MultiPoisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation
Abstract
A multiPoisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form partial X_λ /partial t = [X_λ, A_λ ] in the sense of the isospectral deformation, where X_λ, A_λ \in 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 partial X_λ/partial t = [X_λ, A_λ] + partial A_λ/partial 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.
 Publication:

SIGMA
 Pub Date:
 April 2017
 DOI:
 10.3842/SIGMA.2017.025
 arXiv:
 arXiv:1604.07847
 Bibcode:
 2017SIGMA..13..025C
 Keywords:

 Painlevé equations; Lax equations; multiPoisson structure;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 SIGMA 13 (2017), 025, 27 pages