Resonant averaging for small solutions of stochastic NLS equations

Abstract

We consider the free linear Schrödinger equation on a torus ${\mathbb{T}}^d$, perturbed by a hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t-i\Delta u+i\nu \rho |u{|}^{2q_{\ast }}u=-\nu f(-\Delta )u+\sqrt{\nu }\frac{d}{dt}\sum _{k\in {\mathbb{Z}}^d}{b}_l{\beta }^k(t)e^{ik\cdot x}.$$ Here $u=u(t,x),x\in {\mathbb{T}}^d$, $0<\nu \ll 1$, $q_{\ast }\in \mathbb{N}$, $f$ 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 $\nu \to 0$, behaviour of distributions 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 $|u{|}^{2q_{\ast }}u$. We explain the relevance of this result for the problem of weak turbulence

arXiv admin note: substantial text overlap with arXiv:1309.5022

References in corpus (2)

• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
• Derivation of the Kolmogorov-Zakharov equation from the resonant-averaged stochastic NLS equation

Cited by in corpus (3)

• The effective equation method
• The large-period limit for equations of discrete turbulence
• Derivation of the Kolmogorov-Zakharov equation from the resonant-averaged stochastic NLS equation