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On solutions of the Schlesinger Equations in Terms of $Θ$-Functions

A. V. Kitaev·D. A. Korotkin
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Abstract

In this paper we construct explicit solutions and calculate the corresponding $τ$-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 Painlevé 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 Painlevé equation obtained earlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions.

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A. V. Kitaev, D. A. Korotkin. 1998-10-08. On solutions of the Schlesinger Equations in Terms of $Θ$-Functions. https://arxiv.org/abs/math-ph/9810007

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