# Lax Representations for Matrix Short Pulse Equations

## TL;DR

This paper develops Lax representations for various matrix generalizations of Short Pulse Equations, including four-component systems, their reductions, and associated bi-Hamiltonian structures, advancing the mathematical understanding of these integrable models.

## Contribution

It introduces new four-component Lax representations for matrix SPE generalizations and explores their reductions and bi-Hamiltonian structures, extending prior two-component models.

## Key findings

- Four-dimensional Lax representations for four-component equations obtained.
- Reductions to two-component and original equations demonstrated.
- Bi-Hamiltonian structure identified for a parameter family of the four-component SPE.

## Abstract

The Lax representation for different matrix generalizations of Short Pulse Equations (SPE) is considered. The four-dimensional Lax representations of four-component Matsuno, Feng and Dimakis-M\"{u}ller-Hoissen-Matsuno equations is obtained. The four-component Feng system is defined by generalization of the two-dimensional Lax representation to the four-component case. This system reduces to the original Feng equation or to the two-component Matsuno equation or to the Yao-Zang equation. The three component version of Feng equation is presented. The four-component version of Matsuno equation with its Lax representation is given . This equation reduces the new two-component Feng system. The two-component Dimakis-M\"{u}ller-Hoissen-Matsuno equations are generalized to the four parameter family of the four-component SPE. The bi-Hamiltonian structure of this generalization, for special values of parameters, is defined. This four-component SPE in special case reduces to the new two-component SPE.