# 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.

## Key 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