Hamiltonian system for the elliptic form of Painlevé VI equation
Journal
Journal des Mathematiques Pures et Appliquees
Journal Volume
106
Journal Issue
3
Pages
546-581
Date Issued
2016
Author(s)
Abstract
In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on CP1. In this paper, we extend this isomonodromy theory on CP1 to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodromy theory of the generalized Lamé equation greatly simplifies the computation of the monodromy group in CP1. This is one of the advantages of the elliptic form.
Type
journal article