Painleve-type asymptotics for the defocusing Hirota equation in transition region

TL;DR

This paper derives Painleve-type asymptotics for the Hirota equation's solutions in a critical transition region using Riemann-Hilbert problem analysis and nonlinear steepest descent methods.

Contribution

It provides the first detailed asymptotic description of the Hirota equation in the transition region via Painleve II functions.

Findings

Asymptotics expressed in terms of Painleve II solutions

Application of nonlinear steepest descent to Hirota equation

Long-time behavior characterized in the critical transition region

Abstract

We consider the Cauchy problem for the classical Hirota equation on the line with decaying initial data. Based on the spectral analysis of the Lax pair of the Hirota equation, we first expressed the solution of the Cauchy problem in terms of the solution of a Riemann-Hilbert problem. Further we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region $|\frac{x}{t} - \frac{\alpha^2}{3\beta}|t^{2/3} \leq M$, $M$ is a positive constant. Our result shows that the long time asymptotics of the Hirota equation can be expressed in terms of the solution of Painlev\'{e} $\mathrm{II}$ equation. Keywords: Hirota equation, steepest descent method, Painlev\'{e} $\mathrm{II}$ equation, long-time asymptotics.