Phase space models for stochastic nonlinear parabolic waves: wave spread and collapse

We derive several kinetic equations to model the large scale, low Fresnel number behaviour of the generalized nonlinear Schrödinger (NLS) equation with a rapidly fluctuating random potential. Depending on the relative scale of fluctuation in the longitudinal and transverse directions, we classify the kinetic equations into the longitudinal, the transverse and the isotropic case. The principal assumption of our derivation is that the rapid fluctuation in the linear potential does not give rise to rapid oscillations in the modulus of the wave amplitude. For the longitudinal and the transverse cases we address two problems, the rate of dispersion and the singularity formation, using the nonlinear kinetic equations. The main technique is the variance identities derived for the nonlinear kinetic equations. For the problem of dispersion, we show that in the longitudinal case the spread scales like (time)3/2 whereas in the transverse case the spread is linearly proportional to time. For the problem of singularity formation, we show that the collapse conditions in the transverse case remain the same as those for the homogeneous NLS equation with critical or supercritical self-focusing nonlinearity whereas in the longitudinal case the small-scale medium fluctuations tend to enhance the energy of the system and thus raise the energy barrier to wave collapse. PACS numbers: 02.30.Sa, 05.45.Mt, 46.65.+g