On Turbulence in Nonlinear Schrödinger Equations

Abstract.

We consider the small-dispersion and small-diffusion nonlinear Schrödinger 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.