Matrix solitons solutions of the modified Korteweg-de Vries equation

TL;DR

This paper constructs explicit matrix soliton solutions for the modified Korteweg-de Vries equation using Baecklund transformations and operator methods, revealing structural properties of non-commutative integrable systems.

Contribution

It develops a method to generate matrix soliton solutions for the matrix mKdV equation, extending previous explicit solutions and demonstrating their solitonic behavior.

Findings

Explicit 2x2 and 3x3 matrix soliton solutions obtained.

Structural properties like recursion operators are identified.

Visualizations confirm solitonic behavior of solutions.

Abstract

Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equations. Matrix equation can be viewed as a specialisation of operator equations in the finite dimensional case when operators are finite dimensional and, hence, admit a matrix representation. Baecklund transformations allow to reveal structural properties [S. Carillo and C. Schiebold, J. Math. Phys. 50 (2009), 073510] enjoyed by non-commutative KdV- type equations, such as the existence of a recursion operator. Operator methods are briefly recalled aiming to show how they can be applied to construct soliton solutions. These methods, combined with Baecklund transformations, allow to obtain solutions of matrix soliton equations. Explicit solution formulae previously constructed [C. Schiebold, Glasgow Math. J. 51, 147-155 (2009)], [S. Carillo and C. Schiebold, J. Math. Phys. 52 (2011), 053507] are used to obtain 2 x 2 and 3 x 3 matrix mKdV solutions. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit

Figures

Equations

$$w_t=w_{xxx}+3w_x^2,u_t=u_{xxx}+6u{u}_x\text{and}{v}_t=v_{xxx}+6v^2{v}_x$$

$$W_t=W_{xxx}+3W_x^2,U_t=U_{xxx}+3\{U,U_x\},\text{and}{V}_t=V_{xxx}+3\{V^2,V_x\},$$

$$B(u(x,t),v(x,t)){|}_{t=0}=0\text{implies}B(u(x,t),v(x,t)){|}_{t=\tau }=0,\forall \tau >0.$$

$$B_1:U-W_x=0M:U-i{V}_x-V^2=0.$$

$$V_t=V_{xxx}+3\{V^2,V_x\}.$$

$$AB+BA=\sum _{j=1}^{\text{d}}{d}^{(j)}\otimes c^{(j)}$$

$$V=\frac{\rm i}{2}\ \biggm{(}\frac{\delta\bigm{(}I+\mbox{\rm i}(L+L^{(i,j)})\bigm{)}}{\delta\bigm{(}I+\mbox{\rm i}L\bigm{)}}-\frac{\delta\bigm{(}I-\mbox{\rm i}(L+L^{(i,j)})\bigm{)}}{\delta\bigm{(}I-\mbox{\rm i}L\bigm{)}}\biggm{)}_{i,j=1}^{\textsf{d}},$$

$$L^{(i,j)}=d^{(i)}\otimes\Big{(}\exp\big{(}Ax+A^{3}t\big{)}\,c^{(j)}\Big{)},\quad L=\exp\big{(}Ax+A^{3}t\big{)}B,$$

Full text

1 Introduction

So called soliton equations are widely investigated since the late 1960’s when the first exact solution of the Korteweg-de Vries equation was obtained, as reported, for instance, in the well known book by Calogero and Degasperis [2]. The name soliton equations is generally used to indicate nonlinear evolution equations which admit exact solutions in the Schwartz space of smooth *rapidly decreasing functions222It is generally assumed that $M$ is space of functions $u(x,t)$ which, $\forall$ fixed $t$, belong to the Schwartz space $S$ of rapidly decreasing functions on ${{\mbox{R\hskip-8.99994pt{}I \ }}}^{n}$, in the i.e. $S({{\mbox{R\hskip-8.99994pt{}I \ }}}^{n}):=\{f\in C^{\infty}({{\mbox{R\hskip-8.99994pt{}I \ }}}^{n})\!\!:\!\!|\!|f|\!|_{\alpha,\beta}<\infty,\forall\alpha,\beta\}$, where $|\!|f|\!|_{\alpha,\beta}:=sup_{x\in{{\mbox{R\hskip-6.29996pt{}I \ }}}^{n}}\left|x^{\alpha}D^{\beta}f(x)\right|$, $D^{\beta}:\!\!=\!\!\partial^{\beta}/{\partial x}^{\beta}$. *. The relevance of Bäcklund transformations in studying nonlinear evolution equations is well known both under the viewpoint of finding solutions to given initial boundary value problems, see [25] - [27] as well as in giving insight in the study of their structural properties, such as symmetry properties, admitted conserved quantities and Hamiltonian structure, see e.g. [6] and references therein. The important role played by Bäcklund transformations to investigate properties enjoyed by nonlinear evolution equations is based on the fact that most of the properties of interest are preserved under Bäcklund transformations [16], [17]. Thus, the construction of a net of Bäcklund transformation to connect nonlinear evolution equations allowed to prove new results both in the case of scalar equations [3]-[5], [18]-[19], as well as in the generalised case of operator equations [7]-[12] 333An overview on non-commutative equations is given in [22]. A comparison between the scalar (Abelian) and the operator (non-Abelian) cases referring to third order KdV-type equations, Bäcklund transformations connecting them and related properties is comprised in [9]. Notably, Bäcklund transformations indicate a way to construct solutions to nonlinear evolution equations [20] and also to nonlinear ordinary differential equations, [13] and [14]. Then, the operator approach [1], [15], [24], [28], [29], allows to construct solutions admitted by the whole hierarchies of nonlinear operator equations which are connected via Bäcklund transformations, see [3]-[11] and references therein. The special case under investigation concerns solutions of the $2\times 2$ and $3\times 3$ mKdV matrix equation. Matrix equations are studied in [23] and in [21], where, respectively, solutions admitted by Burgers and KdV equations are obtained. A motivation for investigations on matrix equations is connected to quantum mechanics.

The material is organised as follows. Section 2 briefly reminds the needed notions on Bäcklund transformation, recalling also the links among KdV-type equations. In the subsequent Section 3, a short overview is provided on the method to obtain matrix solutions in the finite dimensional case. Then, some solutions of the matrix mKdV equation are graphically represented and, finally, relevant remarks and research perspectives are in the closing Section 4.

2 Noncommutative potential KdV, KdV and modified KdV hierarchies

The well known potential Korteweg deVries (pKdV), Korteweg deVries (KdV) and modified Korteweg deVries (mKdV) equations, in turn

[TABLE]

are nonlinear evolution equations in the unknown real functions, respectively $w$, $u$ and $v$. All the equations (1) admit soliton solutions, see, among a wide literature, e.g. [2], namely solutions which represent a nonlinear waves which propagate preserving energy and shape. The aim of the present investigation is to consider a generalisation of equations (1), on introduction of operator valued equations according to the approach in [1], [15]. Thus, the non-commutative equations, counterpart, respectively, of (1), are given by:

[TABLE]

where the square and curly brackets denote, in turn, the commutator and the anti-commutator, that is $\forall T,S$, $[T,S]:=TS-ST$ and $\{T,S\}:=TS+ST$.

According to the definition, [16], [17], two different evolution equations, e.g. (1)1 and (1)2, are termed *connected via the Bäcklund transformation * $B{(u,v)}=0$ whenever given two solutions they admit, say, $u(x,t)$ and $v(x,t)$, if

[TABLE]

Well known examples of Bäcklund transformations are the introduction of a bona fide potential and the Miura transformation which, respectively, relate the pKdV to the KdV and the latter to the mKdV equation. The non-commutative extensions are

[TABLE]

Notably, [10], [31], the links via Bäcklund transformations, combined with the knowledge of the hereditary recursion operator of the KdV equation, allow to construct the recursion operator admitted by the pKdV as well as by mKdV equation and, hence, to extend the same links to the whole corresponding hierarchies (see [7] and references therein for details).

That is, given the KdV recursion operator $\Phi(U)$ then, the mKdV recursion operator and the pKdV recursion operator are obtained [10]. All the corresponding members in the three hierarchies of pKdV, KdV and mKdV equations follows to be linked via the transformations (4) Remarkably, these connections [10] allow, given a solution of the noncommutative pKdV equation, to construct the corresponding solutions of the noncommutative KdV and mKdV equations.

3 Matrix soliton solutions

This section aims to summarise the essential steps in the construction of solutions on application of the operator method devised in [1], [15], further developed in [29], [30], and extended to hierarchies in [11]. A very short outline on how this method can be adopted to construct solutions of the matrix mKdV equation is provided. Indeed, the study in [10], [7], [8], is devoted to study nonlinear evolution equations in which the unknown is an operator on a Banach space. The idea of the method can be sketched as follows:

• $\bullet$

Consider the operator method to obtain pKdV solutions.

• $\bullet$

Use the links, via Bäcklund transformations, between the pKdV, KdV and mKdV equations to construct solutions of the KdV and mKdV equations.

• $\bullet$

Implementation and visualisation of the explicit solutions via computer algebra routines.

The quite involved technical details can be found in [11]. The following first two subsections provide the schematic idea of the adopted method. In the third subsection some explicit soliton solutions of the $2\times 2$ and $3\times 3$-matrix mKdV equation are presented.

3.1 Operator method

The solution method, [1], [29], can be sketched via the following diagram

original

soliton equation

solution

$u=u(x,t;a)$, $a\in\mathbb{C}$

solution formula with

an operator-valued

parameter

$\hat{u}=\tau(U(x,t;A))$

operator-valued

soliton equation

solution

$U=U(x,t;A)$, $A\in{\cal L}(E)$

appropriate

translation

scalarization

[0.3cm]

where $a$ and $A$ are parameters which, in turn, represent a generally complex number and an operator.

3.2 Soliton solutions of mKdV matrix equation

The following theorem (cf. Theorem 11 b) in [11] where the matrix mKdV hierarchy is treated) allows to construct noncommutative soliton solutions of the mKdV equation

[TABLE]

More precisely, (5) is to be read as an equation in which the dependent variable $V=V(x,t)$ takes values in the ${\sf d}\times{\sf d}$-matrices. For the proof and the notation related to Banach spaces we refer to [11].

Theorem 3.1**.**

Let $E$ be a Banach space and $\cal A$ a quasi-Banach ideal equipped with a continuous determinant $\delta$. Let $A\in{\cal L}(E)$ with $\mbox{\rm spec}(A)\subseteq\{\lambda\in\mathbb{C}|\mbox{\rm Re}(\lambda)>0\}$. Moreover, assume that $B\in{\cal A}(E)$ satisfies the d-dimensionality condition

[TABLE]

with linearly independent $d^{(j)}\in E^{\prime}$ and $c^{(j)}\in E$, $j=1,\ldots,{\textsf{d}}$. Then the matrix-function

[TABLE]

where

[TABLE]

and $I=I_{E}$ denotes the identity operator on $E$, solves the matrix mKdV (5) with values in the ${\sf d}\times{\sf d}$-matrices on every product domain $(-\infty,c)\times G$ ($c\in\mathbb{R}$, $G$ a domain in $\mathbb{R}^{n-1}$) on which $\delta(I\pm\mbox{\rm i}L)\not=0$.

3.3 Some explicit solutions of the matrix modified KdV equation

The final step consists in the implementation, via computer algebra, of suitable routines amenable to visualise the matrix solutions of the mKdV equation. Indeed, multisoliton solutions admitted by the matrix KdV equation are given by Goncharenko in [21]: these solutions are obtained via a generalisation of the Inverse Scattering Method. As shown in [11], the solution class of the matrix KdV equation that corresponds to the class constructed in Theorem 3.1 comprises Goncharenko’s multisoliton solutions. In the present subsection, soliton solutions of the matrix modified KdV are visualised using Mathematica. Some of such explicit 2-soliton solutions of the matrix mKdV, in the present subsection, are visualised using Mathematica.

Without going into details, we briefly explain the choices in Theorem 3.1 to realize 2-soliton solutions of the ${\sf d}\times{\sf d}$-matrix mKdV. To reflectionless spectral data with discrete eigenvalues $k_{1},k_{2}$ and spectral matrices $B_{1},B_{2}$ of size ${\sf d}\times{\sf d}$, we associate A=\begin{pmatrix}k_{1}I_{d}&0\\ 0&k_{2}I_{d}\end{pmatrix},\qquad B=\Big{(}\frac{{\rm i}}{k_{i}+k_{j}}B_{j}\Big{)}_{i,j=1}^{2}, In particular, $E=\mathbb{C}^{2\sf d}$. In the plots below the case $B_{1}=B_{2}$ is considered.

The examples of solutions of the mKdV equation, graphically represented in the Fig.s 2 - 4, represent different $3\times 3$ matrix solutions: the elements of the spectral matrices are indicated in the caption. The pictures show the behaviour of two-soliton solutions admitted by the matrix mKdV equation.

Note that the crucial features of interaction between two different solitons, well known on the scalar case seems to characterise also matrix solutions. These pictures represents only an example of a study, currently in progress, aiming to construct additional solutions as well as their interpretation. In particular, if the spectral matrices are diagonal, then also the solution enjoys the same property, see Fig. 2. Conversely, Fig. 4, shows that, when off diagonal elements in the spectral matrix are different from zero, the situation changes. For example, given the initial datum $b_{1,3}=0$, in the spectral matrices, the corresponding element of the two-soliton solution, depicted in Fig. 4, is not zero. A detailed study on these solutions is currently under investigation. One of the main issues concerns the energy conservation and its partition among the matrix elements. The appropriate functional which represents the energy of the interacting solitons is expected to play a crucial role to understand the phenomenology under investigation.

4 Concluding remarks

The aim of the present study is to emphasise some of the properties of solutions admitted by matrix mKdV equation. Notably, depending on the spectral data, a variety of soliton solutions may be observed. The crucial feature seem to be the appearance, also in the matrix case, of localised solutions which can be termed solitons on the basis of their interaction properties. A much richer interaction phenomenology, with respect to the scalar case, can be observed when matrix solutions are investigated. Indeed, as soon as the spectral matrices have non-zero off-diagonal terms, the solution exhibits non-zero solutions in further matrix elements. In addition, a variety of different solutions can be observed. However, in all cases, a form of energy distribution seems to be observed: the most appropriate way to define a functional suitable to represent the energy related to a matrix soliton solution is, in the authors opinion, one of the interesting questions this work arises. The obtained results motivate further investigations to provide a better understanding of the interesting phenomenology already observed.

Acknowledgements

Under the financial support of G.N.F.M.-I.N.d.A.M., I.N.F.N. and Università di Roma La Sapienza, Rome, Italy. C.Schiebold acknowledges Dip. S.B.A.I., Università di Roma La Sapienza, for the kind hospitality.