Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions

Abstract

The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold ["Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods," J. Math. Phys. 50, 073510 (2009)]. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies. (C) 2011 American Institute of Physics. [doi:10.1063/1.3576185

Structural properties of the noncommutative KdV recursion operator

“…Proof of Theorem 15: Since the operator KdV is hereditary by Theorem 1, the proof of the commutativity of the vector fields in the KdV hierarchy reduces to verify (31) for the very first field K(u) = K 1 (u) = u x in (3). To this end, we proceed in two steps: First, we show…”

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

“…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.…”

“…The quite involved technical details can be found in [11]. The following first two subsections provide the schematic idea of the adopted method.…”

“…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 × 2 and 3 × 3 mKdV matrix equation.…”

“…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.…”