Geometric Optics and Instability for Semi-Classical Schrödinger Equations

We prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the problem cannot be reduced to the study of an ordinary differential equation.