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

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.