High Frequency Behavior of the Focusing Nonlinear Schr[o-umlaut]dinger Equation with Random Inhomogeneities

We consider the effect of random inhomogeneities on the focusing singularity of the nonlinear Schrodinger equation in three dimensions, in the high frequency limit. After giving a phase space formulation of the high frequency limit using the Wigner distribution, we derive a nonlinear diffusion equation for the evolution of the wave energy density when random inhomogeneities are present. We show that this equation is linearly stable even in the case of a focusing nonlinearity, provided that it is not too strong. The linear stability condition is related to the variance identity for the nonlinear Schrodinger equation in an unexpected way. We carry out extensive numerical computations to get a better understanding of the interaction between the focusing nonlinearity and the randomness.

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