We consider the free linear Schroedinger equation on a torus $\mathbb T^d$, perturbed by a Hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t -i\Delta u +i\nu \rho |u|^{ 2q_*}u = - \nu f(-\Delta) u + \sqrt\nu\,\frac{ d}{ d t}\sum_{ k\in \mathbb Z^d} b_k\beta^k(t)e^{ ik\cdot x} \ . $$ Here $u=u(t,x),\ x\in\mathbb T^d$, $0<\nu\ll1$, $q_*\in\mathbb N\cup\{ 0\}$, $f$ is a positive continuous function, $\rho$ is a positive parameter and $\beta^k(t)$ are standard independent complex Wiener processes. We are interested in limiting, as $\nu\to0$, behaviour of solutions for this equation and of its stationary measure. Writing the equation in the slow time $\tau=\nu t$, we prove that the limiting behaviour of the both is described by the effective equation $$ u_\tau+ f(-\Delta) u = -iF(u)+\frac{ d}{ d\tau}\sum b_k\beta^k(\tau)e^{ ik\cdot x} \, $$ where the nonlinearity $F(u)$ is made out of the resonant terms of the monomial $ |u|^{ 2q_*}u$. We explain the relevance of this result for the problem of weak turbulence.
Categories: math-ph math.MP