Matrix Soliton 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 (Carillo and Schiebold, J Math Phys 50:073510, 2009) enjoyed by noncommutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations allow to construct explicit solution formulae (Carillo and Schiebold, J Math Phys 52:053507, 2011). 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.