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 \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\simeq Ae^{2\I Bx}$ as $x\to \infty$, where $A>0$ and $B\in \mathbb{R}$. 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 $q_{0,R}(x)= \begin{cases} 0, &x\leq R,\\ Ae^{2\I Bx}, &x>R, \end{cases}$ with $R>0$. A special attention is paid to how $B\ne 0$ affects the asymptotics.