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

Albert Fannjiang
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Abstract

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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Albert Fannjiang. 2005-03-19. Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity. https://doi.org/10.1088/0305-4470%2F39%2F37%2F004

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