Global well-posedness for the cubic nonlinear Schrödinger equation with initial data lying in L p-based Sobolev spaces

Abstract

In this paper, we continue our study [B. Dodson, A. Soffer, and T. Spencer, J. Stat. Phys. 180, 910 (2020)] 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 < ∞, provided that the initial data are sufficiently smooth. We do not use the complete integrability of the cubic NLS.