Large-time asymptotics to the focusing nonlocal modified Kortweg-de Vries equation with step-like boundary conditions

TL;DR

This paper analyzes the large-time behavior of solutions to the focusing nonlocal MKdV equation with step-like initial data, revealing different asymptotic behaviors in various space-time sectors using Riemann-Hilbert problem techniques.

Contribution

It develops a direct scattering theory and applies steepest descent analysis to derive explicit large-time asymptotics for the nonlocal MKdV equation with step-like boundary conditions.

Findings

Different asymptotic behaviors in four space-time sectors.

Explicit solutions obtained via Riemann-Hilbert problem deformation.

Asymptotics valid for both positive and negative infinite times.

Abstract

We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$, $u_0(x)\rightarrow A$ as $x\rightarrow+\infty$,where $A$ is an arbitrary positive real number. We firstly develop the direct scattering theory to establish the basic Riemann-Hilbert (RH) problem associated with step-like initial data. Thanks to the symmetries $x\rightarrow-x$, $t\rightarrow-t$ of nonlocal MKdV equation, we investigate the asymptotics for $t\rightarrow-\infty$ and $t\rightarrow+\infty$ respectively. Our main technique is to use the steepest descent analysis to deform the original matrix-valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally we obtain the different large-time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space-time sectors $\mathcal{R}_{I}$, $\mathcal{R}_{II}$, $\mathcal{R}_{III}$ and $\mathcal{R}_{IV}$ on the whole $(x,t)$-plane.