The Painlev\'{e}-type asymptotics of defocusing complex mKdV equation with finite density initial data

TL;DR

This paper analyzes the long-time behavior of solutions to the defocusing complex mKdV equation with finite density initial data, revealing asymptotics related to Painlevé-II transcendents using advanced Riemann-Hilbert techniques.

Contribution

It introduces a novel asymptotic analysis in the transition region employing $ar ext{ extit{d}}$-techniques and double scaling, connecting the solution's behavior to Painlevé-II transcendents.

Findings

Asymptotic behavior described by Painlevé-II transcendents

Application of $ar ext{ extit{d}}$-steepest descent method

Identification of the transition region dynamics

Abstract

We consider the Cauchy problem for the defocusing complex mKdV equation with finite density initial data \begin{align*} &q_t+\frac{1}{2}q_{xxx}-3|q|^2q_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*} which can be formulated into a Riemann-Hilbert (RH) problem. With $\bar\partial$-generation of the nonlinear steepest descent approach and a double scaling limit technique, in the transition region $$\mathcal{D}:=\left\{(x,t)\in\mathbb{R}\times\mathbb{R}^+\big|-C< \left(x/(2t)+3/2\right) t^{2/3}<0, C\in\mathbb{R}^+\right\},$$ we find that the long-time asymptotics of the solution $q(x,t)$ to the Cauchy problem is associated with the Painlev\'{e}-II transcendents.