Asymptotic stage of modulation instability for the nonlocal nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the long-time behavior of solutions to the nonlocal nonlinear Schrödinger equation, revealing that its asymptotics depend on initial data details, unlike the classical NLS equation.

Contribution

It provides the first detailed asymptotic analysis of the nonlocal NLS equation using inverse scattering and Riemann-Hilbert techniques, highlighting non-universal behavior.

Findings

Asymptotics depend on initial data details

Contrasts with classical NLS where asymptotics depend only on phase

Uses nonlinear steepest descent for Riemann-Hilbert problem analysis

Abstract

We study the initial value problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric boundary conditions: $q(x,t)\to Ae^{2iA^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution $q(x,t)$ of this initial value problem. We shown that it exhibits a non-universal, in a sense, behavior: the asymptotics of $|q(x,t)|$ depends on details of the initial data $q(x,0)$. This is in a sharp contrast with the local classical NLS equation, where the long-time asymptotics of the solution depends on the initial value through the phase parameters only. The main tool used in this work is the inverse scattering transform method applied in the form of the matrix Riemann-Hilbert problem. The Riemann-Hilbert problem associated with the original initial value problem is analyzed asymptotically by the nonlinear steepest decent method.