Small data blow-up of L^{2}-solution for the nonlinear Schr\"odinger   equation without gauge invariance
  We study the initial value problem for the nonlinear Schr\"odinger equation. We will prove that the blow-up of the L^{2}-norm of solutions with suitable initial data. We impose a condition related to the sign of the data but put no restriction on their size. 
Masahiro Ikeda and Yuta Wakasugi
math.AP