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High Frequency Behavior of the Focusing Nonlinear Schrödinger Equation with Random Inhomogeneities

2003
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SIAM Journal on Applied Mathematics
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We consider the effect of random inhomogeneities on the focusing singularity of the nonlinear Schrödinger 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

doi:10.1137/s003613999935559x
fatcat:fn3uhf6xpfafzpstewmftx3fgu