On The Energy Transfer To High Frequencies In The Damped/Driven Nonlinear Schrödinger Equation (Extended Version)

Abstract

We consider a damped/driven nonlinear Schrödinger equation in an $n$-cube $K^n\subset {\mathbb{R}}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-νΔu+i|u|^2u=\sqrtνη(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad ν>0, \] where $\eta (t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $\Vert u(t){\Vert }_m^2\le C{\nu }^{-m},$ uniformly in $t\ge 0$ and $\nu >0$. In this work we prove that for small $\nu >0$ and any initial data, with large probability the Sobolev norms $\Vert u(t,\cdot ){\Vert }_m$ of the solutions with $m>2$ become large at least to the order of ${\nu }^{-{\kappa }_{n,m}}$ with ${\kappa }_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu })$.