Focusing nonlocal nonlinear Schr\"odinger equation with asymmetric boundary conditions: large-time behavior

TL;DR

This paper analyzes the large-time behavior of solutions to a focusing nonlocal nonlinear Schrödinger equation with asymmetric boundary conditions, revealing three distinct asymptotic regions with modulated and unmodulated parameters.

Contribution

It provides a detailed asymptotic analysis of the focusing nonlocal NLS with asymmetric boundary conditions, identifying three different large-time regimes and their dependence on initial data.

Findings

Three asymptotic zones in the (x,t) plane identified

Existence of modulated and unmodulated parameter regions

Solution behavior depends on initial data details

Abstract

We consider the focusing integrable nonlocal nonlinear Schr\"odinger equation \[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric nonzero boundary conditions: $q(x,t)\to\pm A\mathrm{e}^{-2\mathrm{i}A^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. The goal of this work is to study the asymptotics of the solution of the initial value problem for this equation as $t\to+\infty$. For a class of initial values we show that there exist three qualitatively different asymptotic zones in the $(x,t)$ plane. Namely, there are regions where the parameters are modulated (being dependent on the ratio $x/t$) and a central region, where the parameters are unmodulated. This asymptotic picture is reminiscent of that for the defocusing classical nonlinear Schr\"odinger equation, but with some important differences. In particular, the absolute value of the solution in all three regions depends on details of the initial data.