Global conservative solutions of the nonlocal NLS equation beyond blow-up

TL;DR

This paper develops a framework for continuing solutions of the nonlocal nonlinear Schrödinger equation beyond singularities using inverse scattering and Riemann-Hilbert methods, addressing solutions with potential blow-up points.

Contribution

It introduces a novel concept for extending weak solutions of the nonlocal NLS equation beyond blow-up, combining inverse scattering transform with PDE existence theory.

Findings

Proposes a continuation method for solutions beyond blow-up

Integrates inverse scattering transform with PDE theory

Addresses long-time soliton resolution scenarios

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2\sigma q^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.