Triangular Schlesinger systems and superelliptic curves
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 oneparameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 DOI:
 10.48550/arXiv.1812.09795
 arXiv:
 arXiv:1812.09795
 Bibcode:
 2018arXiv181209795D
 Keywords:

 Mathematical Physics
 EPrint:
 41 pages, 3 figures