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

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.