Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations
Published online by Cambridge University Press: 20 November 2017
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.
MSC classification
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 2 , April 2018 , pp. 357 - 394
- Copyright
- Copyright © Royal Society of Edinburgh 2018
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