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

In this paper we continue our study [DSS20] of the nonlinear Schr"odinger 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 $2<p<\mathrm{\infty }$, provided the initial data is sufficiently smooth. We do not use the complete integrability of the cubic nonlinear Schr{"o}dinger equation.