The master Painlev\'e VI heat equation

TL;DR

This paper constructs a generalized heat equation with rational coefficients from the scalar Lax pair of the sixth Painlevé equation, removing dependence on the Painlevé variable, thus providing a new analytical framework.

Contribution

It introduces a direct construction of a generalized heat equation derived from the Painlevé VI Lax pair, independent of the Painlevé variable, advancing analytical methods in integrable systems.

Findings

Derived a rational coefficient heat equation from Painlevé VI Lax pair

Removed dependence on Painlevé variable in the heat equation

Provided a new analytical tool for studying Painlevé equations

Abstract

Given the second order scalar Lax pair of the sixth Painlev\'e equation, we build a generalized heat equation with rational coefficients which does not depend any more on the Painlev\'e variable. \'Etant donn\'e la paire de Lax scalaire de la sixi\`eme \'equation de Painlev\'e, nous donnons une construction directe de l'\'equation de la chaleur g\'en\'eralis\'ee \`a coefficients rationnels qui ne d\'epend plus de la variable de Painlev\'e.