# Scattering for the $L^2$ supercritical point NLS

## TL;DR

This paper proves that solutions to the 1D focusing point nonlinear Schrödinger equation in the $L^2$ supercritical case scatter as time approaches infinity, using a modified Kenig-Merle approach.

## Contribution

It demonstrates scattering for the $L^2$ supercritical focusing point NLS, adapting the Kenig-Merle method with a specialized function space.

## Key findings

- Global solutions scatter as t approaches infinity
- The method adapts Kenig-Merle to non-Strichartz spaces
- Provides insight into asymptotic behavior of supercritical NLS

## Abstract

We consider the 1D nonlinear Schr\"odinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters as t tends to minus/plus infinity in the $L^2$ supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.