynamical connection between continuous and tropical discretized dynamical systems 57 dF (¯ x) dx

(2.1)

· x = D(¯ x)x, where D(¯ x) =

df (¯ x) dg(¯ x) − . dx dx

Whereas, the stability of x ¯ for eq. (1.2) is determined by the absolute value of differential coefficient of Fτ at x ¯. At the fixed point x ¯, the relation f (¯ x) = g(¯ x) holds, and the first derivative of Fτ at x ¯ can be represented as (2.2)

dFτ (¯ x) = 1 + Zτ (¯ x)D(¯ x), dx

where (2.3)

Zτ (¯ x) =

τx ¯ . x ¯ + τ f (¯ x)

x) τ (¯ Since x ¯, f , and τ > 0, then Zτ (¯ x) > 0 holds. When | dFdx | < 1, namely,

(2.4)

−2 < Zτ (¯ x)D(¯ x) < 0,

x ¯ is (asymptotically) stable. When D(¯ x) > 0, we have Zτ (¯ x)D(¯ x) > 0. Therefore, it is confirmed that when x ¯ is unstable in eq. (1.1), it is also unstable in eq. (1.2). For D(¯ x) < 0, x ¯ is stable in eq. (1.1). However, its stability for eq. (1.2) depends on the relationship between τ and κ(¯ x), where κ(¯ x) is defined by (2.5)

κ(¯ x) = −

2¯ x . x ¯D(¯ x) + 2f (¯ x)

Note that κ(¯ x) is independent of τ . When κ(¯ x) < 0, the relation −2 < Zτ (¯ x)D(¯ x) < 0 holds for any τ > 0. When κ(¯ x) > 0, −2 < Zτ (¯ x)D(¯ x) holds only for τ satisfying 0 < τ < κ(¯ x). Therefore, relation for the stability of the fixed point x ¯ between eqs. (1.1) and (1.2) can be summarized as follows. Prop. 2 (Stability Conditions) (a) When x ¯ is a stable fixed point of eq. (1.1), (a-i) if κ(¯ x) < 0, x ¯ is stable in eq. (1.2) for any τ , (a-ii) if κ(¯ x) > 0 and 0 < τ < κ(¯ x), x ¯ is stable in eq. (1.2), (a-iii) if κ(¯ x) > 0 and τ > κ(¯ x), x ¯ is unstable in eq. (1.2). (b) When x ¯ is an unstable fixed point of eq. (1.1), x ¯ is also unstable in eq. (1.2) for any τ . By Prop. 2, the stable fixed point x ¯ for eq. (1.1) retains its stability in eq. (1.2) when τ < κ(¯ x). Thus, κ(¯ x) is considered as a threshold of stability. However, when x ¯ is unstable in eq. (1.1), its stability does not change in eq. (1.2) for any τ .

58

Shousuke Ohmori and Yoshihiro Yamazaki

§ 2.3.

Flip Bifurcation

Let us suppose that D(¯ x) < 0 and κ(¯ x) > 0. In this case, x ¯ becomes nonhyperbolic at τ = κ(¯ x); ∂Fτ (¯ x) (2.6) = −1 ∂x τ =κ(¯x) holds. Then, it is possible to exhibit the flip bifurcation when eq. (1.2) satisfies the following four conditions[8] at the bifurcation point (x, τ ) = (¯ x, κ(¯ x)): ∂ 2 Fτ2 (x) ∂ 2 Fτ2 (x) ∂Fτ2 (x) ∂ 3 Fτ2 (x) = 0, ̸= 0, = 0, and ̸= 0, ∂τ ∂τ ∂x ∂x2 ∂x3 where Fτ2 = Fτ ◦ Fτ . Among these conditions, the first three conditions are always satisfied at (x, τ ) = (¯ x, κ(¯ x)). Thus, the emergence condition of the flip bifurcation is obtained as the following proposition. Prop. 3 (Flip Bifurcation Condition) Suppose that a fixed point x ¯ of eq. (1.2) has D(¯ x) < 0 and κ(¯ x) > 0. Equation (1.2) exhibits the flip bifurcation at the bifurcation point τ = κ(¯ x) when x) ∂ 3 Fτ2 (¯ ̸= 0. ∂x3 τ =κ(¯ x)

Since the bifurcation parameter τ is given as the time interval in the tropical discretization, this bifurcation occurs even when eq. (1.1) does not possess a bifurcation structure. Note that other local bifurcations, e.g., saddle node, transcritical, and pitchfork bifurcations are not found. § 2.4.

Preservation of Original Bifurcations

We consider the case where the original differential equation (1.1) has one of saddle node, transcritical, and supercritical pitchfork bifurcations. Set the bifurcation parameter c > 0 in eq. (1.1), (2.7)

dx = F (x, c) = f (x, c) − g(x, c), dt

and suppose that the bifurcation occurs at (¯ x, c¯). Then, the tropical discretization of eq. (2.7) gives the discrete dynamical system with the bifurcation parameter c, (2.8)

xn+1 = Fτ (xn , c) = xn

xn + τ f (xn , c) . xn + τ g(xn , c)

At the bifurcation point (x, c) = (¯ x, c¯), Fτ (x, c) connects to F (x, c) by the following equalities ∂Fτ (¯ x, c¯) ∂F (¯ x, c¯) = Zτ (¯ x) , ∂c ∂c

∂F 2 (¯ x, c¯) ∂Fτ2 (¯ x, c¯) = Z (¯ x ) , τ 2 ∂x ∂x2

Dynamical connection between continuous and tropical discretized dynamical systems 59

∂ 2 Fτ (¯ x, c¯) ∂ 2 F (¯ x, c¯) ∂ 3 Fτ (¯ x, c¯) ∂ 3 F (¯ x, c¯) = Zτ (¯ , = (¯ ) . x) Z x τ 3 ∂x∂c ∂x∂c ∂x ∂x3 Therefore, we obtain the following proposition for preservation of the saddle-node, transcritical, and pitchfork bifurcations in eq. (2.8) based on the bifurcation conditions in eq. (2.7) [8]. Prop. 4 (Saddle-node, Transcritical, and Pitchfork Bifurcation Conditions) (a: saddle-node) When eq. (2.7) satisfies the condition for the saddle-node bifur2 (¯ x,¯ c) cation at the bifurcation point (x, c) = (¯ x, c¯): ∂F∂c ̸= 0 and ∂ F∂x(¯x2 ,¯c) ̸= 0, eq. (¯ x,¯ c) (2.8) also satisfies the saddle-node bifurcation conditions at (¯ x, c¯): ∂Fτ∂c ̸= 0 and

∂ 2 Fτ (¯ x,¯ c) ∂x2

̸= 0.

(b: transcritical) When eq. (2.7) satisfies the transcritical bifurcation conditions 2 2 (¯ x,¯ c) F (¯ x,¯ c) = 0, ∂ F∂x(¯x2 ,¯c) ̸= 0, and ∂ ∂x∂c ̸= 0, eq. (2.8) also satisfies at (¯ x, c¯): ∂F∂c the transcritical bifurcation conditions at (¯ x, c¯): and

∂ 2 Fτ (¯ x,¯ c) ∂x∂c

∂Fτ (¯ x,¯ c) ∂c

= 0,

∂ 2 Fτ (¯ x,¯ c) ∂x2

̸= 0,

̸= 0.

(c: supercritical pitchfork) When eq. (2.7) satisfies the pitchfork bifurcation 2 2 3 (¯ x,¯ c) F (¯ x,¯ c) conditions at (¯ x, c¯): ∂F∂c = 0, ∂ F∂x(¯x2 ,¯c) = 0, ∂ ∂x∂c ̸= 0, and ∂ F∂x(¯x3 ,¯c) ̸= 0, (¯ x,¯ c) = eq. (2.8) also satisfies the pitchfork bifurcation conditions at (¯ x, c¯): ∂Fτ∂c 0,

∂ 2 Fτ (¯ x,¯ c) ∂x2

= 0,

∂ 2 Fτ (¯ x,¯ c) ∂x∂c

̸= 0, and

∂ 3 Fτ (¯ x,¯ c) ∂x3

̸= 0.

Note that each direction of these three bifurcations in eq. (2.8) coincides with that in eq. (2.7). § 3. § 3.1.

Examples

Saddle-node bifurcation

For the saddle-node bifurcation, we focus on the nonlinear differential equation with positive x(t)[4], (3.1)

dx = c + x(x − 2), dt

where c is the bifurcation parameter; the saddle-node bifurcation occurs at (¯ x, c¯) = √ (1, 1). When 0 < c < 1, there are two fixed points x ¯± = 1 ± 1 − c, where x ¯− and x ¯+ are stable and unstable, respectively. When c > 1, eq. (3.1) has no fixed points. The tropically discretized equation for eq. (3.1) is (3.2)

xn+1 = Fτ (xn , c) =

xn + τ (c + x2n ) . 1 + 2τ

60

Shousuke Ohmori and Yoshihiro Yamazaki

From Prop. 4 (a), eq. (3.2) exhibits the saddle-node bifurcation at (¯ x, c¯) = (1, 1). Also, it is shown, from Props. 1 and 2 (b), that x ¯+ also becomes the unstable fixed point of eq. (3.2) for any τ . For x ¯− , we have κ(¯ x− ) = −

2¯ x− x ¯− 1 √ =− 2 =− , x ¯− D(¯ x− ) + 2f (¯ x− ) 2¯ x− − x ¯− + c 2− 1−c

which becomes negative when c < 1. From Prop. 2 (a-i), x ¯− is the stable fixed point of eq. (3.2) for any τ . Therefore, eq. (3.2) has only the saddle-node bifurcation. Note that the bifurcation diagram of eq. (3.2) coincides with of the original differential equation (3.1). § 3.2.

Pitchfork bifurcation

For the supercritical pitchfork bifurcation, we consider (3.3)

dx = 3cx(x − 1) + 1 − x3 . dt

The supercritical pitchfork bifurcation occurs at (¯ x, c¯) = (1, 1) (c > 0). When, 0 < c < 1, this equation has one unique√stable fixed point x ¯ = 1. When c > 1, we find three 3c−1±

(1−3c)2 −4

fixed points x ¯ = 1, x ¯± = ; x ¯ = 1 is unstable and x ¯± are stable. The 2 tropical discretizate equation of eq. (3.3) is (3.4)

xn+1 = Fτ (xn , c) =

xn + τ (3cx2n + 1) . 1 + τ (x2n + 3c)

From Prop. 4 (c), the supercritical pitchfork bifurcation occurs at (¯ x, c¯) = (1, 1). Furthermore, it is shown that eq. (3.4) possesses the flip bifurcation for τ . For c > 1, eq. (3.4) has three fixed points x ¯ = 1, x ¯± from Prop. 1. Their stabilities are found from Prop. 2 and κ(¯ x± ) < 0 such that x ¯ = 1 is unstable and x ¯± are stable for any τ > 0. When 0 < c < 1, Prop. 1 shows that x ¯ = 1 is also the only fixed point for eq. (3.4). 2 However, the stability of x ¯ for 0 < c < 1 varies with the sign of κ(¯ x) = − 9c−1 . Actually 1 x) < 0. Then from Prop. 2, x ¯ is stable for any τ > 0. If 0 < c < 91 , on when 9 ≤ c, κ(¯ the other hand, κ(¯ x) becomes positive. Then, the fixed point x ¯ = 1 becomes stable and unstable for τ < κ(¯ x) and τ > κ(¯ x), respectively. ∂ 3 Fτ2 (¯ x,c) 8(18c2 −3c+1) Considering = − ̸= 0 for 0 < c < 19 , it is found from 3 ∂x 3(c−1)2 τ =κ(¯ x)

Prop. 3 that the flip bifurcation occurs at τ = κ(¯ x). Actually for τ > κ(¯ x), there is an f f f attracting cycle C = {¯ x+ , x ¯− } with period 2 around x ¯, where p −6cτ + τ 2 − 9c2 τ 2 ± (6cτ − τ 2 + 9c2 τ 2 )2 − 4(τ + 3cτ 2 + 9c2 τ 2 )(2 + 9cτ + 3cτ 2 + 9c2 τ 2 ) f . x ¯± = 2(τ + 3cτ 2 + 9c2 τ 2 ) −2 are the Figure 1 shows τ -c diagram for the dynamics of eq. (3.4). c = 1 and τ = 9c−1 bifurcation curves on which the supercritical pitchfork and the flip bifurcations occur,

Dynamical connection between continuous and tropical discretized dynamical systems 61

Figure 1. τ -c diagram for eq. (3.4). c = 1 and τ = and the flip bifurcation curves, respectively.

−2 9c−1

are the supercritical pitchfork

respectively. Figure 2 shows the graphs of eq. (3.4) with τ = 3 for (a) c = 2, (b) c = 0.5, (c) c = 0.15, and (d) c = 0.01. For (a), from intersection of the curve and the diagonal xn+1 = xn , it is found that there are three fixed points xn = 1, x ¯± , where x ¯ = 1 is unstable and x ¯± are stable. For (b) and (c), the curves intersect the diagonal only at the stable fixed point x ¯, which becomes node in (b) and focus in (c). For (d), the fixed point x ¯ becomes unstable and an attracting cycle Cf = {¯ xf+ , x ¯f− } with period 2 emerges. Figure 3 shows the bifurcation diagram for τ = 3; the supercritical pitchfork bifurcation 1 . occurs at c = 1 and the flip bifurcation occurs at c = 27 § 4.

Conclusion

In this article, we have focused on the one-dimensional continuous and its tropically discretized dynamical systems and shown their relationships regarding stability and local bifurcations as the general propositions. In particular, we have obtained the occurrence condition of the flip bifurcation caused by the discretized time interval τ , which is introduced in tropical discretization. Some application examples have been treated. Note that the tropically discretized equations depend on decomposition of F (x) into the positive smooth functions f (x) and g(x) in the original differential equation (1.1). The propositions given in Sec. 2 tell us the suitable choice of f (x) and g(x) in the sense that the original bifurcations retain or additional flip bifurcations emerge.

62

Shousuke Ohmori and Yoshihiro Yamazaki

(a)

(b)

(c)

(d)

Figure 2. The graphs of eq. (3.4) for τ = 3. (a) c = 2, (b) c = 0.5, (c) c = 0.15, and (d) c = 0.01.

For the case of τ → ∞, we can also discuss whether the tropically discretized dynamical systems retains the bifurcations of the original continuous dynamical system and, furthermore, the derived max-plus equations (ultradiscrete equations) from the tropically discrete equations also retain the bifurcations of the original continuous dynamical system. Regarding the details of these topics, see [9]. Acknowledgement The authors are grateful to Prof. D. Takahashi, Prof. T. Yamamoto, and Prof. Emeritus A. Kitada at Waseda University, Associate Prof. K. Matsuya at Musashino University, Prof. M. Murata at Tokyo University of Agriculture and Technology for useful comments and encouragements. This work was supported by JSPS KAKENHI Grant Numbers 22K13963 and 22K03442.

Dynamical connection between continuous and tropical discretized dynamical systems 63

Figure 3. The bifurcation diagram of eq. (3.4) for τ = 3.

References [1] M. Murata, J. Differ. Equations Appl. 19 1008 (2013). [2] R. E. Mickens, Nonstandard finite difference models of differential equations, (World Scientific, Singapore, 1994). [3] M. E. Alexander and S. M. Moghadas, Electron. J. Differ. Eqn. Conf. 12 9 (2005). [4] S. Ohmori and Y. Yamazaki, J. Math. Phys. 61 122702 (2020). [5] Y. Yamazaki and S. Ohmori, J. Phys. Soc. Jpn. 90 103001 (2021). [6] S. Ohmori and Y. Yamazaki, in preparation (arXiv:2103.16777v1). [7] S. Ohmori and Y. Yamazaki, JSIAM Letters 14 127 (2022). [8] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (SpringerVerlag, New York. 1990). [9] S. Ohmori and Y. Yamazaki, J. Math. Phys. 64 042704 (2023).

RIMS Kˆ okyˆ uroku Bessatsu B94 (2023), 065–083

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions By

Shin Isojima ∗ and Seiichiro Suzuki

∗∗

Abstract The p-ultradiscrete procedure (ultradiscretization with parity variables) enables to ultradiscretize an equation with subtraction. Its feature is that a solution may have an infinite branches under certain conditions. Recently, an “approximative” technique by which such infinite branches may reduce to finite ones is proposed. In this article, a complicated situation for solutions of the p-ultradiscrete hard-spring equation is investigated, in which an infinite types of branching conditions appear. The approximative technique fairly summarizes the solutions and extends an understanding of the structure of solutions.

§ 1.

Introduction

Ultradiscretization (UD) is a limiting procedure which transforms a given difference equation into a piecewise linear equation [1]. If we write a dependent variable of the given equation as xn , we first replace it by xn = eXn /ε ,

(1.1)

where Xn is a new dependent variable and ε > 0 is a parameter. Then, we apply ε log to both sides of the equation and take the limit ε → +0. By using a key formula   X Y (1.2) lim ε log e ε + e ε = max(X, Y ), ε→+0

Received Feburary 11, 2023. Revised May 15, 2023. 2020 Mathematics Subject Classification(s): 34A34, 39A10 Key Words: ultradiscretization, phase plane, cellular automaton, oscillator Supported by JSPS KAKENHI (grant number JP22K03407) ∗ Department of Industrial and Systems Engineering, Faculty of Science and Engineering, Hosei University, 3-7-2, Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan. e-mail: [email protected] ∗∗ Graduate School of Science and Engineering, Hosei University, 3-7-2, Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

66

Shin Isojima and Seiichiro Suzuki

addition, multiplication, and division for xn are replaced with max operation, addition, and subtraction for Xn , respectively. Namely, the resulting piecewise linear equation is written in max-plus algebra [2]. If we assign integer values to initial values and system parameters for the resulting equation, its dependent variable takes integer values only. Therefore, it can be regarded as time evolution rule of a cellular automaton [3]. Hence, UD may be a procedure which transforms a difference equation into a cellular automaton. The procedure of UD has been applied to many difference equation and its solutions and the constructed cellular automata have been studied. Also, various applications have been reported (for example, [4]–[14]). From this perspective, UD acts as a mediator between continuous (differential or difference equation) and discrete (cellular automaton) mathematics. Since cellular automata have good compatibility with digital computers, UD may be a field with high future growth potential. However, UD has a strong restriction called “negative difficulty.” That is, a given difference equation must be subtraction-free for taking the limit (1.2). Solutions must be positive for applying (1.1). Some attempts have been reported to solve this issue [15]– [18]. The UD with parity variables (pUD, p-ultradiscretization) is one of such attempts [19]. In this method, the “parity (sign)” and “amplitude” variables are introduced. The method enables us to treat a difference equation with subtraction or negative-valued solutions. Its review will be given in the next section. This method has been applied some equations and studied sequentially unlike other ones (for example, [20], [21]). In this meaning, pUD is one of the active methods to aiming to solve the negative difficulty. A feature of pUD is appearance of an “indeterminate solution.” That is, uniqueness of solution may be lost in a specific situation, in exchange for handling subtraction in a equation or negative values for its solution. For example, the successive solution Xn+1 is just restricted by an inequality such as Xn+1 ≤ F (Xn , Xn−1 ), which can be take an arbitrary value as long as it satisfies this inequality. As a result, the solution has an infinite number of branches at indeterminate step. As might be expected, such indeterminacy makes analysis of a solution difficult. On the other hand, it is necessary for capturing all solutions of the p-ultradiscrete system. For example, the p-ultradiscrete analog of the Airy function is constructed through a specific choice of indeterminate solutions [22]. For this usefulness, we cannot discard indeterminate solutions at all. Hence, it is an important problem how to express the indeterminate solutions. Recently, a new approximative expression for indeterminate solutions casts light on this problem in [23]. In this method, a kind of coarse graining is introduced. That is, a p-ultradiscrete system is reinterpreted as the mapping which maps a set on the phase plane to the other. By this method, an infinite number of branches can be reduced to a finite number of branches, and indeterminate solutions are efficiently understood than before. In this article, we shall report additional contents for [24], which gives another application of

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 67

the approximative method to p-ultradiscrete analog of the hard-spring equation [25]. The remainder of this article is organized as follows. In Section 2, we review the hard-spring equation and its p-ultradiscrete analog. We express the p-ultradiscrete equation in the conditional explicit forms which include indeterminate cases. Then, in Section 3, we illustrates the transition of the amplitude on the phase plane. A complicated situation is mainly discussed, which was omitted in [24]. Although the indeterminate solutions still have an infinite number of branches after approximative expression, its complexness is relieved. Finally, we give concluding remarks in Section 4. § 2.

Ultradiscrete hard-spring equation with parity variables

We consider the hard-spring equation d2 x + kx + lx3 = 0, dt2

(2.1)

where x = x(t) and k, l > 0 are constants. This equation has the conserved quantity (2.2)

1 H(t) = 2



dx dt

2

1 1 + kx2 + lx4 = 0. 2 2

Therefore, (2.1) is an integrable equation. Its integrable discrete analog is presented in the Japanese book [26] as (2.3) xn+1 −2xn +xn−1 +2c1 δ 2 (xn+1 + xn−1 )+4c2 δ 2 xn +2c3 δ 2 x2n (xn+1 + xn−1 ) = 0. This difference equation (2.3) is also integrable because it has the conserved quantity (2.4) Hn =

xn 2 − 2xn xn−1 + xn−1 2 + c1 (xn 2 + xn−1 2 ) + 2c2 xn xn−1 + c3 xn 2 xn−1 2 . 2δ 2

See [26] for details. We shall ultradiscretize (2.3) with parity variables. We introduce the sign variable ξn and the “amplitude” variable Xn for xn by (2.5)

ξn =

xn , |xn |

e

We define a function s as (2.6)

 1 (ξ = +1) s(ξ) = 0 (ξ = −1),

Xn ε

= |xn |.

68

Shin Isojima and Seiichiro Suzuki

and we replace xn with (2.7)

xn = ξn e

Xn ε


Xn = s(ξn ) − s(−ξn ) e ε .

We transform the positive parameters ci and δ by (2.8)

e

αi ε

∆

= ci (i = 1, 2, 3),

e ε = δ.

If we substitute (2.7) and (2.8) into (2.3), we have (2.9)


Xn
Xn−1
Xn+1 s(ξn+1 ) − s(−ξn+1 ) e ε − 2 s(ξn ) − s(−ξn ) e ε + s(ξn−1 ) − s(−ξn−1 ) e ε n
Xn+1
Xn−1 o α1 +2∆ ε ε + 2e s(ξn+1 ) − s(−ξn+1 ) e + s(ξn−1 ) − s(−ξn−1 ) e ε
Xn α2 +2∆ + 4e ε 2 s(ξn ) − s(−ξn ) e ε n
Xn+1
Xn−1 o α3 +2∆+2Xn ε + 2e s(ξn+1 ) − s(−ξn+1 ) e ε + s(ξn−1 ) − s(−ξn−1 ) e ε = 0.

For simplicity of notation, we write Xn+1 , Xn , Xn−1 as X+ , X, X− , respectively, and put αi + 2∆ = αbi . Then, we move the negative terms to the other side of the equation and take the limit ε → +0. Here, we shall utilize a formula   A B lim ε log s(ξ)e ε + e ε = max(S(ξ) + A, B), (2.10) ε→+0

where a function S is defined by (2.11)

S(ξ) =

 0

(ξ = +1)

−∞

(ξ = −1).

The resulting equation  max S(ξn+1 ) + X+ , S(ξn+1 ) + α c1 + X+ , S(ξn+1 ) + α c3 + 2X + X+ , S(−ξn ) + X, S(ξn ) + α c2 + X, S(ξn−1 ) + X− , S(ξn−1 ) + α c1 + X− , S(ξn−1 ) + α c3 + 2X + X−



 = max S(−ξn+1 ) + X+ , S(−ξn+1 ) + α c1 + X+ , S(−ξn+1 ) + α c3 + 2X + X+ , S(ξn ) + X, S(−ξn ) + α c2 + X, (2.12)

S(−ξn−1 ) + X− , S(−ξn−1 ) + α c1 + X− , S(−ξn−1 ) + α c3 + 2X + X−



has the implicit form of max[. . . ] = max[. . . ]. We shall rewrite this form into explicit forms with introducing some cases. Here, we consider only the case α c2 > 0, which we are focusing in this article, and give the result only (See [24] for details). We introduce notations   (2.13) a = max 0, α c1 , α c3 + 2X ,

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 69

which includes the dependent variable X, and   c2 > X− + a   Cg : X + α (2.14) Ce : X + α c2 = X− + a    Cℓ : X + α c2 < X− + a, which are used to describe the condition for each case. These notations are convenient to illustrate the cases for the amplitude on X− v.s. X plane (See Figure 1). The amplitude of the next step X+ is calculated from ξn−1 , ξn , Xn−1 and Xn as follows. (i) ξn−1 = ξn = ξn+1 : a solution does not exist. (ii) ξn−1 = ξn = −ξn+1 : we have the following amplitude.   c2 − a (Cg )  X + α (2.15)

X+ =

  

X−

(Ce )

X−

(Cℓ )

(iii) −ξn−1 = ξn = ξn+1 : we have the following amplitude including indeterminate solutions. (2.16)

X+ ≤ X−

(Ce )

(2.17)

X+ = X−

(Cℓ )

(iv) ξn−1 = −ξn = ξn+1 : we have the following amplitude including indeterminate solutions. (2.18)

X+ = X + α c2 − a

(Cg )

(2.19)

X+ ≤ X−

(Ce )

We comment that we need further cases to obtain the expression without max in a:        max 0, α c1 − α c3   max 0, α c1 X<  2           max 0, α c1 − α c3 (2.20) a = max 0, α c1 = α c3 + 2X X= 2         max 0, α c1 − α c3   α c3 + 2X X> . 2 Because of this, X+ = X + α c2 − a is not a line    X +α c2 − max [0, α c1 ]        max 0, α c1 + α c3 X+ = − (2.21) +α c2 2       −X + α c2 − α c3

but a polygonal line on the phase plane:     max 0, α c1 − α c3 X< 2     max 0, α c1 − α c3 X= 2     max 0, α c1 − α c3 . X> 2

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Shin Isojima and Seiichiro Suzuki

Based on this evolution rule, we analyze solutions on the phase plane for the amplitude. Each condition appears on the phase plane as a domain, for example, {(X− , X) | Cg } (see Figure 1 (a)). Note that the polygonal line which gives the boundary draws the shape “>.” The coordinate of the indifferentiable point P 1 is   max[0, α c1 ] + α c3 max[0, α c1 ] − α c3 (2.22) P1 − +α c2 , . 2 2 Then, we calculate X+ and examine the mapped domain {(X, X+ )}. Here, mirrorimage domains about the line X = X− become important on the X v.s. X+ plane. Such domains are shown in Figure 1 (b). The boundary is observed as the polygonal line of the shape “∧” whose indifferentiable point P 2 is   max[0, α c1 ] − α c3 max[0, α c1 ] + α c3 (2.23) P2 ,− +α c2 . 2 2

(a)

(b) Figure 1. (a) conditions Cg , Ce , Cℓ and domains

(b) mirror-image domains

By the result in [24], the positional relationship between “>” and “∧” is quite important to classify the behavior of solutions. In this article, we focus on the “overlapping” type of relationship as shown in Figure 2. It was found in [24] that this positional type appears if and only if α c2 > 0 and α c2 > α c1 . Note that we discuss only the amplitude variables on the above phase plane. For including information of the sign variables, in [24], four amplitude phase planes which correspond to the four pairs of signs (ξn , ξn+1 ) = (+, +), (+, −), (−, +), (−, −), respectively, were introduced. If we use the (ξn eXn ) v.s. (ξn+1 eXn+1 ) coordinates, and place the four planes on the corresponding quadrants (See Fig. 8 in [24]), we can discuss the global dynamics of the solutions. However, in this article, we mainly discuss the amplitude phase plane, because the behavior of solutions for “overlapping” type becomes rather complicated on the (ξn eXn ) v.s. (ξn+1 eXn+1 ) plane.

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 71

Figure 2. “overlapping” type of domains

§ 3.

Transition on the amplitude phase plane

§ 3.1.

Notation

For explanation in this section, we define some objects on the amplitude phase plane. We shall use (x, y) as the generic coordinate for the phase plane. We denote the midpoint between P 1 and P 2 as P 0, whose coordinate is  P0

(3.1)

α c2 − α c3 α c2 − α c3 , 2 2

 .

We define points P (2k − 1) (k = 1, 2, 3, . . . ) whose coordinate (P (2k − 1)x , P (2k − 1)y ) is given by (3.2) (3.3)

max[0, α c1 ] − α c2 max[0, α c1 ] + α c3 (k − 1) − +α c2 , 2 2 max[0, α c1 ] − α c2 max[0, α c1 ] − α c3 P (2k − 1)y = (k − 1) + 2 2

P (2k − 1)x =

and P (2k) (k = 1, 2, 3, . . . ) whose coordinate is (P (2k − 1)y , P (2k − 1)x ). Note that {P (2k − 1)} and {P (2k)} are on L4 and L5 (defined by (3.15) and (3.14)), respectively, and that P (2k) and P (2k − 1) for a fixed k are symmetric about y = x. Note that we may omit the round brackets for P if we substitute a specific value to k. For example, P (2k − 1) k=1 = P 1, P (2k) k=1 = P 2, which are consistent with (2.22), (2.23), respectively.

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We further define “open” half lines

(3.4) (3.5)

LP (2k−1) = {(x, y) | x = P (2k − 1)x , y < P (2k − 1)y } , LP (2k) = {(x, y) | x < P (2k)x , y = P (2k)y } .

Here, “open” means that each half line does not include its end point. Moreover, we define open segments

(3.6)

P (2k − 1)P (2k + 1) = {(x, y) ∈ L4 | P (2k + 1)x < x < P (2k − 1)x } ,

(3.7)

P (2k)P (2k + 2) = {(x, y) ∈ L5 | P (2k + 2)x < x < P (2k)x } .

These objects are illustrated in Figure 3.

Figure 3. Points, half lines, and “open segments”

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 73

Finally, we define the following domains. (3.8)

S1 = {(x, y) | y > x, y > −x + α c2 − α c3 }

(3.9)

S2 = {(x, y) | y > P 2y , y < −x + α c2 − α c3 }

(3.10)

S5 = {(x, y) | x > P 1x , y < −x + α c2 − α c3 }

(3.11)

S6 = {(x, y) | y < x, y > −x + α c2 − α c3 }

(3.12)

L1 = {(x, y) | y = x}

(3.13)

L2 = {(x, y) | y = −x + α c2 − α c3 , x < P 2x }

(3.14)

L5 = {(x, y) | y = x − max[0, α c1 ] + α c2 , x < P 1y }

(3.15)

L4 = {(x, y) | y = x + max[0, α c1 ] − α c2 , x < P 1x }

(3.16) (3.17) (3.18)

L7 = {(x, y) | y = −x + α c2 − α c3 , x > P 1x }   α c2 − α c3 < x < P 1x P 0P 1 = (x, y) | y = −x + α c2 − α c3 , 2   α c2 − α c3 P 0P 2 = (x, y) | y = −x + α c2 − α c3 , P 1x < x < 2

See Figure 4 in which these domains are illustrated. In the approximative method, we trace the time evolution of the amplitude as transition among these domains, not among points. Its detail is explained in the following subsections.

Figure 4. domains

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Shin Isojima and Seiichiro Suzuki

§ 3.2.

ξn−1 = ξn = ξn+1

This case has no solution in the “overlapping” relationship. § 3.3.

ξn−1 = ξn = −ξn+1

Transition of the amplitude on the X− v.s. X phase plane is given in Table 3 of   [24]. First, we study the case Cℓ : X < X− + a − α c2 (a = max 0, α c1 , α c3 + 2X ). Such points are on the right of “>” (see Figure 1 (a)). In this case, we have Xn+1 = X− , and therefore the unique mapping (X− , X) 7→ (X, X− ), which is the reflection about the line X = X− , is obtained. Therefore, the right of “>” is mapped to the upper of “∧.” As specific cases, LP (2k−1) is mapped to LP (2k) . Second, we study the case Ce : X = X− + a − α c2 . Such points are on L4 ∪ L2 ∪ P 0P 1 ∪ P 0P 2 ∪ P 0 ∪ P 1 ∪ P 2. We again have the unique reflection about the line X = X− . As preparation to discuss other cases, we illustrate the approximative method in terms of (3.6) and (3.7). In this method, the mapping is considered to map a set on the phase plane to the other. By (3.6) and (3.7), L4 7→ L5 is segmentalized as P (2k −1)P (2k +1) 7→ P (2k)P (2k +2) (k = 1, 2, 3, . . . ). We shall need this segmentation in later cases. We also obtained P (2k − 1) 7→ P (2k). Moreover, we obtain P 0P 1 7→ P 0P 2, P 0P 2 7→ P 0P 1, P 1 7→ P 2, and P 2 7→ P 1, which are bijections. Finally, P 0 is a fixed point. Third, we study the case Cg : X > X− + a − α c2 . Such points are on the left of “>.” In this case, we have Xn+1 = X − a + α c2 , and therefore the non-injective mapping (X− , X) 7→ (X, X − a + α c2 ) ∈ L5 ∪ L7, which is independent in X− , is obtained. If we introduce horizontal band domains (see Figure 5) (3.19)

˜ Y˜ ) ∈ P (2k − 1)P (2k + 1), x < X, ˜ y = Y˜ }, HBk = {(x, y) | (X,

any points (X− , X) ∈ HBk , that is, (X− , Y˜ ) is mapped to the point (Y˜ , Y˜ − a + α c2 ) on P (2k)P (2k+2). This mapping can be visually understood as follows. The point (X− , Y˜ ) is temporally mapped to a mirror image about X = X− , and the mirror image is further projected to vertical direction onto P (2k)P (2k + 2). We often use this “reflection and projection” hereafter. By the approximative method, we have HBk 7→ P (2k)P (2k + 2). If we further introduce (3.20)

˜ Y˜ ) ∈ P 0P 1, x < X, ˜ y = Y˜ }, HB1′ = {(x, y) | (X,

we obtain the approximative mapping HB1′ 7→ P 0P 2. Similarly, introducing (3.21)

˜ Y˜ ) ∈ P 0P 2, x < X, ˜ y = Y˜ }, HB2′ = {(x, y) | (X,

we obtain HB2′ 7→ P 0P 1.

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 75

Now, we have traced the image of all points on the X− v.s. X plane for ξn−1 = ξn = −ξn+1 .

(a)

(b) Figure 5. (a) “horizontal band” domains (b) “vertical band” domains

§ 3.4.

−ξn−1 = ξn = ξn+1

First, we check the case Cℓ : X < X− + a − α c2 . As a consequence, we have the same results as the first case in Subsection 3.3. Second, we consider the case Ce : X = X− + a − α c2 . In this case, we have the indeterminate solution X+ ≤ X− . We firstly consider a point on L4. If we take a point on a open segment P (2k − 1)P (2k + 1), it can be mapped on the open segment P (2k)P (2k +2) by reflection and projection or on the vertical-band domains (see Figure 5 (b)) (3.22)

˜ Y˜ ) ∈ P (2k)P (2k + 2), x = X, ˜ y < Y˜ }. V Bk = {(x, y) | (X,

By approximative method, P (2k − 1)P (2k + 1) 7→ P (2k)P (2k + 2) ∪ V Bk . Secondly, a point on P 0P 1 is mapped on P 0P 2 by reflection and projection or V B2′ defined by (3.23)

˜ Y˜ ) ∈ P 0P 2, x = X, ˜ y < Y˜ }. V B2′ = {(x, y) | (X,

Points P 0 and P 1 are respectively mapped on P 0 ∪ P 3 ∪ P 0P 3 ∪ LP 3 and P 0 ∪ P 5 ∪ P 2P 5 ∪ LP 5 , where we define open segments (3.24)

P (2k)P (2k + 3) = {(x, y) | x = P (2k)x , P (2k + 3)y < y < P (2k)y },

and P 0P 3 and P 2P 5 are obtained their specific cases for k = 0, 1. Thirdly, a point on P 0P 2 is mapped on P 0P 1 by reflection and projection or V B1′ defined by (3.25)

˜ Y˜ ) ∈ P 0P 1, x = X, ˜ y < Y˜ }. V B1′ = {(x, y) | (X,

76

Shin Isojima and Seiichiro Suzuki

Similarly, a point on P 2 is mapped on P 1 ∪ LP 1 . Fourthly, we consider a point on the half line L2. It is mapped on L7 by reflection and projection or on S5. § 3.5.

ξn−1 = −ξn = ξn+1

First, we check the case Ce . We have the same results as the second case in Subsection 3.4. Second, we check the case Cg . We have the same results as the third case in Subsection 3.3. § 3.6.

Examples of solutions

In this section, some examples of solutions are illustrated. When ξn−1 6= ξn and (X− , X) ∈ L4∪L2∪P 0P 1∪P 0P 2∪P 0∪P 1∪P 2, we encounter indeterminate solutions. By choosing a possible pair of sign and amplitude, the successive solutions are calculated. However, we may encounter indeterminate solutions again. Its indeterminacy may generally be different from the first one. We use notation Zn = (ξn−1 , ξn , Xn−1 , Xn ) for representing a pair of successive two points. Here, we assign initial values and study the initial value problem. In the first example, we do not encounter any indeterminate solution. We start from Z1 = (+, +, X0 , X1 ) and (X0 , X1 ) ∈ S1. We obtain the following transition: Z1 7→ Z2 = (+, −, X1 , X0 ), (X1 , X0 ) ∈ S6, 7→ Z3 = (−, −, X0 , X1 ), (X0 , X1 ) ∈ S1, 7→ Z4 = (−, +, X1 , X0 ), (X1 , X0 ) ∈ S6, 7→ Z5 = (+, +, X0 , X1 ) = Z1 . Note that these transitions are unique. In “approximative method” which was proposed in [24], information on a domain in the phase plane to which a pair of amplitude (Xn−1 , Xn ) belongs is more important than values of amplitudes. Therefore, we reinterpret this mapping that from a domain to another one, not from a point to another. For convenience, we introduce further short expression, “Zn ∈ (ξn−1 , ξn , domain).” For example, the above examples are written as Z1 ∈ (+, +, S1) 7→ Z2 ∈ (+, −, S6), 7→ Z3 ∈ (−, −, S1), 7→ Z4 ∈ (−, +, S6), 7→ Z5 ∈ (+, +, S1). In the concept of approximative method, this transition should be expressed as (+, +, S1) 7→ (+, −, S6) 7→ (−, −, S1) 7→ (−, +, S6) 7→ (+, +, S1) · · · .

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 77

We combine use of the original form Zn = (ξn−1 , ξn , Xn−1 , Xn ) and the short form Zn ∈ (ξn−1 , ξn , domain) for understanding the discussion. Since all evolutions are unique in this example, advantage of this approximation is not clear. However, it is convenient to understand indeterminate solutions as shown in the next example. The authors comment that they expressed the information of two signs (ξn−1 , ξn ) by some shapes in [24]. That is, a pair of signs (+, +) is represented by a white circle, (+, −) by a black circle, (−, −) by a black-painted diamond, and (−, +) by a white-painted diamond. The information of the amplitudes is represented by writing the set which a pair of two amplitudes belongs into the shape. For example, the above Z1 –Z4 are represented as follows. Z1 :

Z2 :

Z3 :

Z4 :

Moreover, by using these shapes, the transition diagrams were drawn in [24]. In the second example, we encounter a infinite number of indeterminate solutions but they are just two types. We start from Z1 = (+, −, X0 , X1 ) and (X0 , X1 ) ∈ L2. Note that this initial value belongs to the domain (+, −, L2), and that actually X1 = −X0 + α c2 − α c3 holds. At the next step, we encounter indeterminate solutions. Hence, in the usual sense, an infinite number of pairs of sign and amplitude can be chosen. However, we classify such solutions into four types as (3.26)

Z2 = (−, +, X1 , X0 ), (X1 , Y2 ) ∈ L7,

(3.27)

Z2′ = (−, +, X1 , Y2 ), (X1 , Y2 ) ∈ S5 (Y2 < −X1 + α c2 − α c3 ),

(3.28)

Z2′′ = (−, −, X1 , Y2 ), (X1 , Y2 ) ∈ S5,

(3.29)

Z2′′′ = (−, −, X1 , X0 ), (X1 , X0 ) ∈ L7.

Here, Y2 denotes a chosen value for X2 which is less than −X1 + α c2 − α c3 . Note that X2 = −X1 + α c2 − α c3 = X0 holds in (3.26) and (3.29). We calculate successive evolutions for each type. The results are as follows. (i) When we choose Z2 , Z2 7→ Z3 = (+, +, X0 , X1 ), (X0 , X1 ) ∈ L2, 7→ Z4 = (+, −, X1 , X0 ), (X1 , X0 ) ∈ L7, 7→ Z5 = (−, −, X0 , X1 ), (X0 , X1 ) ∈ L2, 7→ Z6 = Z2 .

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(ii) When we choose Z2′ , Z2′ 7→ Z3′ = (+, +, Y2 , X1 ), (Y2 , X1 ) ∈ S2, 7→ Z4′ = Z4 , 7→ Z5′ = Z5 , 7→ Z6′ = Z6 = Z2 . (iii) When we choose Z2′′ , Z2′′ 7→ Z3′′ = (−, +, Y2 , X1 ), (Y2 , X1 ) ∈ S2, 7→ Z4′′ = Z4 , 7→ Z5′′ = Z5 , 7→ Z6′′ = Z6 = Z2 . (iv) When we choose Z2′′′ , Z2′′′ 7→ Z3′′′ = (−, +, X0 , X1 ), (X0 , X1 ) ∈ L2, 7→ Z4′′′ becomes indeterminate. We again classify indeterminate solutions into four types as Zˆ4 = (+, +, X1 , X0 ), (X1 , X0 ) ∈ L7, Zˆ4′ = (+, −, X1 , X0 ), (X1 , X0 ) ∈ L7, ˆ 2 ), (X1 , Y ˆ 2 ) ∈ S5, Zˆ4′′ = (+, +, X1 , Y ˆ 2 ), (X1 , Y ˆ 2 ) ∈ S5. Zˆ4′′′ = (+, −, X1 , Y ˆ 2 denotes a chosen value for X4 which is less than −X1 + α Here, Y c2 − α c3 and may be ′′′ ˆ different from Y2 . If we choose Z4 = Z4 , we obtain the following evolution: Z4′′′ = Zˆ4 , 7→ Z5′′′ = (+, −, X0 , X1 ), (X0 , X1 ) ∈ L2, which is identical to the initial value Z1 . For the other cases, we obtain Z4′′′ = Zˆ4′ (= Z4 ), 7→ Zˆ5′ = Z5 , 7→ Zˆ6′ = Z2 , and Z4′′′ = Zˆ4′′ , ˆ 2 , X1 ), (Y ˆ 2 , X1 ) ∈ S2, 7→ Zˆ5′′ = (+, −, Y 7→ Zˆ6′′ = Z2 ,

Phase plane analysis for p-ultradiscrete system: infinite types of branching conditions 79

and Z4′′′ = Zˆ4′′′ , ˆ 2 , X1 ), (Y ˆ 2 , X1 ) ∈ S2, 7→ Zˆ5′′′ = (−, −, Y 7→ Zˆ ′′′ = Z2 . 6

Note that the domains L2, L7, S2, and S5 appear with all pair of signs (ξn−1 , ξn ). This means that these solution-orbits are closed in these domains. The behavior of this solution is summarized by the transition diagram in Figure 6. Four arrows which mean the indeterminate solutions leave from (+, −, L2) and (−, +, L2). Note that, although ˆ 2 denote an infinite number of values in usual sense, they are expressed by Y2 and Y one domain S5 or S2. (+, +, 𝐿𝐿𝐿)

(+, −, 𝑆𝑆𝑆)

(−, −, 𝑆𝑆𝑆)

(−, +, 𝐿𝐿𝐿)

(+, −, 𝐿𝐿𝐿)

(−, −, 𝐿𝐿𝐿)

(+, +, 𝐿𝐿𝐿)

(−, +, 𝑆𝑆𝑆)

(−, −, 𝑆𝑆𝑆)

(+, +, 𝐿𝐿𝐿)

(+, +, 𝑆𝑆𝑆)

(+, −, 𝑆𝑆𝑆)

(−, +, 𝐿𝐿𝐿)

(+, −, 𝐿𝐿𝐿)

(+, +, 𝑆𝑆𝑆)

(−, +, 𝑆𝑆𝑆)

Figure 6. Example of transition diagram In the third example, we study the most complicated case. We study the transition for the domain on the amplitude phase plane (3.30)

{(x, y) | y < P 2y , y ≤ −x + α c2 − α c3 , x < P 1x } ,

whose discussion has been omitted in [24]. Note that we have already introduced some subsets included in (3.30). We first consider the horizontal-band domains defined by (3.19), (3.20), and (3.21). If we start from a point (X0 , X1 ) ∈ HBk , we obtain (±, ±, HBk ), (±, ∓, HBk ) 7→ (±, ∓, P (2k)P (2k + 2)), (∓, ±, P (2k)P (2k + 2)) 7→ (∓, ±, P (2k − 4)P (2k − 2)), (±, ∓, P (2k − 4)P (2k − 2)) 7→ · · · . If we put it into words, a horizontal band is mapped to the corresponding open segment, and after this, goes up to every other open segment in order, and reach P2P4 or P4P6.

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Shin Isojima and Seiichiro Suzuki

Note that both HB1′ and P 4P 6 are mapped to P 0P 2, and similarly, both HB2′ and P 2P 4 are mapped to P 0P 1. Moreover, the solution (ξn−1 , ξn , Xn−1 , Xn ) on an open segment has different signs, that is, ξn−1 ξn = −1. Therefore, a point (X0 , X1 ) ∈ HBk will be eventually included in one of (±, ∓, P 0P 1), (±, ∓, P 0P 2). Then, at the next step, we encounter indeterminate solutions. Next, we define the “bottom” part of each vertical band by (3.31) ˜ Y˜ ) ∈ P (2k)P (2k + 2), x = X, ˜ y

1 γ − β βδk

which must be satisfied when we use the gdSIR model as a numerical scheme. If we set ϵn = ϵ, i.e., integrable case, the above exact solution takes the following simple form: )n ( 1 − (1 − p)βϵ (3.138) Sn = S0 , 1 + pβϵ (3.139) In = I0 − γϵn  + βϵS0 p

1 − (1 − p)βϵ 1 −

(

1−(1−p)βϵ 1+pβϵ

)n

1 + pβϵ 1 − 1−(1−p)βϵ 1+pβϵ ( )n 1 − (1 − p)βϵ = S0 + I0 − γϵn − S0 , 1 + pβϵ (3.140) Rn = R0 + γϵn , (3.141) tn = t0 +

n−1 ∑ k=0

1 ϵ. Ik

+ (1 − p)

1−

(

1−

1−(1−p)βϵ 1+pβϵ

)n 

1−(1−p)βϵ 1+pβϵ



I NTEGRABLE DISCRETIZATIONS OF THE SIR MODEL

105

Note that Sn is written in the form of a power function which includes the infection rate β, the lattice parameter ϵ and the initial value S0 , and In is a linear combination of a power function and a linear function. This drastic simplification is due to integrability. Since Sn and In always take positive values in the SIR model, we find two inequalities (1 − p)βϵ < 1 for p ̸= 1 , ( )n 1 − (1 − p)βϵ S0 + γϵn < S0 + I0 1 + pβϵ

(3.142) (3.143)

which must be satisfied when we use the gdSIR model with ϵn = ϵ as a numerical scheme. If we set δn = δ, the above solution leads to (3.144)

Sn = S0

n−1 ∏ k=0

(3.145)

In = I0 + β

1 − (1 − p)βδIk , 1 + pβδIk n−1 ∑

δIk (pSk+1 + (1 − p)Sk ) − γδ

k=0

= I0 + βS0

= I0

Ik

k=0

n−1 ∑ k=0

n−1 ∏

n−1 ∑

δIk 1 + pβδIk

k−1 ∏ l=0

∑ 1 − (1 − p)βδIl − γδ Ik 1 + pβδIl n−1

k=0

(1 − γδ + βδSk+1 ) ,

k=0

(3.146)

Rn = R0 + γδ

n−1 ∑

Ik .

k=0

In figure 4, we show the graphs of the exact solution to the initial value problems for the dSIR1 model, the dSIR2 model and the gdSIR model with p = 0.5 in the case of ϵn = ϵ, where the horizontal axis is τ . Among these three cases, the gdSIR model with p = 0.5 gives the exact solution which is very close to the exact solution of the SIR model. To find the best value of p for which the second conserved quantities of the gdSIR model and the SIR model coincide, the relation (3.147)

1 1 − (1 − p)βϵ β(S + I) − γ log S = − (S + I) log − γ log S . ϵ 1 + pβϵ

must be satisfied for any solutions. This leads to 1 − (1 − p)βϵ = e−βϵ , 1 + pβϵ

(3.148)

and solving this equation, we obtain (3.149)

p=

βϵ − 1 + e−βϵ (βϵ − 1)eβϵ + 1 = . βϵ(1 − e−βϵ ) βϵ(eβϵ − 1)

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Y UTA TANAKA AND K EN - ICHI M ARUNO

S

R

I

Figure 4. The graph of an exact solution to the initial value problem for the gdSIR model with p = 0 (the dSIR1 model), p = 0.5, p = 1 (the dSIR2 model). The parameters and initial values are β = 0.0005, γ = 0.1, S(0) = 997, I(0) = 3, R(0) = 0, ϵk = 500. The horizontal axis is τ .

Thus by using this formula to determine the value p, the second conserved quantity of the gdSIR model coincides with the one of the SIR model. In the case of figure 4, the best value of p is 0.520812 · · · . If we substitute the formula (3.149) into the exact solution (3.138), we find )n ( 1 − (1 − p)βϵ (3.150) Sn = S0 = S0 e−βϵn = S0 e−β(τ −τ0 ) , 1 + pβϵ where τ = τ0 + ϵn. This means that Sn in the gdSIR model coincides with S(τ ) in the SIR model. Thus Sn , In , Rn in the gdSIR model coincide with S(τ ), I(τ ), R(τ ) in the SIR model respectively if we choose the best value of p. However, the discrete time variable tn in the gdSIR model does not coincide with the time variable t in the SIR model. Thus the solution set (tn , Sn ), (tn , In ), (tn , Sn ) of the gdSIR model is different from the solution set (t, S(t)), (t, I(t)), (t, R(t)) of the SIR model. To obtain numerical solutions that are close to exact solutions, it is necessary to choose ϵ as small as possible. In figure 5, we show the graphs of a numerical computation by the gdSIR model with the best value of p. In this case, i.e., β = 0.0005, ϵk = ϵ = 0.5, the best value is p = 0.50021 · · · which is very close to 0.5. This is because (3.151)

(βϵ − 1)eβϵ + 1 1 = . ϵ→0 βϵ(e1βϵ − 1) 2 lim

I NTEGRABLE DISCRETIZATIONS OF THE SIR MODEL

107

Thus we can choose p = 0.5 for numerical computations if we choose small enough ϵ.

Figure 5. The graph of a numerical solution to the initial value problem for the gdSIR model with the best value p = 0.50021 · · · . The parameters and initial values are β = 0.0005, γ = 0.1, S(0) = 997, I(0) = 3, R(0) = 0, ϵk = 0.5.

§ 3.4.

The nonautonomous generalized discrete SIR model

In this subsection, we consider the nonautonomous gdSIR model. We discretize the system of linear differential equations (2.9), (2.10), (2.11) in the following form: (3.152) (3.153) (3.154)

Sn+1 − Sn = −β(pn Sn+1 + (1 − pn )Sn ) , ϵn In+1 − In = β(pn Sn+1 + (1 − pn )Sn ) − γ , ϵn Rn+1 − Rn =γ, ϵn

where Sn = S(τn ), In = I(τn ), Rn = R(τn ), and the parameter pn are real numbers between 0 and 1 depending on n. Let us define τn and tn as (3.4) and (3.5). Then we consider the discrete hodograph transformation (3.6) and the inverse discrete hodograph transformation (3.7). Substituting ϵn = δn In into the system of linear difference equations (3.152), (3.153),

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(3.154), we obtain the nonautonomous gdSIR model (3.155) (3.156) (3.157)

Sn+1 − Sn = −β(pn Sn+1 + (1 − pn )Sn )In , δn In+1 − In = β(pn Sn+1 + (1 − pn )Sn )In − γIn , δn Rn+1 − Rn = γIn , δn

where Sn = S(tn ) = S(τn ), In = I(tn ) = I(τn ), Rn = R(tn ) = R(τn ). Note that the nonautonomous gdSIR model (3.155), (3.156), (3.157) is rewritten as 1 − (1 − pn )βδn In , 1 + pn βδn In = (1 + βδn (pn Sn+1 + (1 − pn )Sn ) − γδn ) In ,

(3.158)

Sn+1 = Sn

(3.159)

In+1

(3.160)

Rn+1 = Rn + γδn In .

Note that the set of (tn , Sn ), (tn , In ), (tn , Rn ) provides the approximate solution of the SIR model. Next we consider conserved quantities. We can easily see that Sn + In + Rn is a conserved quantity of the nonautonomous gdSIR model (3.155), (3.156), (3.157). From equation (3.152) we obtain Sn+1 =

(3.161)

1 − (1 − pn )βϵn Sn 1 + pn βϵn

which leads to (3.162)

log Sn+1 − log Sn = log

1 − (1 − pn )βϵn . 1 + pn βϵn

Adding (3.152) and (3.153), we obtain Sn+1 + In+1 − Sn − In = −γϵn

(3.163)

From (3.162) and (3.163), we obtain (3.164)

β((Sn+1 + In+1 ) − (Sn + In )) − γ(log Sn+1 − log Sn ) 1 − (1 − pn )βδn In . = −βγϵn − γ log 1 + pn βδn In

If we require log

(3.165)

1 − (1 − pn )βδn In = −βϵn 1 + pn βδn In

to (3.164), then we obtain (3.166)

β(Sn+1 + In+1 ) − γ log Sn+1 = β(Sn + In ) − γ log Sn ,

I NTEGRABLE DISCRETIZATIONS OF THE SIR MODEL

109

which indicates that β(Sn + In ) − γ log Sn

(3.167)

is a conserved quantity of the nonautonomous gdSIR model (3.155), (3.156), (3.157), but it is also a conserved quantity of the continuous SIR model. Solving (3.165), we obtain (3.168)

pn =

βϵn − 1 + e−βϵn (βϵn − 1)eβϵn + 1 = , βϵn (1 − e−βϵn ) βϵn (eβϵn − 1)

and substituting ϵn = δn In into (3.168), this formula is written as (3.169)

pn =

βδn In − 1 + e−βδn In (βδn In − 1)eβδn In + 1 = . βδn In (1 − e−βδn In ) βδn In (eβδn In − 1)

This means that the nonautonomous gdSIR model is integrable when pn is given by (3.169). Next we verify directly that (3.167) is a conserved quantity of the gdSIR model when pn is given by (3.169). From (3.155), we have (3.170)

Sn+1 = Sn

1 − (1 − pn )βδn In 1 + pn βδn In

and by taking the logarithm of both sides of this equation we obtain (3.171)

log Sn+1 − log Sn = log

1 − (1 − pn )βδn In 1 + pn βδn In

Adding (3.155) and (3.156), we obtain (3.172)

Sn+1 + In+1 − (Sn + In ) = −γδn In .

Combining (3.171) and (3.172), we obtain (3.173)

β(Sn+1 + In+1 − (Sn + In )) − γ(log Sn+1 − log Sn ) 1 − (1 − pn )βδn In = −βγδn In − γ log . 1 + pn βδn In

If (3.174)

log

1 − (1 − pn )βδn In = −βδn In 1 + pn βδn In

is satisfied, i.e., pn is given by (3.169), then (3.167) is a conserved quantity of the nonautonomous gdSIR model (3.155), (3.156), (3.157). In other words, ) ( Sn+1 − Sn (βδn In − 1)eβδn In + 1 (Sn+1 + Sn ) + Sn In , (3.175) = −β δn βδn In (eβδn In − 1) ) ( In+1 − In (βδn In − 1)eβδn In + 1 (3.176) (Sn+1 + Sn ) + Sn In − γIn , =β δn βδn In (eβδn In − 1) Rn+1 − Rn (3.177) = γIn , δn

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Y UTA TANAKA AND K EN - ICHI M ARUNO

has the same conservation quantities as the SIR model (1.1), (1.2), (1.3), thus we can think that the system of difference equations (3.175), (3.176), (3.177) with the hodograph transformation is an integrable discrete analogue of the SIR model. § 4.

Integrability of an ultradiscretizable SIR model

Sekiguchi et al. presented an ultradiscrete SIR model with time delay and studies its analytical property [17]. Although they presented some special solutions, they did not mention about integrability of the ultradiscrete SIR model. In this section, we consider an ultradiscretizable SIR model and its ultradiscretization from the point of view of integrability. Let us consider the following discrete SIR model: (4.1) (4.2) (4.3)

Sn+1 − Sn = −βSn+1 In , δn In+1 − In = βSn+1 In − γIn+1 , δn Rn+1 − Rn = γIn+1 , δn

where Sn = S(τn ), In = I(τn ), Rn = R(τn ). If δn is a constant, this discrete SIR model is a special case of the discrete SIR model considered in Sekiguchi et al. This can be written as (4.4) (4.5) (4.6)

Sn , 1 + βδn In (1 + βδn Sn+1 )In In+1 = , 1 + γδn Rn+1 = Rn + γδn In+1 . Sn+1 =

Let us define τn and tn as (3.4) and (3.5). Then we consider the discrete hodograph transformation (3.6) and the inverse discrete hodograph transformation (3.7). By using the relation ϵn = δn In , the discrete SIR model (4.1), (4.2), (4.3) is transformed to the following form: (4.7) (4.8) (4.9)

Sn+1 − Sn = −βSn+1 , ϵn In+1 − In In+1 = βSn+1 − γ , ϵn In Rn+1 − Rn In+1 =γ . ϵn In

We note that (4.7) is a linear difference equation but (4.8) is a nonlinear difference equation. Next we consider conserved quantities of the discrete SIR model (4.1), (4.2), (4.3). We can easily see that Sn + In + Rn is a conserved quantity of the discrete SIR model (4.1), (4.2), (4.3).

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111

By taking the logarithm of both sides of (4.5), we obtain (4.10)

log Sn+1 − log Sn = − log(1 + βδn In ) .

Adding (4.1) and (4.2), we obtain (4.11)

Sn+1 + In+1 − (Sn + In ) = −γδn In+1 .

Combining (4.10) and (4.11), we obtain (4.12) −(Sn+1 + In+1 − (Sn + In )) log(1 + βδn In ) + γδn In+1 (log Sn+1 − log Sn ) = 0 is zero for any n, this is invariant for (4.1), (4.2), (4.3). Setting ϵn = δn In = ϵ and δn In+1 = µ, where ϵ and µ are constant, we obtain (4.13) −(Sn+1 +In+1 ) log(1+βϵ)+γµ(log Sn+1 ) = −(Sn +In ) log(1+βϵ)+γµ(log Sn ) , which indicates that (4.14)

(Sn + In ) log(1 + βϵ) − γµ(log Sn )

is a conserved quantity of the discrete SIR model (4.1), (4.2), (4.3). This means that the discrete SIR model (4.1), (4.2), (4.3) is integrable when ϵn and In+1 /In are constants. Other cases including δn = δ, where δ is a constant, are nonintegrable. In the case of ϵn = ϵ and In+1 /In = µ/ϵ, i.e., integrable case, the discrete SIR model (4.1), (4.2), (4.3) is written as (4.15) (4.16) (4.17)

Sn+1 − Sn = −βSn+1 In , δn µ In+1 − In = βSn+1 In − γ In , δn ϵ Rn+1 − Rn µ = γ In , δn ϵ

which is equivalent to the dSIR2 model. Note that one of the conditions of integrability, In+1 /In = µ/ϵ, is too strong because this condition indicates that In is a geometric sequence but this is incompatible with the conservation of population. Let us consider the solution to the initial value problem for the discrete SIR model (4.1), (4.2), (4.3). For the initial value S(t0 ) = S(τ0 ) = S0 , I(t0 ) = I(τ0 ) = I0 , R(t0 ) = R(τ0 ) =

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R0 , the solution of (4.7), (4.8), (4.9) is given by (4.18)

S n = S0

n−1 ∏ k=0

(4.19)

1 , 1 + βϵk

In = I0 + βS0

n−1 ∑

ϵk

k=0

= I0 + β

n−1 ∑

k ∏ l=0

∑ Ik+1 1 −γ ϵk 1 + βϵl Ik

ϵk Sk+1 − γ

k=0

(4.20)

k=0

n−1 ∑

ϵk

k=0

n−1 ∑

Rn = R 0 + γ

n−1

ϵk

k=0

Ik+1 , Ik

Ik+1 . Ik

Substituting ϵn = δn In into this solution, the solution to the initial value problem for the discrete SIR model (4.1), (4.2), (4.3) is given by (4.21)

Sn = S0

n−1 ∏ k=0

(4.22)

1 , 1 + βδk Ik

In = I0 + βS0

n−1 ∑

ϵk

k=0

= I0 + β

n−1 ∑

k ∏ l=0

∑ 1 δk Ik+1 −γ 1 + βδl Il n−1

k=0

δk Ik Sk+1 − γ

Rn = R0 + γ

δk Ik+1 ,

k=0

k=0

(4.23)

n−1 ∑

n−1 ∑

δk Ik+1 ,

k=0

(4.24)

tn = t0 +

n−1 ∑

δk .

k=0

From (4.22), In can be written as I0 + β (4.25)

δk Ik Sk+1 − γ

k=0

In =

n−2 ∑

δk Ik+1

k=0

1 + γδn−1 I0 + β

=

n−1 ∑

n−1 ∑

δ k I k S0

k=0

k ∏ l=0

∑ 1 −γ δk Ik+1 1 + βδl Il

1 + γδn−1

From (4.5), In can be also written as (4.26)

In = I0

n−1 ∏ k=0

1 + βδk Sk+1 . 1 + γδk

n−2

k=0

.

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113

If we set ϵn = δn In = ϵ, the above exact solution takes the following simpler form: (4.27)

Sn =

S0 , (1 + βϵ)n I0 + βϵ

(4.28)

n−1 ∑ k=0

In =

∑ Ik+1 S0 − γ ϵ (1 + βϵ)k+1 Ik n−2

k=0

1 + γδn−1 S0 + I0 − S0 (1 + βϵ)−n − γ

Ik+1 Ik = I0

1 + γδn−1 n−1 ∑

Rn = R0 + γ

ϵ

k=0

(4.30)

ϵ

k=0

=

(4.29)

n−2 ∑

tn = t0 +

n−1 ∑

n−1 ∏

S0 (1 + βϵ)k+1 , Ik + γϵ

Ik + βϵδk

k=0

Ik+1 , Ik

δk .

k=0

Note that Sn is written in the form of a power function which includes the infection rate β, the lattice parameter ϵ and the initial value S0 , but In is not simple as the previous discrete SIR models, i.e., for getting a solution, we need to compute the above formula recursively. This is due to the lack of the second conserved quantity. If we set δn = δ, the above solution leads to (4.31) Sn = S0

n−1 ∏ k=0

1 , 1 + βδIk

(4.32) In = I0 + βδ

n−1 ∑

Ik Sk+1 − γδ

k=0

= I0 + βδS0

k=0

(4.33) Rn = R0 + γδ

Ik

k=0

n−1 ∑

n−1 ∑

n−1 ∑

Ik

k ∏ l=0

1 − γδ 1 + βδIl

n−1 ∑ k=0

Ik = I0

n−1 ∏

1 + βδS0

k=0

k ∏ l=0

1 1 + βδIl

1 + γδ

,

Ik+1 .

k=0

Setting Sn = exp(Sn /h), In = exp(In /h), Rn = exp(Rn /h), β = exp(B/h), γ = exp(Γ/h), ϵn = exp(En /h), δn = 1, we obtain (4.34) (4.35) (4.36)

exp(Sn /h) , 1 + exp((B + In )/h) (1 + exp((B + Sn+1 )/h)) exp(In /h) exp(In+1 /h) = , 1 + exp(Γ/h) exp(Rn+1 /h) = exp(Rn /h) + exp((Γ + In+1 )/h) .

exp(Sn+1 /h) =

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Y UTA TANAKA AND K EN - ICHI M ARUNO

Taking the logarithm of both sides, we obtain (4.37)

Sn+1 = Sn − h log(1 + exp((B + In )/h)) ,

(4.38)

In+1 = In + h log(1 + exp((B + Sn+1 )/h)) − h log(1 + exp(Γ/h)) ,

(4.39)

Rn+1 = Rn + exp((Γ + In+1 )/h)) .

Taking the ultradiscrete limit h → +0, we obtain the ultradiscrete SIR model (4.40)

Sn+1 = Sn − max(0, B + In ) ,

(4.41)

In+1 = In + max(0, B + Sn+1 ) − max(0, Γ) ,

(4.42)

Rn+1 = max(Rn , Γ + In+1 )

which is a special case of the ultradiscrete SIR model with time-delay [17]. Here we used the following formula [25]: ) ( B A (4.43) lim log exp + exp = max(A, B) , A, B > 0 . h→+0 h h Note that the above ultradiscrete SIR model is nonintegrable because the discrete SIR model (4.1), (4.2), (4.3) does not have the second conserved quantity, but we can think that this is very close to an integrable system. § 5. Conclusions We have presented structure-preserving discretizations of the SIR model, namely the dSIR1, dSIR2, gdSIR and nonautonomous gdSIR models, and their conserved quantities and exact solution to the initial value problem. For these discretizations of the SIR model, the conditions for integrability have been presented. By choosing the best value of the parameter p (for the gdSIR model) or pn (for the nonautonomous gdSIR model), the gdSIR model and the nonautonomous gdSIR mode conserve the conserved quantities of the continuous SIR model. This fact suggests that the gdSIR model and the nonautonomous gdSIR model are very powerful when used numerically. We have also investigated an ultradiscretizable discrete SIR model and its ultradiscretization, and we conclude that the ultradiscretizable SIR model and its ultradiscretization are not integrable. However, it has some good properties that might make it near-integrable. To the best of our knowledge, structure-preserving discretizations of the SIR model focusing on hodograph transformations have been previously unknown. Our results may shed new light on the study of structure-preserving discretization of mathematical models such as infectious diseases. It is very interesting to extend our discretization method to construct structurepreserving discretizations of other epidemic models such as the SEIR model.

I NTEGRABLE DISCRETIZATIONS OF THE SIR MODEL

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One of the authors (K.M.) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme ”Dispersive hydrodynamics: mathematics, simulation and experiments”, with applications in nonlinear waves where work on this paper was undertaken. This work was partially supported by JSPS KAKENHI Grant Numbers 18K03435, 22K03441, JST/CREST, and EPSRC grant no EP/R014604/1. This work was also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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