1 results
Resonant averaging for small-amplitude solutions of stochastic nonlinear Schrödinger equations
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 148 / Issue 2 / April 2018
- Published online by Cambridge University Press:
- 20 November 2017, pp. 357-394
- Print publication:
- April 2018
-
- Article
- Export citation
-
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.
are standard independent complex Wiener processes. We are interested in limiting, as 
