Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 023, 14 pages      arXiv:1503.03959      https://doi.org/10.3842/SIGMA.2015.023

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

Yulia Bibilo a and Galina Filipuk b
a) Department of Theory of Information Transmission and Control, Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny per. 19, Moscow, 127994, Russia
b) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

Received November 20, 2014, in final form March 04, 2015; Published online March 13, 2015

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.

Key words: Middle convolution; isomonodromic deformation; non-Schlesinger isomonodromic deformation.

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