Global well-posedness for the cubic nonlinear Schr{ö}dinger equation with initial lying in $L^p$-based Sobolev spaces

Abstract

In this paper we continue our study [DSS20] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove global well-posedness for the 1D NLS with initial data lying in $L^p$ for any $2<p<\infty$, provided the initial data is sufficiently smooth. We do not use the complete integrability of the cubic nonlinear Schr{ö}dinger equation.

12 pages