On some Hamiltonian properties of the isomonodromic tau functions

TL;DR

This paper reveals that the logarithmic derivatives of Painlevé tau functions align with classical action differentials, simplifying asymptotic evaluations and highlighting the Hamiltonian structure's importance in integrable systems.

Contribution

It demonstrates that, up to total differentials, the tau function derivatives match classical action differentials, aiding asymptotic analysis of Painlevé equations.

Findings

Logarithmic derivatives of tau functions coincide with classical action differentials.

Simplification of constant factor evaluation in tau-function asymptotics.

Emphasizes the Hamiltonian role in integrable systems theory.

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 Painlev\'e equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlev\'e 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 Painlev\'e 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.