The non-linear steepest descent approach to the singular asymptotics of the sinh-Gordon reduction of the Painlev\'e III equation

TL;DR

This paper develops a non-linear steepest descent method to analyze the asymptotic behavior of solutions to the sinh-Gordon reduction of Painlevé III, establishing connection formulas between local behaviors at zero and infinity.

Contribution

It introduces a novel non-linear steepest descent approach for the sinh-Gordon Painlevé III equation, linking monodromy data to asymptotic solutions.

Findings

Derived large and small x asymptotics for solutions

Established connection formulas between behaviors at zero and infinity

Unified approach via Riemann-Hilbert problem

Abstract

Motivated by the simplest case of tt*-Toda equations, we study the large and small $x$ asymptotics for $x>0$ of real solutions of the sinh-Godron Painlev\'e III($D_6$) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann-Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.