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

TL;DR

This paper establishes global well-posedness for the 1D cubic nonlinear Schrödinger equation with initial data in certain $L^{p}$-based Sobolev spaces, extending previous local results without relying on integrability.

Contribution

It proves global well-posedness for initial data in $L^{p}$ spaces for $2 < p < olinebreak \infty$, broadening the class of initial conditions beyond previous analytic or vanishing-at-infinity assumptions.

Findings

Global well-posedness for initial data in $L^{p}$ spaces with $2 < p < olinebreak \infty$.

No reliance on the complete integrability of the equation.

Extension of local results to global for a wider class of initial data.

Abstract

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 $2 < p < \infty$, provided the initial data is sufficiently smooth. We do not use the complete integrability of the cubic nonlinear Schr{\"o}dinger equation.