In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic nonlinear Schrödinger equation (NLS) on the real line and on the circle for solutions below the -threshold. We point out common results for NLS on and the so-called Wick-ordered NLS (WNLS) on , suggesting that WNLS may be an appropriate model for the study of solutions below . 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 to the space of distributions, we show that this is not the case for WNLS.
"On the one-dimensional cubic nonlinear Schrödinger equation below ." Kyoto J. Math. 52 (1) 99 - 115, Spring 2012. https://doi.org/10.1215/21562261-1503772