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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 

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