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

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., $u_0(x)\to 0$ as $x\to -\infty$, $u_0(x)\to A$ as $x\to +\infty$,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 $x\to -x$, $t\to -t$ of nonlocal MKdV equation, we investigate the asymptotics for $t\to -\infty$ and $t\to +\infty$ 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.

44 pages