Long time and Painlev\'{e}-type asymptotics for the defocusing Hirota equation with finite density initial data

TL;DR

This paper analyzes the long-time asymptotics of the defocusing Hirota equation with finite density initial data, revealing different asymptotic behaviors in oscillating and transition regions using Riemann-Hilbert techniques.

Contribution

It provides the first detailed asymptotic analysis of the defocusing Hirota equation with nonzero background, employing a $ar{ ext{D}}$-RH problem approach to derive Painlevé and oscillatory asymptotics.

Findings

In the oscillating region, the solution approaches a nonzero background with specific decay terms.

In the transition region, Painlevé II asymptotics describe the solution behavior.

The analysis distinguishes different asymptotic regimes based on the spatial variable.

Abstract

In this work, we consider the Cauchy problem for the defocusing Hirota equation with a nonzero background \begin{align} \begin{cases} iq_{t}+\alpha\left[q_{xx}-2\left(\left\vert q\right\vert^{2}-1\right)q\right]+i\beta\left(q_{xxx}-6\left\vert q\right\vert^{2}q_{x}\right)=0,\quad (x,t)\in \mathbb{R}\times(0,+\infty),\\ q(x,0)=q_{0}(x),\qquad \underset{x\rightarrow\pm\infty 1}{\lim} q_{0}(x)=\pm 1, \qquad q_{0}\mp 1\in H^{4,4}(\mathbb{R}). \end{cases} \nonumber \end{align} According to the Riemann-Hilbert problem representation of the Cauchy problem and the $\bar{\partial}$ generalization of the nonlinear steepest descent method, we find different long time asymptotics types for the defocusing Hirota equation in oscillating region and transition region, respectively. For the oscillating region $\xi<-8$, four phase points appear on the jump contour $\mathbb{R}$, which arrives at an asymptotic expansion,given by \begin{align} q(x,t)=-1+t^{-1/2}h+O(t^{-3/4}).\nonumber \end{align} It consists of three terms. The first term $-1$ is leading term representing a nonzero background, the second term $t^{-1/2}h$ originates from the continuous spectrum and the third term $O(t^{-3/4})$ is the error term due to pure $\bar{\partial}$-RH problem. For the transition region $\vert\xi+8\vert t^{2/3}<C$, three phase points raise on the jump contour $\mathbb{R}$. Painlev\'{e} asymptotics expansion is obtained \begin{align} q(x,t)=-1-(\frac{15}{4}t)^{-1/3}\varrho+O(t^{-1/2}),\nonumber \end{align} in which the leading term is a solution to the Painlev\'{e} II equation, the last term is a residual error being from pure $\bar{\partial}$-RH problem and parabolic cylinder model.