Duke Mathematical Journal

Monodromy dependence and connection formulae for isomonodromic tau functions

A. R. Its, O. Lisovyy, and A. Prokhorov

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We discuss an extension of the Jimbo–Miwa–Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for the generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painlevé II tau function.

Article information

Duke Math. J., Volume 167, Number 7 (2018), 1347-1432.

Received: 27 November 2016
Revised: 17 August 2017
First available in Project Euclid: 9 March 2018

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Zentralblatt MATH identifier

Primary: 33E17: Painlevé-type functions
Secondary: 34E05: Asymptotic expansions

connection problem tau function isomonodromy Painlevé equations Riemann–Hilbert method


Its, A. R.; Lisovyy, O.; Prokhorov, A. Monodromy dependence and connection formulae for isomonodromic tau functions. Duke Math. J. 167 (2018), no. 7, 1347--1432. doi:10.1215/00127094-2017-0055. https://projecteuclid.org/euclid.dmj/1520586158

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