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

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.

References in corpus (2)

• The Third, Fifth and Sixth Painlevé Equations on Weighted Projective Spaces
• Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two

Cited by in corpus (2)

• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Weights, Kovalevskaya exponents and the Painlevé property