Riemann–Hilbert problem to the modified Korteveg–de Vries equation: Long-time dynamics of the steplike initial data

Abstract

We consider the modified Korteveg–de Vries equation on the line. The initial data are the pure step function, i.e., q(x,0)=0 for x≥0 and q(x,0)=c for x<0, where c is an arbitrary real number. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t→∞. Using the steepest descent method and the so-called g-function mechanism, we deform the original oscillatory matrix Riemann–Hilbert problem to explicitly solving model forms and show that the solution of the initial-value problem has a different asymptotic behavior in different regions of the xt-plane. In the regions x<−6c2t and x>4c2t, the main terms of asymptotics of the solution are equal to c and 0, respectively. In the region −6c2t<x<4c2t, asymptotics of the solution takes the form of a modulated elliptic wave of finite amplitude.