Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 025, 27 pages      arXiv:1604.07847      https://doi.org/10.3842/SIGMA.2017.025
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation

Hayato Chiba
Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

Received October 11, 2016, in final form April 11, 2017; Published online April 15, 2017

Abstract
A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a systematic way to construct completely integrable Hamiltonian systems on $\mathfrak{g}$ expressed in Lax form $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ]$ in the sense of the isospectral deformation, where $X_\lambda , A_\lambda \in \mathfrak{g}$ depend rationally on the indeterminate $\lambda $ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ] + \partial A_\lambda /\partial \lambda $ in the sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed. This method gives a few new Painlevé systems of dimension four.

Key words: Painlevé equations; Lax equations; multi-Poisson structure.

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