Triangular isomonodromic solutions to a Fuchsian system from superelliptic curves

TL;DR

This paper constructs explicit fundamental solutions for certain upper triangular Fuchsian systems derived from superelliptic curves, demonstrating their isomonodromic nature through monodromy analysis.

Contribution

It introduces a method to explicitly solve matrix Fuchsian systems with coefficients from the Schlesinger system on superelliptic curves, highlighting their isomonodromic properties.

Findings

Fundamental solutions are explicitly constructed for the systems.

Monodromy matrices confirm the solutions are isomonodromic.

Solutions involve contour integrals on superelliptic curves.

Abstract

We give fundamental solutions of arbitrarily sized matrix Fuchsian linear systems, in the case where the coefficients $B^{(i)}$ of the systems are matrix solutions of the Schlesinger system that are upper triangular, and whose eigenvalues follow an arithmetic progression of a rational difference. The values on the superdiagonals of the matrices $B^{(i)}$ are given by contour integrals of meromorphic differentials defined on Riemann surfaces obtained by compactification of superelliptic curves. We show that our fundamental solutions are isomonodromic by obtaining their monodromy matrices.