Global well-posedness for the 1D cubic nonlinear Schr\"odinger equation in $L^p,\,p>2$

TL;DR

This paper proves global well-posedness of the 1D cubic nonlinear Schrödinger equation in $L^p$ spaces for $p$ between 2 and 13/6, extending classical results beyond $L^2$ using advanced data decomposition techniques.

Contribution

It extends the known global well-posedness results of the 1D cubic NLS from $L^2$ to a broader range of $L^p$ spaces, employing a novel data decomposition approach.

Findings

Global well-posedness in $L^p$ for $2 \\le p < 13/6$

Persistence of solutions for twisted variables at any time

Extension of classical $L^2$ results to $L^p$ spaces

Abstract

In this paper, we show that the one dimensional cubic nonlinear Schr\"odinger equation is globally well posed in $L^p$ for $2\le p <13/6$. In particular, we prove that the global solution enjoys the persistence property for a twisted variable at any time, which implies the result is a natural exetension of the classical global well-posedness in $L^2$ to $L^p$. The proof exploits the data-decomposition argument originally developed by Vargas-Vega in the functional framework introduced by Zhou.