On matrix Painlev\'e II equations

TL;DR

This paper introduces three matrix versions of the Painlevé II equation derived via the Painlevé–Kovalevskaya test, and constructs their isomonodromic Lax pairs by interpreting them as reductions of integrable matrix evolution equations.

Contribution

It presents new matrix Painlevé II equations and provides their Lax pairs, expanding the understanding of matrix integrable systems.

Findings

Three matrix Painlevé II equations identified

Lax pairs constructed for each matrix equation

Equations derived as reductions of integrable matrix systems

Abstract

The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to construct isomonodromic Lax pairs for them.