Nonlinear Schrodinger equation with a white-noise potential: Phase-space approach to spread and singularity

[摘要] We propose a phase-space formulation for the nonlinear Schrodinger equation with a white-noise potential in order to shed light on two issues: the rate of spread and the singularity formation in the average sense. Our main tools are the energy law and the variance identity. The method is completely elementary. For the problem of wave spread, we show that the ensemble-averaged 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 bound respectively. We have also found that in the critical and supercritical focusing cases the presence of a white-noise random potential results in different conditions for singularity-with-positive-probability from the homogeneous case but does not prevent singularity formation. We show that in the supercritical focusing case the ensemble-averaged self-interaction energy and the momentum variance can exceed any fixed level in a finite time with positive probability. (c) 2005 Elsevier B.V All rights reserved.

[发布日期] 2005-12-15

[关键词] nonlinear Schrodinger equation;random potential;wave spread;blow-up