Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T23:40:54.150Z Has data issue: false hasContentIssue false

Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations

Published online by Cambridge University Press:  20 November 2017

Sergei Kuksin
Affiliation:
CNRS and IMJ, Université Paris Diderot-Paris 7, Paris, France ([email protected])
Alberto Maiocchi
Affiliation:
Laboratoire de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, Cergy-Pontoise, France ([email protected])

Abstract

We consider the free linear Schrödinger equation on a torus 𝕋d, perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping:

Here u = u(t, x), x ∈ 𝕋d, 0 < ν ≪ 1, q ℕ, f is a positive continuous function, ρ is a positive parameter and are standard independent complex Wiener processes. We are interested in limiting, as ν → 0, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time τ = νt, we prove that the limiting behaviour of them both is described by the effective equation

where the nonlinearity F(u) is made out of the resonant terms of the monomial |u|2q∗u.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)