Symmetries in the third Painlev\'e equation arising from the modified Pohlmeyer-Lund-Regge hierarchy

TL;DR

This paper introduces a modified hierarchy related to the Pohlmeyer-Lund-Regge system, revealing new symmetries of the third Painlevé equation and connecting tau-functions to Painlevé equations.

Contribution

It presents a modified AKNS hierarchy including the mPLR equation, a new Lax representation, and a comprehensive symmetry analysis of the third Painlevé equation.

Findings

New Lax representation for the third Painlevé equation

Complete symmetry description via similarity reduction

Relation established between tau-functions and Painlevé equations

Abstract

We propose a modification of the AKNS hierarchy that includes the "modified" Pohlmeyer-Lund-Regge (mPLR) equation. Similarity reductions of this hierarchy give the second, third, and fourth Painlev\'e equations. Especially, we present a new Lax representation and a complete description of the symmetry of the third Painlev\'e equation through the similarity reduction. We also show the relation between the tau-function of the mPLR hierarchy and Painlev\'e equations.