Existence of global solutions to the nonlocal Schr\"{o}dinger equation on the line

TL;DR

This paper proves the existence of global solutions for the nonlocal nonlinear Schrödinger equation on the real line using inverse scattering and Riemann-Hilbert methods under small initial data assumptions.

Contribution

It establishes the global well-posedness of the nonlocal NLS equation with small initial data in $H^{1,1}( )$ by analyzing spectral properties and employing inverse scattering theory.

Findings

Spectral problem admits no eigenvalues or resonances under small data.

Inverse scattering map is bijective and Lipschitz continuous.

Global solutions exist for the nonlocal NLS with small initial data.

Abstract

In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schr\"{o}dinger (nonlocal NLS) equation with the initial data $q_0(x)\in H^{1,1}(\R)$ with the $L^1(\R)$ small-norm assumption. We rigorously show that the spectral problem for the nonlocal NLS equation admits no eigenvalues or resonances, as well as Zhou vanishing lemma is effective under the $L^1(\R)$ small-norm assumption. With inverse scattering theory and the Riemann-Hilbert approach, we rigorously establish the bijectivity and Lipschitz continuous of the direct and inverse scattering map from the initial data to reflection coefficients.By using reconstruction formula and the Plemelj projection estimates of reflection coefficients,we further obtain the existence of the local solution and the priori estimates, which assure the existence of the global solution to the Cauchy problem for the nonlocal NLS equation.