The cubic NLS on the line with an inverse square potential

TL;DR

This paper proves modified scattering for the cubic nonlinear Schrödinger equation on the real line with a repulsive inverse square potential, using advanced analytical techniques involving wave packets and vector fields.

Contribution

It introduces a novel approach combining free and distorted Galilei vector fields and wave packet transforms to analyze the cubic NLS with inverse square potential.

Findings

Established modified scattering for small localized data

Developed a new analytical framework for inverse square potential problems

Extended understanding of long-time behavior of solutions

Abstract

We establish modified scattering for solutions of the cubic NLS on the line with a repulsive inverse square potential and small localized data. The method is based on a comparison between the free and distorted Galilei vector fields and a wave packet transform.