On some Hamiltonian properties of the isomonodromic tau functions

Abstract

We discuss some new aspects of the theory of the Jimbo-Miwa-Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau-functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. Furthermore, we believe that this observation is yet another manifestation of L. D. Faddeev's emphasis of the key role which the Hamiltonian aspects play in the theory of integrable system. This article will appear in the WSPC memorial volume dedicated to Ludwig Faddeev.

arXiv commentCorrected typos and mistakes in section 3.7

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• Hamiltonian structure of rational isomonodromic deformation systems
• Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type
• Short distance asymptotics for a generalized two-point scaling function in the two-dimensional Ising model
• Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation
• Painlevé/CFT correspondence on a torus
• On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation