Nonlinear Schrödinger Equation with a White-noise Potential: Phase-space Approach to Spread and Singularity

We propose a phase-space formulation for the nonlinear Schrödinger equation with a white-noise potential in order to shed light on two problems: the rate of dispersion and the singularity formation. Our main tools are the energy laws and the variance identity. The method is completely elementary. For the problem of dispersion, we show that in the absence of dissipation the ensemble-average dispersion in the critical or defocusing case follows the cubic-in-time law while in the supercritical and subcritical focusing cases the cubic law becomes an upper and lower bounds respectively. In the presence of dissipation the cubic law is replaced by the linear-in-time law. We have also found that the presence of a white-noise random potential merely changs the singularity condition but does not prevent singularity formation in the critical and supercritical focusing cases. We show that the finite-time singularity in the supercritical focusing case is of the blow-up type.