On Turbulence in Nonlinear Schrödinger Equations

Abstract

We consider the small-dispersion and small-diffusion nonlinear Schrodinger equation $ -i \dot{u} = -\delta_{1} \Delta u - i \delta _{2} \Delta u + \mid u \mid ^{2}u + \zeta _{\omega} (t, x)$ , $ 1 \geq \delta : = \sqrt{\delta^{2}_{1} + \delta^{2}_{2}} > 0$ , where the space-variable x belongs to the unit n-cube ( $ n \leq 3 $ ) and u satisfies Dirichlet boundary conditions. Assuming that the force $ \zeta $ 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 $ m \geq 3 $ of solutions u, averaged in ensemble and locally averaged in time, are larger than $ \delta ^{-\kappa m} $ , $ \kappa \approx 1/5 $ . This means that the length-scale of a solution u decays with $ \delta $ as its positive degree (at least, as $ \delta^{\kappa} $ and - in a sense - proves existence of turbulence for this equation.