Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions

We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time T N → 0 T_N\to 0 and solution u N ( t ) u_N(t) such that u N ( T N ) → ∞ u_N(T_N)\to \infty in the Besov space B 2 , σ − 1 ( R ) B_{2,\sigma }^{-1}(\mathbb {R}) ( σ > 2 \sigma >2 ) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H − 1 ( R 2 ) H^{-1}(\mathbb {R}^2) . We systematically utilize the modulation space M 2 , 1 0 M_{2,1}^0 for one dimension and the scaled modulation space ( M 2 , 1 0 ) N (M_{2,1}^0)_N for two dimensions.