# Cauchy matrix structures and solutions to the nonisospectral three-component mKdV equations

## TL;DR

This paper constructs and analyzes solutions for nonisospectral three-component mKdV equations using the Cauchy matrix approach, revealing how nonisospectral terms influence soliton and double-pole dynamics.

## Contribution

It introduces a novel application of the Cauchy matrix approach to nonisospectral matrix mKdV equations and derives explicit solutions and their behaviors.

## Key findings

- Explicit soliton solutions are obtained.
- Double-pole solutions are derived.
- Nonisospectral terms significantly affect solution dynamics.

## Abstract

Nonisospectral integrable systems can describe solitary waves in nonuniform media. In this paper, we apply the Cauchy matrix approach to construct three types of nonisospectral matrix modified Korteweg-de Vries (mKdV) eqautions and present their Cauchy matrix structures and solutions. Further, through complex reduction, we further obtain three nonisospectral three-component mKdV (NTCmKdV) equations, which can be regarded as novel members of the nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. In particular, the explicit solutions are given for the soliton solutions, and the double-pole solutions, respectively. The dynamical behaviors of these solutions are analyzed to reveal the influence of nonisospectral terms on the solution structure.