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

TL;DR

This paper establishes local well-posedness for the 1D cubic nonlinear Schrödinger equation in L^p spaces for p between 2 and 4, extending previous results and allowing for slower decay data.

Contribution

It generalizes well-posedness results to a broader range of L^p spaces, including p>2, and analyzes regularity properties of solutions in Sobolev and Strichartz spaces.

Findings

Proves local well-posedness in L^p for 2<p<4.

Extends classical results from p=2 to a wider p-range.

Analyzes regularity of solutions in Sobolev and Strichartz spaces.

Abstract

In this paper, local well-posedness is shown for the one dimensional cubic nonlinear Schr\"odinger equation in $L^p$-spaces for $2<p<4$, which generalizes a classical result for $p=2$ by Y. Tsutsumi and recent work for $1<p<2$ by Y. Zhou. As a consequence, a local theory of solutions is established for a class of data which decay more slowly than square integrable functions. Regularity properties of the local solutions in the $L^p$-based Sobolev spaces and Stricharz spaces are also proved.