On the ill-posedness of the cubic nonlinear Schr\"odinger equation on the circle

Abstract

In this note, we consider the ill-posedness issue for the cubic nonlinear Schr\"odinger equation (NLS) on the circle. In particular, adapting the argument by Christ-Colliander-Tao [14] to the periodic setting, we exhibit a norm inflation phenomenon for both the usual cubic NLS and the Wick ordered cubic NLS for $s \leq s_\text{crit} :=- \frac 12$. We also discuss norm inflation phenomena for general cubic fractional NLS on the circle.