Integration of the matrix KdV equation with self-consistent source

Chaos, Solitons & Fractals 49 (2013) 21–27

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Integration of the matrix KdV equation with self-consistent source N. Bondarenko a,⇑, G. Freiling b, G. Urazboev c a

Department of Mathematics of Saratov State University, Astrakhanskaya 83, 410012 Saratov, Russia Department of Mathematics of the University Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany c Department of Mathematics of the Urgench State University, 14 Kh.Alimdjan, 220100 Urgench, Uzbekistan b

a r t i c l e

i n f o

Article history: Received 18 April 2012 Accepted 18 February 2013 Available online 21 March 2013

a b s t r a c t In this work, it is shown that the solutions of the matrix KdV equation with self-consistent source can be found by the inverse scattering transform for the matrix Sturm–Liouville operator. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction We study the matrix Korteweg-de Vries (KdV) equation with self-consistent source. Our goal is to describe how this equation can be integrated by the inverse scattering transform. This method was first proposed in 1967 by Gardner, Green, Kruskal, and Miura (GGKM) [1] for solving the Cauchy problem for the classical KdV equation

ut ¼ 6uux þ uxxx ¼ 0: Their approach was based on the connection between the KdV equation and the spectral theory for the Sturm–Liouville operator on the line. Shortly thereafter, Lax [2] pointed out the general character of the inverse scattering method. A few years later, Zakharov and Shabat [3] managed to solve another important nonlinear evolution equation, the so-called nonlinear Schrödinger equation, using a nontrivial extension of the methods used in [1,2]. Thus, the way for constructing some other classes of equations solvable by these methods was opened. A detailed exposition of the relations between the inverse problems and nonlinear equations of mathematical physics is provided, for example, in the monographs [4–7]. In the attempts to construct a wider class of integrable systems, the important role has been played by the squares of eigenfunctions of Sturm–Liouville eigenvalue problems; ⇑ Corresponding author. Tel.: +7 78452515538. E-mail addresses: [email protected] (N. Bondarenko), [email protected] (G. Freiling), [email protected] (G. Urazboev). 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.02.010

this was revealed in [8]. Newell [9] showed that precisely the squares of eigenfunctions rather than the eigenfunctions themselves are essential in integrating by the inverse scattering method for the Sturm–Liouville equation. This fact was rigorously proved by Calogero and Degasperis [10]. The present work illustrates a similar fact for the matrix Sturm–Liouville equation. Since the late 1980s and the early 1990s, integrable hamiltonian ODEs as well as integrable symplectic maps were constructed by taking ‘‘restricted flows’’ or ‘‘Bargmann constrained flows’’ of integrable nonlinear evolution equations, either continuous and discrete. These finite dimensional systems have been readily recognized to be stationary flows of nonlinear evolution equations with self-consistent sources. As noted in [11], some generalizations of the Lax equations in the theory of integrable equations have been proposed by Melnikov [12]. Later they were also derived by Zakharov and Kuznetsov [13]. The general form of these equations is the following extension of the Lax form:

Lt ¼ ½L; A þ C; where L and A are operators dependent on time and acting on a fixed Hilbert space, and C is the sum of differential operators with coefficients dependent on the solutions of the spectral problem for the operator L. Usually these equations are called equations with self-consistent sources. Soliton equations with self-consistent sources have received much attention in the recent literature. Physically,

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N. Bondarenko et al. / Chaos, Solitons & Fractals 49 (2013) 21–27

the sources appear in solitary waves with non-constant velocity and lead to a variety of dynamics of physical models. For applications, these kinds of systems are usually used to describe interactions between different solitary waves and are relevant in some problems related with, among others, hydrodynamics, solid state physics or plasma physics [14–18]. Different techniques have been used to construct their solutions, such as inverse scattering [15,16,19,20], Darboux transformation [21–24] or Hirota bilinear methods [25–27]. The matrix KdV equation

U t  3UU x  3U x U þ U xxx ¼ 0; where U = U(x, t) is d  d matrix, was introduced by Lax [2]. In the work [28] Goncharenko considered multisoliton solutions of the matrix KdV equation. In the works [29,30] several types of nonlinear evolution equations that can be integrated by the inverse problem method for the matrix Sturm–Liouville operator were considered. We study here the following problem

8 N > < U  3UU  3U U þ U ¼ 2X @ u u ; t x x xxx n n @x n¼1 > : u00n þ U un ¼ kn un ; n ¼ 1; 2; . . . ; N;

ð1Þ

ð2Þ

and the normalizing conditions

Z

un ðx; tÞun ðx; tÞdx ¼ a2n ðtÞ  Imn ; n ¼ 1; 2; . . . ; N:

ð3Þ

Here U ¼ Uðx; tÞ ¼ ðujk ðx; tÞÞdj;k¼1 is a Hermitian d  d matrix (U = U⁄); N P 1 and mn, n = 1, 2, . . . , N, 1 6 mn 6 d, are arbitrary fixed integers; un(x,t), n = 1, 2, . . . , N are d  mn matrix functions. We use the following notation. Denote by L2 ðð1; 1Þ; Cd Þ the space of complex-valued vector functions of size d with components belonging to L2(1, 1). P Define the matrix norm kXk :¼ maxj dk¼1 jxjk j, where d X ¼ ðxjk Þj;k¼1 . The initial condition (2) is given by the Hermitian d  d matrix U0(x) having the following properties: R1 1. 1 ð1 þ jxjÞkU 0 ðxÞkdx < 1; 2. The operator generated by the differential expression

Lð0Þ :¼ D2 þ U 0 ðxÞ;

1

ð1 þ jxjÞkUðx; tÞkdx < 1:

1

2. The matrix Sturm–Liouville operator generated by the differential expression

LðtÞ :¼ D2 þ Uðx; tÞ;

D :¼

d ; dx

x 2 R;

in the domain L2 ðð1; 1Þ; Cd Þ for any t P 0 has exactly N eigenvalues k1(t), k2(t), . . . , kN(t) with multiplicities m1, m2, . . . , mN, respectively. 3. The functions U(x, t) and un(x, t), n = 1, 2, . . . , N, satisfy the relations (1)–(3). 4. For n = 1, 2, . . . , N, the columns of un(x, t) belong to L2 ðð1; 1Þ; Cd Þ. Therefore, if aj(t) – 0 for some t P 0, these columns are linearly independent eigenfunctions corresponding to the eigenvalue kn(t), otherwise un(x, t) = 0 for all x 2 (1, 1). We assume that the solution {U(x, t), u1(x, t), . . . , uN(x, t)} of the problem (1)–(3) exists in the sense described above. Our main purpose is to obtain a representation for this solution in the framework of the inverse scattering method for the operator L(t).

In this section we give basic information about the scattering theory for the matrix Sturm–Liouville equation on the real line (1 < x < 1), 2

Lw  w00 þ UðxÞw ¼ kw;

1

1

Z

2. Facts from scattering theory

under the initial condition

Ujt¼0 ¼ U 0 ðxÞ;

1. For all t P 0

x 2 R;

d

in the domain L2 ðð1; 1Þ; C Þ has exactly N eigenvalues k1(0), k2(0), . . . , kN(0) with multiplicities m1, m2, . . . , mN, respectively. The normalizing conditions (3) are given by the continuous scalar functions an(t); Imn are the unit mn  mn matrices. A solution of the problem (1)–(3) is a collection {U(x, t), u1(x, t), u2(x, t), . . . , uN(x, t)} of matrix functions, satisfying the following conditions.

k¼k ;

ð4Þ

ðwj ðx; kÞÞdj¼1

where w ¼ wðx; kÞ ¼ is a column vector (see, for example, [29,32,33,31]). In this section the potential U(x) is assumed to be a Hermitian d  d matrix such that

Z

1

ð1 þ jxjÞkUðxÞkdx < 1:

1

Lemma 1. Let the vector functions X(x, k) and Y(x, l) be  Y, respectively. solutions of the equations LX = kX and LY ¼ l Then the following relation holds

ðl  kÞY  X ¼

d WfY  ; Xg; dx

where WfY  ; Xg ¼ Y  X 0  Y 0 X. For all k in the upper half-plane Im k P 0, Eq. (4) has unique solutions F(x, k) and G(x, k) with the following asymptotics

Fðx; kÞ ¼ eikx ðI þ oð1ÞÞ; ikx

Gðx; kÞ ¼ e

ðI þ oð1ÞÞ;

x ! 1; x ! 1:

ð5Þ

These solutions are called the Jost solutions for x ? 1 and x ? 1, respectively. For real k the functions F(x, k) and G(x, k) are also solutions of Eq. (4), and therefore for Im k = 0, k – 0 we have the representations

Gðx; kÞ ¼ Fðx; kÞAðkÞ þ Fðx; kÞBðkÞ; Fðx; kÞ ¼ Gðx; kÞCðkÞ þ Gðx; kÞDðkÞ;

ð6Þ

where A(k), B(k), C(k), D(k) are some d  d matrices. One can easily check that

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N. Bondarenko et al. / Chaos, Solitons & Fractals 49 (2013) 21–27

 Gðx; kÞg;  2ikAðkÞ ¼ WfF  ðx; kÞ;   Fðx; kÞg: 2ikCðkÞ ¼ WfG ðx; kÞ;

Then the first equation in (1) is equivalent to the operator equation

The matrix function A(k) can be analytically continued to the half-plane Im k > 0, and the equation detA(k) = 0 has there a finite number of zeros kj = ivj, j = 1, 2, . . . , N, with kj ¼ v2j , being eigenvalues of L. The matrix function (A(k))1 has simple poles in the points kj, j = 1, 2, . . . , N. Let N j ¼ Resk¼kj ðAðkÞÞ1 ; j ¼ 1; 2; . . . ; N. Then there are matrices Rj such that

Gðx; kj ÞNj ¼ iFðx; kj ÞRj ;

j ¼ 1; 2; . . . ; N:

ð7Þ

Lt ¼ ½B; L þ 2

N X  @  un un ; @x n¼1

½B; L :¼ BL  LB;

in the sense that both sides of Eq. (13) turn out to be operators of multiplication by a matrix function. Lemma 2. Let F0(x, k, t) be a d  d matrix solution of the equation 2

The ranks of the matrices Nj and Rj are equal to the multiplicity of the corresponding eigenvalue kj. The matrix

RðkÞ ¼ BðkÞðAðkÞÞ1 ;

Im k ¼ 0;

ð8Þ

is called the reflection matrix. The set {R(k), v1, v2, . . . , vN, R1, R2, . . . , RN} represents the scattering data associated with Eq. (4). Consider the following. Inverse scattering problem. Given the scattering data {R(k), v1, v2, . . . , vN, R1, R2, . . . , RN}, find the potential U(x) of Eq. (4). This problem was studied by Wadati and Kamijo [29] for the case of the Hermitian potential (U = U⁄) and by Olmedilla [33] for the non-self-adjoint case. Here we provide their results for convenience of the reader. The Jost solution can be represented in the following form:

Fðx; kÞ ¼ eikx I þ

Z

Kðx; yÞeiky dy:

ð9Þ

Here K(x, y) is a kernel independent of k and related to U(x) by the formula

ð10Þ

For each fixed x, the kernel K(x, y) of the integral operator in (9) satisfies the Gel’fand–Levitan–Marchenko equation

Kðx; yÞ þ Hðx þ yÞ þ

Z

1

Kðx; zÞHðy þ zÞdz ¼ 0;

and let Fn(x, k, t) be any mn  d matrices, n = 1, 2, . . . , N, satisfying the following equations

@F n ¼ un F 0 ; @t

n ¼ 1; 2; . . . ; N

ð15Þ

Then the matrix function

H0 :¼

N X @F 0 un F n  BF 0  @t n¼1

ð16Þ

is also a solution of Eq. (14). Proof. Define mn  d matrices

Hn :¼ un

@F 0 @ un 2  F 0 þ ðk  kn ÞF n ; n ¼ 1; 2; .. .; N: @x @x

ð17Þ

According to Lemma 1

  @Hn @ @F n 2 ¼ W un ; F 0 þ ðk  kn Þ @x @x @x   2 2 ¼ kn  k un F 0 þ ðk  kn Þun F 0 ¼ 0;

n

ð11Þ where N X RðkÞe dk  i Rj eikj x ikx

1

Therefore, Hn(x, k, t) does not depend on x. Calculating the limit of the function Hn(x, k, t) as x ? 1 (or x ? 1), we get

Hn  0;

x 6 y;

x

1

ð14Þ

¼ 1; 2; . . . ; N:

d 1 Kðx; xÞ ¼  UðxÞ: dx 2

1 HðxÞ ¼ 2p

LF 0 ¼ k F 0 ;

1

x

Z

ð13Þ

n ¼ 1; 2; . . . ; N:

Now we calculate LH0. Using (16), we obtain

LH0 ¼ LF 0  LBF 0  ð12Þ

j¼1

The matrix function H(x) can be determined via (12) by the scattering data. Solving the Gel’fand–Levitan–Marchenko Eq. (11), find K(x, y) and then obtain U(x) via (10).

ð18Þ

N X Lðun F n Þ:

ð19Þ

n¼1

Differentiating the equality L F0 = k2F0 with respect to t, we get

Lt F 0 þ L

@F 0 2 @F 0 ¼k : @t @t

Using (13), we derive 3. Evolution of the scattering data In this section we develop and extend methods of Mel’nikov‘s works [14–16]. This allows us to obtain the evolution of the scattering data and to provide the algorithm for solution of the problem (1)–(3). We set

B :¼ 4D3 þ 6UD þ 3U x :

L

@F 0 2 @F 0 ¼k  Lt F 0 @t @t 2

¼k

N X  @F 0 @  @F un un F 0 k2 0  k2 BF 0  ½B; LF 0  2 @x @t @t n¼1

þ LBF 0  2

N X  @  un un F 0 : @x n¼1

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N. Bondarenko et al. / Chaos, Solitons & Fractals 49 (2013) 21–27

Substituting this relation into (19), we have

Proof. Put

N  X @u @ un 2 2 LH0  k H0 ¼ 2 n un F 0  2un F 0 þ k un F n  Lðun F n Þ : @x @x n¼1

Hðx; k; tÞ : ¼ H0 ðx; k; tÞ  Hþ0 ðx; k; tÞAðk; tÞ

ð20Þ

Further, we use the explicit expression for L to calculate L(unFn):

@2 ðu F n Þ þ Uðun F n Þ @x2 n @ un @F n @2Fn ¼ ðLun ÞF n  2  un @x @x @x2 @ un    @ ¼ kn un F 0  2 un F 0  un ðun F 0 Þ: @x @x

2

N X

un un

n¼1

¼

N X

@F 0 @ un 2  F 0 þ ðk  kn ÞF n @x @x



 þF þn ðx; k; tÞAðk; tÞ þ F þn ðx; k; tÞBðk; tÞ : Using the relations

@Gðx; k; tÞ @t @Fðx; k; tÞ @Aðk; tÞ Aðk; tÞ þ Fðx; k; tÞ ¼ @t @t @Fðx; k; tÞ @Bðk; tÞ Bðk; tÞ þ Fðx; k; tÞ ; þ @t @t

Gðx; k; tÞ ¼ Fðx; k; tÞAðk; tÞ þ Fðx; k; tÞBðk; tÞ;

un Hn :

n¼1

Taking Eq. (18) into account, we arrive at (L  k2) H0 = 0. h Corollary 1. Let G(x, k, t) and F(x, k, t) be the Jost solutions of Eq. (14) for x ? 1 and x ? 1, respectively. Then the matrix functions

F n ðx; k; tÞ

@Gðx; k; tÞ @Fðx; k; tÞ  BGðx; k; tÞ  Aðk; tÞ @t @t @Fðx; k; tÞ þ BFðx; k; tÞAðk; tÞ  Bðk; tÞ @t N X   un F n ðx; k; tÞ þ BFðx; k; tÞBðk; tÞ þ n¼1

Substituting this expression into (20) and using (17), we obtain

LH0  k H0 ¼

ð25Þ

Substituting (22) into (25), we get

Hðx; k; tÞ ¼

Lðun F n Þ ¼ 



 Hþ0 ðx; k; tÞBðk; tÞ:

we derive

Hðx; k; tÞ ¼ Fðx; k; tÞ þ

N X

@Aðk; tÞ @Bðk; tÞ þ Fðx; k; tÞ @t @t



un F n ðx; k; tÞ þ F þn ðx; k; tÞAðk; tÞ

n¼1

Z

 þF þn ðx; k; tÞBðk; tÞ :

x

u

:¼ 1

F þn ðx; k; tÞ :¼ 

Z

 n ðs; tÞGðs; k; tÞds;

ð21Þ

1

un ðs; tÞFðs; k; tÞds

x

According to (6) and (21)

F n ðx; k; tÞ þ F þn ðx; k; tÞAðk; tÞ þ F þn ðx; k; tÞBðk; tÞ Z 1 un ðs; tÞGðs; k; tÞds; ¼

satisfy (15), therefore the matrices

1

Hþ0 ðx; k; tÞ

N X @F :¼ un F þn ;  BF  @t n¼1

H0 ðx; k; tÞ

N X @G :¼ un F n :  BG  @t n¼1

By virtue of Lemma 1

ð22Þ

un ðs; tÞGðs; k; tÞ ¼ Z

1

1

are solutions of (14).

¼ Remark 1. Since the Jost solutions have the asymptotics (5), we obtain from (22) 3

H0 ðx; k; tÞ ¼ 4ik eikx ðI þ oð1ÞÞ; Hþ0 ðx; k; tÞ

By the uniqueness of the Jost solutions, we conclude that 3

3

@Bðk; tÞ 3 ¼ 8ik Bðk; tÞ: @t

k  kn



un ðs; tÞ

 @Gðs; k; tÞ @ un ðs; tÞ  Gðs; k; tÞ j1 1 : @s @s

Using (5), (6) and the fact that the entries of un(s, t) belong to L2(1, 1), for fixed k 2 R n f0g and t P 0, one can easily

Z

ð23Þ

ð24Þ

@Gðs;k;tÞ @s

are

1

1

Lemma 3. For all k 2 R n f0g, the following relations hold

@Aðk; tÞ ¼ 0; @t

1 2

check that un ðs; tÞ; n@s ! 0 and Gðs; k; tÞ; bounded as s ? ±1. Therefore

¼ 4ik e ðI þ oð1ÞÞ; x ! 1:

Hþ0 ðx; k; tÞ ¼ 4ik Fðx; k; tÞ:

un ðs; tÞGðs; k; tÞ ds

@ u ðs;tÞ

x ! 1;

3 ikx

H0 ðx; k; tÞ ¼ 4ik Gðx; k; tÞ;

 @   W un ðs; tÞ; Gðs; k; tÞ ; @s k  kn 1

2

un ðs; tÞGðs; k; tÞds ¼ 0:

Finally

Hðx; k; tÞ ¼ Fðx; k; tÞ

@Aðk; tÞ @Bðk; tÞ þ Fðx; k; tÞ : @t @t

ð26Þ

Moreover, substituting (23) into (25) and using (6), we obtain

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N. Bondarenko et al. / Chaos, Solitons & Fractals 49 (2013) 21–27

Z

3

H ¼ 4ik ½Fðx; k; tÞAðk; tÞ þ Fðx; k; tÞBðk; tÞ 3

1

3

 4ik Fðx; k; tÞAðk; tÞ þ 4ik Fðx; k; tÞBðk; tÞ 3

¼ 8ik Fðx; k; tÞBðk; tÞ:

ð27Þ

Comparing (26) and (27), we arrive at (24).

Corollary 3. Since A(k, t) does not depend on t, its determinant detA(k, t), its zeros kn and the eigenvalues 2 kn ¼ kn ; n ¼ 1; 2; . . . ; N, also do not depend on t. Lemma 4. The matrix functions Rj(t), j = 1, 2, . . . , N, satisfy the equations

j ¼ 1; 2; . . . ; N;

ð28Þ

where a2j ðtÞ are the given functions from the normalizing conditions (3).

j ¼ 1; 2;. .. ;N;

ð29Þ

Substituting (22), we get





un iF þn ðx; kj ; tÞRj ðtÞ  F n ðx; kj ; tÞNj :

ð30Þ

ð31Þ

Using (21) and (6), we obtain

1

Substituting (31) and (32) into (30), we get

dRj ðtÞ dt Z N X  i un ðx; tÞ

Hj ¼ iFðx; kj ; tÞ

1

1

un ðs; tÞFðs; kj ; tÞ ds Rj ðtÞ:

According to Lemma 1, for n – j we have

therefore

where ej(t) are mj  d matrices. Using this fact and the relations (3), (33), we get

Z



1

uj ðx; tÞ

1

uj ðs; tÞFðs; kj ; tÞds Rj ðtÞ

Z

1

 1 @   W un ðs; tÞ; Fðs; kj ; tÞ ; kj  kn @s



uj ðs; tÞuj ðs; tÞds ej ðtÞ

¼ Fðx; kj ; tÞcj ðtÞa2j ðtÞImj ej ðtÞ ¼ a2j ðtÞFðx; kj ; tÞcj ðtÞej ðtÞ ¼ a2j ðtÞFðx; kj ; tÞRj ðtÞ:

j ð34Þ

According to (23)

H0 ðx; kj ; tÞ ¼ 4v3j Gðx; kj ; tÞ; Hþ0 ðx; kj ; tÞ ¼ 4v3j Fðx; kj ; tÞ; and from (29) we obtain

Hj ¼ 8iv3j Fðx; kj ; tÞRj :

ð35Þ

Comparing (34) and (35), we arrive at (28). h

þ

iF n ðx; kj ; tÞRj ðtÞ  F n ðx; kj ; tÞNj Z 1 ¼ i un ðs; tÞFðs; kj ; tÞ ds Rj ðtÞ:

un ðs; tÞFðs; kj ; tÞ ¼

Rj ðtÞ ¼ cj ðtÞej ðtÞ;

dRj ðtÞ 2  iaj Fðx; kj ; tÞRj ðtÞ; dt ¼ 1; 2; . . . ; N:

It follows from Lemma 3 that Nj does not depend on t, so differentiating (7), we get

@Gðx; kj ; tÞ @Fðx; kj ; tÞ dRj ðtÞ Nj ¼ i Rj ðtÞ þ iFðx; kj ; tÞ : @t @t dt

ð33Þ

where cj(t) are some d  mj matrices, j = 1, 2, . . . , N. Suppose that aj(t) – 0 (the case aj(t) = 0 is trivial). Then the columns of cj(t) are linearly independent, so it has rank mj. On the other hand, the columns of the matrix F(x, kj, t) Rj(t) are also eigenfunctions of L. Therefore the columns of Rj(t) can be represented as linear combinations of the columns of cj(t):

Hj ¼ iFðx; kj ; tÞ

n¼1

n¼1

The matrix of eigenfunctions can be represented in the form

Therefore

@Gðx; kj ; tÞ @Fðx; kj ; tÞ Nj  BGðx; kj ; tÞNj  i Rj ðtÞ @t @t þ iBFðx; kj ; tÞRj ðtÞ N X

1

1

þ Hj :¼ H0 ðx; kj ;tÞNj  iH0 ðx;kj ;tÞRj ðtÞ;

þ

dRj ðtÞ dt Z 1 uj ðs; tÞFðs; kj ; tÞ ds Rj ðtÞ:  iuj ðx; tÞ

¼ uj ðx; tÞ

Proof. Denote

Hj ¼

Hence

uj ðx; tÞ ¼ Fðx; kj ; tÞcj ðtÞ; j ¼ 1; 2; . . . ; N;

Im k ¼ 0:

 dRj  3 ¼ 8vj þ a2j Rj ; dt

un ðs; tÞFðs; kj ; tÞds ¼ 0; n – j:

Hj ¼ iFðx; kj ; tÞ

h

Corollary 2. Taking (8) into account, one can easily derive that

@Rðk; tÞ 3 ¼ 8ik Rðk; tÞ; @t

1

ð32Þ

Thus we have proved the following. Theorem 1. If U(x, t), un(x, t), n = 1, 2, . . . , N form a solution of the problem (1)–(3), then the scattering data for the operator

LðtÞ ¼ D2 þ Uðx; tÞ acting on L2 ðð1; 1Þ; Cd Þ satisfy the relations

@Rðk; tÞ 3 ¼ 8ik Rðk; tÞ; Im k ¼ 0; @t dvj ¼ 0; dt  dRj  3 ¼ 8vj þ a2j ðtÞ Rj ; j ¼ 1; 2; . . . ; N: dt

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N. Bondarenko et al. / Chaos, Solitons & Fractals 49 (2013) 21–27

The obtained relations completely specify the evolution of the scattering data for L(t) and this allows using the inverse scattering method to find solutions of the problem (1)–(3). In order to solve this problem, one can use the following. Algorithm 1. Suppose that U0(x) and an(t), n = 1, 2, . . . , N, are given. 1. Solving the direct scattering problem for the initial matrix U0(x), obtain the scattering data

fRðk; 0Þ; v1 ð0Þ; v2 ð0Þ; . . . ; vN ð0Þ; R1 ð0Þ; R2 ð0Þ; . . . ; RN ð0Þg of the operator

Lð0Þ ¼ D2 þ U 0 ðxÞ; acting on L2 ðð1; 1Þ; Cd Þ. 2. Using the result of Theorem 1, find the scattering data

fRðk; tÞ; v1 ðtÞ; v2 ðtÞ; . . . ; vN ðtÞ; R1 ðtÞ; R2 ðtÞ; . . . ; RN ðtÞg for t > 0. 3. Using the method based on the Gel’fand–Levitan– Marchenko equation [29,33], solve the inverse scattering problem, i.e. from the scattering data

fRðk; tÞ; v1 ðtÞ; v2 ðtÞ; . . . ; vN ðtÞ; R1 ðtÞ; R2 ðtÞ; . . . ; RN ðtÞg determine U(x, t). 4. Find the Jost solutions of the operator L(t) with the potential U(x, t), and then using (33), construct the matrices un(t) corresponding to the eigenvalues kn ðtÞ ¼ v2n ðtÞ; n ¼ 1; 2; . . . ; N; satisfying the normalizing conditions (3).

Remark 2. It is worth while mentioning that in step 3 of the preceding algorithm one can alternatively use the method of spectral mappings (see [34,35]) for solving the inverse scattering problem. Remark 3. Note that the system (1), as well as most of the integrable models in [4], have the Hamiltonian structure:

@U @ ¼ rU H; @t @x

rum H ¼ 0;

rum H ¼ 0;

m ¼ 1; 2; . . . ; N;

where H¼

" # N

X 1 @ un @ un dx; Tr U 3 þ U 2x þ 2 U un un þ  kn un un 2 @x @x 1 n¼1

Z

1

the matrix rUH is defined as

ðrU Hjk Þdj;k¼1 ¼



dH dukj

d ; j;k¼1

and

ruk Hst ¼

dH ; duk;ts

ruk Hts ¼

t ¼ 1; 2; . . . ; N;

dH ; duk;st

s ¼ 1; 2; . . . ; mk ;

k ¼ 1; 2; . . . ; N:

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