On the one-dimensional cubic nonlinear Schrödinger equation below L2

Abstract

In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic nonlinear Schrödinger equation (NLS) on the real line $\mathbb{R}$ and on the circle $\mathbb{T}$ for solutions below the $L^2$-threshold. We point out common results for NLS on $\mathbb{R}$ and the so-called Wick-ordered NLS (WNLS) on $\mathbb{T}$, suggesting that WNLS may be an appropriate model for the study of solutions below $L^2\left(\mathbb{T}\right)$. In particular, in contrast with a recent result of Molinet, who proved that the solution map for the periodic cubic NLS equation is not weakly continuous from $L^2\left(\mathbb{T}\right)$ to the space of distributions, we show that this is not the case for WNLS.