Resonant averaging for weakly nonlinear stochastic Schr\"odinger equations

TL;DR

This paper analyzes the limiting behavior of weakly nonlinear stochastic Schrödinger equations on a torus as the small parameter tends to zero, showing it is governed by an effective resonant equation relevant to weak turbulence.

Contribution

The paper introduces a novel averaging method for weakly nonlinear stochastic Schrödinger equations, deriving an effective equation that captures the limiting dynamics and stationary measures as the small parameter approaches zero.

Findings

The solutions converge to an effective resonant equation as the parameter tends to zero.

The stationary measure of the original system converges to that of the effective equation.

Resonant terms dominate the nonlinear dynamics in the limiting regime.

Abstract

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.