Homogenization for nonlinear Schrödinger equations with periodic nonlinearity and dissipation in fractional order spaces

We study the nonlinear Schrödinger equation with time-oscillating nonlinearity and dissipation originated from the recent studies of Bose-Einstein condensates and optical systems which reads iψt+Δψ+φ(ωt)|ψα|ψ+iς(ωt)ψ=0. Under some conditions, we show that as ω→∞, the solution ψω will locally converge to the solution of the averaged equation iψt+Δψ+φ0|ψα|ψ+iζ0ψ=0 with the same initial condition in for all admissible pairs (q,r),where T∈(0,Tmax). We also show that if the dissipation coefficient ζo large enough, then, ψω is global if ω is sufficiently large and ψω converges to for all admissible pairs (q,r) In particular, our results hold for both subcritical and critical nonlinearities.