The integrable nonlocal nonlinear Schr\"odinger equation with oscillatory boundary conditions: long-time asymptotics

TL;DR

This paper analyzes the long-time behavior of solutions to a nonlocal nonlinear Schrödinger equation with step-like initial data, focusing on how oscillatory boundary conditions influence asymptotics.

Contribution

It provides the first detailed asymptotic analysis of the integrable nonlocal NLS with oscillatory boundary conditions and step-like initial data.

Findings

Asymptotic behavior depends on the oscillatory boundary parameter B.

Long-time solutions exhibit distinct regimes influenced by boundary oscillations.

The analysis reveals how nonlocality and boundary conditions shape solution dynamics.

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger 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: $q(x,0)\to0$ 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 $B\ne0$ affects the asymptotics.