Hamiltonian system for the elliptic form of Painlevé VI equation

Zhijie Chen, Ting Jung Kuo, Chang Shou Lin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)546-581
Number of pages36
JournalJournal des Mathematiques Pures et Appliquees
Issue number3
Publication statusPublished - 2016 Sept 1
Externally publishedYes


  • Hamiltonian system
  • Isomonodromy theory
  • Painlevé VI equation
  • The elliptic form

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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