Integral solutions to Schlesinger equations
type: Item
instance of: scholarly article
title: Integral solutions to Schlesinger equations
zbMATH Open document ID: 1339.34097
author: V. P. Leksin
published in: Journal of Mathematical Sciences (New York)
publication date: 2016-01-21T00:00:00Z
review text: The paper discusses the parameterized linear system of differential equations of Fuchsian class \[ \frac{dy}{dz}=\left(\sum_{i=1}^{n}\frac{{{B}_{i}}(a)}{z-{{a}_{i}}}\right)y,\tag{1} \] where \({{a}_{1}},\dots ,{{a}_{n}}\) are parameters, \({{a}_{i}}\neq {{a}_{j}}(i\neq j),\mathbf a=({{a}_{1}},\dots ,{{a}_{n}}),\mathbf B=({{B}_{1}}(a),\dots ,{{B}_{n}}(a))\) is a tuple of \(p\times p\) matrix-valued functions. One needs to find such a tuple of functions \(\mathbf B\) for which the monodromy (Galois) group of system (1) will not depend on the parameters. It is well known if a tuple of functions \(\mathbf B\) satisfies the system of Schlesinger differential equations \[ d{{B}_{i}}(a)=-\sum_{j=1,i\neq j}^{n}{\frac{[{{B}_{i}}(a),{{B}_{j}}(a)]}{{{a}_{i}}-{{a}_{j}}}d({{a}_{i}}-{{a}_{j}})},\,i=1,\dots ,n,\tag{2} \] then \(\mathbf B\) is a solution of the above problem. Unfortunately, if \(n> 3\) and \(p> 1\), it is extremely difficult to find a solution of system (2). The author shows that if all matrices of \(\mathbf B\) have eigenvalues which do not dependent on the parameters and if for each eigenvalue the inequality \(\text{Re}\beta >-1/n(p-1)\) holds, then the system (1) reduces to a triangular form, and the system (2) is reduced to a Pfaffian system. Moreover, for \(p=2,3\) the solution of the problem can be obtained in closed form.
Mathematics Subject Classification ID: 34M56
Mathematics Subject Classification ID: 34M50
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 6534011
zbMATH Keywords: isomonodromic deformations
zbMATH Keywords: Fuchsian system
zbMATH Keywords: Schlesinger equations
zbMATH Keywords: parameterized Picard-Vessiot theory
reviewed by: Mykola Grygorenko
MaRDI profile type: Publication
full work available at URL: s10958-015-2440-3
OpenAlex ID: W808114507
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DOI: 10.1007/S10958-015-2440-3
description: scientific article; zbMATH DE number 6534011