Scattering for the nonlinear Schr\"odinger equation with concentrated nonlinearity

TL;DR

This paper proves global well-posedness and scattering for the one-dimensional cubic defocusing nonlinear Schrödinger equation with nonlinear effects concentrated at a point, extending the results to highly localized nonlinearities.

Contribution

It establishes global well-posedness and scattering for the NLS with concentrated nonlinearity in the critical space, generalizing previous results to highly localized nonlinear effects.

Findings

Global well-posedness in L^2 space

Scattering for solutions with concentrated nonlinearity

Phenomenology extends to highly localized nonlinear effects

Abstract

We consider the cubic defocusing nonlinear Schr\"odinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all such solutions. Moreover, we demonstrate that the same phenomenology holds whenever nonlinear effects are sufficiently concentrated in space.