A note on the R. Fuchs's problem for the Painlev\'e equations

TL;DR

This paper investigates a class of integrable PDE systems related to Painlevé equations, demonstrating a variable change that simplifies the system and confirming Fuchs's conjecture for certain algebraic solutions.

Contribution

It introduces a variable transformation technique for integrable systems and verifies Fuchs's conjecture for specific algebraic solutions of Painlevé equations.

Findings

System reduction via variable change is possible under certain conditions.

Fuchs's conjecture holds for some algebraic solutions of Painlevé equations.

The method simplifies the analysis of integrable PDE systems related to Painlevé equations.

Abstract

In this article we consider a first-order completely integrable system of partial differential equations $\partial \Fi/partial x=A(x, t) \Fi, \partial \Fi/partial t=B(x, t) \Fi$ with $\Fi=(\fi_1, \fi_2)^{\tau}$ where $A(x, t)$ and $B(x, t)$ are 2 by 2 holomorphic matrices functions. Under some assumptions we find a variable change by which the system $\partial \Fi/\partial x=A(x, t) \Fi$ is reduced to an equation independent on the variable $t$. As an application we show that the R. Fuchs's conjecture for the Painlev\'e equations is true for some algebraic solutions.