Resonant averaging for weakly nonlinear stochastic Schr "odinger\n equations

2013/09/19 by Sergei Kuksin, Kuksin, Sergei, Alberto Maiocchi +1
Physics and Astronomy · Computer Science · Mathematics · #Advanced Thermodynamics and Statistical Mechanics #Nonlinear Dynamics and Pattern Formation #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1309.5022

Abstract

We consider the free linear Schroedinger equation on a torus mathbb T^d,\nperturbed by a Hamiltonian nonlinearity, driven by a random force and damped by\na linear damping: u_t -i
Delta u +i
nu
rho |u|^2q_*u = -
nu f(-
Delta) u\n+
sqrt
nu
,
fracdd t
sum_{k
in
mathbb Z^d} b_k
beta^k(t)e^{ik
cdot x}
.\n Here u=u(t,x), x\∈ mathbb T^d, 0<\ν\≪1, q_*\∈ mathbb N\∪ 0 ,\nf is a positive continuous function, \ρ is a positive parameter and\n\β^k(t) are standard independent complex Wiener processes. We are\ninterested in limiting, as \ν\→0, behaviour of solutions for this equation\nand of its stationary measure. Writing the equation in the slow time \τ=\ν\nt, we prove that the limiting behaviour of the both is described by the\neffective equation u_
tau+ f(-
Delta) u = -iF(u)+
fracdd
tau
sum\nb_k
beta^k(
tau)e^{ik
cdot x}
, where the nonlinearity F(u) is made out\nof the resonant terms of the monomial |u|^2q_*u. We explain the relevance\nof this result for the problem of weak turbulence.\n

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