Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

Abstract

Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Bäcklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equation. Matrix equations can be viewed as a specialisation of operator equations in the finite dimensional case when operators admit a matrix representation. Bäcklund transformations allow to reveal structural properties [10] enjoyed by non-commutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations, allow to construct explicit solution formulae [11]. The latter are adapted to obtain solutions admitted by the 2 × 2 and 3 × 3 matrix mKdV equation. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit. A further key tool used to obtain the presented results is an ad hoc construction of computer algebra routines to implement non-commutative computations.

Bäcklund transformations: a tool to study Abelian and non-Abelian nonlinear evolution equations

“…Solutions admitted by the matrix equations are a subject of interest in the literature. The study presented, based on previous results [11,12] further developed in [18,19], take into account multisoliton solutions of the matrix KdV equation obtained by Goncharenko [31], via a generalisation of the Inverse Scattering Method. According to [12], and in particular Theorem 3 therein, generalises Goncharenko's multisoliton solutions which follow as special ones.…”

“…According to [12], and in particular Theorem 3 therein, generalises Goncharenko's multisoliton solutions which follow as special ones. Solutions of matrix mKdV equation are discussed and obtained in [18], where some 2 × 2 and some 3 × 3 examples are provided; in [19] the solution formula, which in [12] is obtained in the general operator case, is discussed referring to the case of a d × d, d ∈ N and further solutions are produced to give an idea of the results in this direction and currently under investigation [20]. Further matrix solutions are obtained in [23,31,41,60,61,35,59].…”

Construction of soliton solutions of the matrix modified Korteweg-de Vries equation

“…The present article is a sequel to [7], where a general approach to the solution theory of the matrix MKdV is outlined and certain solutions are explicitly constructed. Actually these solutions are part of a family for which a complete classification will be given in the forthcoming article [8].…”

“…Here the focus is on solutions beyond the setting of [7,8]. In a somewhat experimental spirit, we will examine ways to weaken the assumptions in [8], and initialize the study of some novel solution classes.…”

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An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in [7]. In fact, the solutions in [7] are part of a subclass studied in detail by the authors in a forthcoming publication. Here several solutions beyond this subclass are constructed and discussed with respect to qualitative properties.

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