The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: Long-time asymptotics

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation $i{q}_t(x,t)+q_{xx}(x,t)+2q^2(x,t)\bar{q}(-x,t)=0,$subject to the step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\simeq A{e}^{2iBx}$ as $x\to \infty$, where $A>0$ and $B\in R$. The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that $q(x,0)$ is close, in a certain spectral sense, to the “step-like” function $q_{0,R}\left(x\right)=\left(\begin{array}{cc}0,& x\le R,\\ A{e}^{2iBx},& x>R,\end{array}\right)$ with $R>0$. A special attention is paid to how $B\ne 0$ affects the asymptotics.