Long time asymptotics for the nonlocal mKdV equation with finite density initial data

TL;DR

This paper analyzes the long-time behavior of solutions to a nonlocal mKdV equation with finite density initial data, revealing soliton structures and decay rates using spectral analysis and Riemann-Hilbert techniques.

Contribution

It provides the first detailed asymptotic analysis of the nonlocal mKdV equation with nonzero boundary conditions using the $ar{ ext{D}}$-steepest descent method.

Findings

Solution characterized by $N( ext{Lambda})$-solitons on discrete spectrum.

Leading order decay rate of $ ext{O}(t^{-1/2})$ on continuous spectrum.

Residual error decays as $ ext{O}(t^{-1})$.

Abstract

In this paper, we consider the Cauchy problem for an integrable real nonlocal (also called reverse-space-time) mKdV equation with nonzero boundary conditions \begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=\delta q_{-}$, $\sigma\delta=-1$. Based on the spectral analysis of the Lax pair, we express the solution of the Cauchy problem of the nonlocal mKdV equation in terms of a Riemann-Hilbert problem. In a fixed space-time solitonic region $-6<x/t<6$, we apply $\bar{\partial}$-steepest descent method to analyze the long-time asymptotic behavior of the solution $q(x,t)$. We find that the long time asymptotic behavior of $q(x,t)$ can be characterized with an $N(\Lambda)$-soliton on discrete spectrum and leading order term $\mathcal{O}(t^{-1/2})$ on continuous spectrum up to an residual error order $\mathcal{O}(t^{-1})$.