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arXiv · 1506.06545

Hamiltonian system for the elliptic form of Painlevé VI equation

Zhijie Chen·Ting-Jung Kuo·Chang-Shou Lin
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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 $\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on $\mathbb{CP}^{1}$ 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 $\mathbb{CP}^{1}$. This is one of the advantages of the elliptic form.

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Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin. 2015-06-22. Hamiltonian system for the elliptic form of Painlevé VI equation. https://arxiv.org/abs/1506.06545

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