On solutions of the Schlesinger Equations in Terms of $\Theta$-Functions

TL;DR

This paper explicitly constructs solutions and calculates the tau-function for Schlesinger equations related to isomonodromy deformations, providing a new theta-function representation for the sixth Painlevé equation with specific parameters.

Contribution

It introduces a novel explicit solution and tau-function calculation for Schlesinger equations, connecting them to elliptic theta-functions for the sixth Painlevé equation.

Findings

New theta-function representation of Painlevé VI solutions

Explicit solutions for Schlesinger equations with four poles

Connection to previously known solutions by Okamoto and Hitchin

Abstract

In this paper we construct explicit solutions and calculate the corresponding $\tau$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose coefficients are rational functions with poles of the first order; in particular, in the case when the coefficients have four poles of the first order and the corresponding Schlesinger system reduces to the sixth Painlev\'e equation with the parameters $1/8, -1/8, 1/8, 3/8$, our construction leads to a new representation of the general solution to this Painlev\'e equation obtained earlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions.