Hamiltonian system for the elliptic form of Painlevé VI equation

Zhijie Chen; Ting Jung Kuo; Chang Shou Lin

doi:10.1016/j.matpur.2016.03.003

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 journal › Article › peer-review

14 Citations (Scopus)

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.

Original language	English
Pages (from-to)	546-581
Number of pages	36
Journal	Journal des Mathematiques Pures et Appliquees
Volume	106
Issue number	3
DOIs	https://doi.org/10.1016/j.matpur.2016.03.003
Publication status	Published - 2016 Sept 1
Externally published	Yes

Keywords

Hamiltonian system
Isomonodromy theory
Painlevé VI equation
The elliptic form

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1016/j.matpur.2016.03.003

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title = "Hamiltonian system for the elliptic form of Painlev{\'e} VI equation",

abstract = "In literature, it is known that any solution of Painlev{\'e} 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 Lam{\'e} equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlev{\'e} VI equation for generic parameters. For Painlev{\'e} VI equation with some special parameters, the isomonodromy theory of the generalized Lam{\'e} equation greatly simplifies the computation of the monodromy group in CP1. This is one of the advantages of the elliptic form.",

keywords = "Hamiltonian system, Isomonodromy theory, Painlev{\'e} VI equation, The elliptic form",

author = "Zhijie Chen and Kuo, {Ting Jung} and Lin, {Chang Shou}",

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T1 - Hamiltonian system for the elliptic form of Painlevé VI equation

AU - Chen, Zhijie

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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.

AB - 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.

KW - Hamiltonian system

KW - Isomonodromy theory

KW - Painlevé VI equation

KW - The elliptic form

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Hamiltonian system for the elliptic form of Painlevé VI equation

Abstract

Keywords

ASJC Scopus subject areas

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