Expansions for solutions of the Schlesinger equation at a singular point

TL;DR

This paper investigates the local behavior of solutions to the Schlesinger equation near singular points, providing convergent expansions, and extends the results to the sixth Painlevé equation using isomonodromic methods.

Contribution

It introduces convergent local expansions for solutions of the Schlesinger equation at singular points and applies these findings to the sixth Painlevé equation.

Findings

Derived convergent expansions near singular points

Extended results to the sixth Painlevé equation

Used isomonodromic approach for analysis

Abstract

A local behavior of solutions of the Schlesinger equation is studied. We obtain expansions for this solutions, which converge in some neighborhood of a singular point. As a corollary the similar result for the sixth Painlev\'e equation was obtained. In our analysis, we use the isomonodromic approach to solve this problem.