Defocusing nonlocal nonlinear Schr\"odinger equation with step-like boundary conditions: long-time behavior for shifted initial data

Abstract

The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr\"odinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\to A$ as $x\to +\infty$. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by $R>0$, with the initial data that can be viewed as perturbations of the "shifted step function" $q_{R,A}(x)$: $q_{R,A}(x)=0$ for $x<R$ and $q_{R,A}(x)=A$ for $x>R$, where $A>0$ and $R>0$ are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the $(x,t)$ plane, the number of which depends on the relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger number of sectors. Moreover, the sectors can be collected into 2 alternate groups: in the sectors of the first group, the solution decays to 0 while in the sectors of the second group, the solution approaches a constant (varying with the direction $x/t=const$).