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

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation $$\text{I}{q}_t(x,t)+q_{xx}(x,t)+2q^2(x,t)\bar{q}(-x,t)=0,$$ subject to the step-like initial data: $q(x,0)\to 0$ as $x\to -\mathrm{\infty }$ and $q(x,0)\simeq A{e}^{2\text{I}Bx}$ as $x\to \mathrm{\infty }$, 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 $q_{0,R}(x)=\{\begin{array}{ll}0,& x\le R,\\ A{e}^{2\text{I}Bx},& x>R,\end{array}$ with $R>0$. A special attention is paid to how $B\mathrm{\ne }0$ affects the asymptotics.