On orders of movable poles of the Schlesinger equation
type: Item
instance of: scholarly article
title: On orders of movable poles of the Schlesinger equation
zbMATH Open document ID: 0961.34077
DOI: 10.1023/A:1009569605675
published in: Journal of Dynamical and Control Systems
publication date: 2001-05-28T00:00:00Z
review text: The author treats movable singular points of the Schlesinger equation of isomonodromic deformations as the points of jumping of the splitting type of the vector bundle constructed from the initial Fuchsian equation. The Fuchsian system considered has the form \[ {dy\over dx}= \Biggl( \sum^n_{i= 1} {B^0_i\over x- a^0_i}\Biggr) y,\;\sum^n_{i=1} B^0_i= 0, \] on the Riemann sphere determined by the Schlesinger equation \[ dB_i(a)=- \sum^n_{j=1} {[B_i(a), B_j(a)]\over a_i- a_j} d(a_i- a_j). \] The main result cleverly constructed including many minor details is that the solutions to the Schlesinger equation of isomonodromic deformations for the case of irreducible monodromy have poles of order two at \({\mathcal M}\). This means that the solutions to the Schlesinger equation being multiplied by \(C^2(t)\) become holomorphic at \({\mathcal M}\), where the local ideal of the analytic subvariety \({\mathcal M}\) is determined by \(C(t)\). A wonderful example of a Schlesinger isomonodromic deformation is included which is closely connected to a Painlevé VI equation.
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 14B07
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 34M35
zbMATH DE Number: 1475292
zbMATH Keywords: isomonodromic deformation
zbMATH Keywords: Fuchsian system
zbMATH Keywords: vector bundle
zbMATH Keywords: logarithmic connection
zbMATH Keywords: Painlevé equation
zbMATH Keywords: Schlesinger equation
author: A. A. Bolibrukh
reviewed by: John Schmeelk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1475292