Global bounds for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension

TL;DR

This paper establishes global existence results for small initial data in a one-dimensional cubic NLS, introduces a simplified proof technique, and discusses modified scattering and asymptotic completeness.

Contribution

It provides a new, simpler method to prove global solutions for small data in $H^{0,1}$ and explores related scattering and asymptotic properties.

Findings

Global solutions exist for small data in $H^{0,1}$.

Introduces a simplified proof approach.

Discusses modified scattering and asymptotic completeness.

Abstract

This article is concerned with the small data problem for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in $H^{0,1}$. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness.