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

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 $x\ge 0$ 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 $x<-6c^{2}t$ and $x>4c^{2}t$ the solution is tend to $c$ and 0 correspondingly. In the region $-6c^{2}t<x<4c^{2}t$ the solution takes the form of a modulated elliptic wave.

31 pages, 8 figures

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• On the Cauchy problem of defocusing mKdV equation with finite density initial data: long time asymptotics in soliton-less regions
• Large-time asymptotics to the focusing nonlocal modified Kortweg-de Vries equation with step-like boundary conditions
• On the form of dispersive shock waves of the Korteweg-de Vries equation
• Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra
• Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
• Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials
• Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions