Shock problem for MKdV equation: Long time Dynamics of the Step-like initial data

TL;DR

This paper analyzes the long-time asymptotic behavior of solutions to the modified Korteweg-de Vries equation with step-like initial data, revealing different regimes including constant states and modulated elliptic waves.

Contribution

It provides a detailed asymptotic analysis of the MKdV equation with step-like initial data using Riemann-Hilbert problem techniques, identifying distinct solution behaviors in different regions.

Findings

Solution tends to c for x < -6c^2 t

Solution tends to 0 for x > 4c^2 t

In between, solution forms a modulated elliptic wave

Abstract

We consider the modified Korteveg de Vriez equation on the whole line. Initial data is real and step-like, i.e. $q(x,0)=0$ for $x\geq0$ and $q(x,0)=c$ for $x<0$, where c is arbitrary real number. The goal of this paper is to study the asymptotic behavior of the initial-value problem's solution by means of the asymptotic behavior of the some Riemann\textendash Hilbert problem. In this paper we show that the solution of this problem has different asymptotic behavior in different regions. In the region $x<-6c^2t$ and $x>4c^2t$ the solution is tend to $c$ and 0 correspondingly. In the region $-6c^2t<x<4c^2t$ the solution takes the form of a modulated elliptic wave.