Long-time asymptotics for the Nonlocal mKdV equation

TL;DR

This paper analyzes the long-time behavior of solutions to the nonlocal mKdV equation using Riemann-Hilbert techniques, revealing new asymptotic results that differ from the classical mKdV case.

Contribution

It formulates the Riemann-Hilbert problem for the nonlocal mKdV and applies the Deift-Zhou method to derive novel long-time asymptotics, highlighting differences from the local mKdV.

Findings

New asymptotic formulas for nonlocal mKdV solutions

Differences in long-time behavior compared to classical mKdV

Additional assumptions on scattering data are required

Abstract

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV) \[q_t(x,t)+q_{xxx}(x,t)-6q(x,t)q(-x,-t)q_x(x,t)=0,\] which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.