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

Albert Fannjiang, George Papanicolaou, Shi Jin
2003 SIAM Journal on Applied Mathematics  
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
more » ... not too strong. The linear stability condition is related to the variance identity for the nonlinear Schrödinger 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. In the focusing case β < 0 and with a negative energy H < 0, the solution cannot remain bounded for all time. More precisely, it follows from the variance identity and the uncertainty inequality that the L 2 norm of the gradient of the solution blows up in finite time [6] . Many other properties of NLS can be found in [17] .
doi:10.1137/s003613999935559x fatcat:fn3uhf6xpfafzpstewmftx3fgu