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

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 riot the case for WNLS.