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

Abstract

A multi-Poisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form ∂Xλ/∂t=[Xλ,Aλ] in the sense of the isospectral deformation, where Xλ,Aλ∈g depend rationally on the indeterminate λ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation ∂Xλ/∂t=[Xλ,Aλ]+∂Aλ/∂λ 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.