Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity

Details Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity

2005-03-09

arxiv.org/abs/nlin/0503021v2

Journal of Physics A.: Math. Gen. Vol 39 (2006), 11383-11398.

Nonlinear Sciences

Chaotic Dynamics

Scientific paper

10.1088/0305-4470/39/37/004

We derive several kinetic equations to model the large scale, low Fresnel number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly fluctuating random potential. There are three types of kinetic equations the longitudinal, the transverse and the longitudinal with friction. For these nonlinear kinetic equations we address two problems: the rate of dispersion and the singularity formation. For the problem of dispersion, we show that the kinetic equations of the longitudinal type produce the cubic-in-time law, that the transverse type produce the quadratic-in-time law and that the one with friction produces the linear-in-time law for the variance prior to any singularity. For the problem of singularity, we show that the singularity and blow-up conditions in the transverse case remain the same as those for the homogeneous NLS equation with critical or supercritical self-focusing nonlinearity, but they have changed in the longitudinal case and in the frictional case due to the evolution of the Hamiltonian.

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