# A novel noncommutative KdV-type equation, its recursion operator, and solitons

## TL;DR

This paper introduces a new noncommutative KdV-type equation called meta-mKdV, explores its integrability via recursion operators, and constructs explicit soliton solutions, extending the noncommutative integrable systems theory.

## Contribution

It presents the first formulation of the meta-mKdV equation, establishes its connection to noncommutative mKdV equations, and derives recursion operators and explicit solutions.

## Key findings

- Derived recursion operators for meta-mKdV and its mirror
- Constructed hierarchies of solutions for the equations
- Established links to noncommutative mKdV equations

## Abstract

A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived.