On Turbulence in Nonlinear Schrödinger Equations

We consider the small-dispersion and small-diffusion nonlinear Schr?dinger equation , , where the space-variable x belongs to the unit n-cube () and u satisfies Dirichlet boundary conditions. Assuming that the force is a zero-meanvalue random field, smooth in x and stationary in t with decaying correlations, we prove that the C ^m -norms in x with of solutions u, averaged in ensemble and locally averaged in time, are larger than , . This means that the length-scale of a solution u decays with as its positive degree (at least, as and - in a sense - proves existence of turbulence for this equation. Submitted: August 1996, revised version: May 1997