Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

TL;DR

This paper constructs a new example of non-Schlesinger isomonodromic deformations for resonant Fuchsian systems using middle convolution, and analyzes how these deformations are affected by this operation.

Contribution

It introduces a novel example of non-Schlesinger deformations for resonant Fuchsian systems and investigates the impact of middle convolution on these deformations.

Findings

Constructed a new non-Schlesinger deformation example for order 5 systems.

Showed that middle convolution does not preserve non-Schlesinger deformations in general.

Highlighted differences between resonant and non-resonant Fuchsian systems regarding deformation preservation.

Abstract

The paper is devoted to non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution.