### 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 CP^{1}. In this paper, we extend this isomonodromy theory on CP^{1} 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 CP^{1}. 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 | |

Publication status | Published - 2016 Sep 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal des Mathematiques Pures et Appliquees*,

*106*(3), 546-581. https://doi.org/10.1016/j.matpur.2016.03.003

**Hamiltonian system for the elliptic form of Painlevé VI equation.** / Chen, Zhijie; Kuo, Ting Jung; Lin, Chang Shou.

Research output: Contribution to journal › Article

*Journal des Mathematiques Pures et Appliquees*, vol. 106, no. 3, pp. 546-581. https://doi.org/10.1016/j.matpur.2016.03.003

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=84964370096&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964370096&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2016.03.003

DO - 10.1016/j.matpur.2016.03.003

M3 - Article

VL - 106

SP - 546

EP - 581

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 3

ER -