On the one-dimensional cubic nonlinear Schrodinger equation below L^2

TL;DR

This paper reviews recent results on the well-posedness of the one-dimensional cubic nonlinear Schrödinger equation below the L^2 threshold, highlighting differences between the standard and Wick ordered versions on the real line and circle.

Contribution

It demonstrates that Wick ordered NLS on the circle is well-posed below L^2, contrasting with the non-continuity results for standard cubic NLS.

Findings

Wick ordered NLS is well-posed below L^2 on the circle.

Standard cubic NLS solution map is not weakly continuous below L^2.

Wick ordering provides a suitable model for solutions below L^2.

Abstract

In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real line R and on the circle T for solutions below the L^2-threshold. We point out common results for NLS on R and the so-called "Wick ordered NLS" (WNLS) on T, suggesting that WNLS may be an appropriate model for the study of solutions below L^2(T). 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(T) to the space of distributions, we show that this is not the case for WNLS.