The complex mKdV equation with step-like initial data: Large time asymptotic analysis

TL;DR

This paper analyzes the large-time behavior of solutions to the complex mKdV equation with step-like initial data, using Riemann-Hilbert problem techniques to derive asymptotics in different spatial regions.

Contribution

It introduces a Riemann-Hilbert problem framework and applies steepest descent methods to obtain detailed asymptotic descriptions for the complex mKdV equation with step-like initial conditions.

Findings

Asymptotics in the Zakharov-Manakov region

Asymptotics in the plane wave region

Asymptotics in the slow decay region

Abstract

In this paper, we study large-time asymptotics for the complex modified Korteveg-de Vries equation \begin{equation} u_t + \frac{1}{2}u_{xxx}+3|u|^2 u_x=0, \end{equation} with the step-like initial data \begin{equation} u(x,0)=u_0(x)= \begin{cases} 0, & {x \ge 0,}\\ A e^{iBx}, &{x < 0.} \end{cases} \end{equation} It is shown that the step-like initial problem can be described by a matrix Riemann-Hilbert problem. We apply the steepest descent method to obtain different large-time asymptotics in the the Zakharov-Manakov region, a plane wave region and a slow decay region.