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arXiv · 1812.09795

Triangular Schlesinger systems and superelliptic curves

Vladimir Dragović·Renat Gontsov·Vasilisa Shramchenko
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Abstract

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size $(p\times p)$ are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference $q$, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference $q$, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the $(2\times2)$-case, we obtain explicit sequences of rational solutions and one-parameter families of rational solutions of Painlev\'e VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.

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Vladimir Dragović, Renat Gontsov, Vasilisa Shramchenko. 2018-12-23. Triangular Schlesinger systems and superelliptic curves. https://arxiv.org/abs/1812.09795

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