HOMOGENIZATION FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH PERIODIC NONLINEARITY AND DISSIPATION IN FRACTIONAL ORDER SPACES

doi:10.1016/S0252-9602(15)30004-7

Abstract

Abstract:

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 iψ_t+Δψ +φ(ωt)|ψ|^αψ+iζ(ωt)ψ= 0.Under some conditions, we show that as ω→∞, the solution ψ_ω will locally converge to the solution of the averaged equation iψ_t+Δψ +φ₀|ψ|^αψ+iζ₀ψ= 0 with the same initial condition in L^q((0, T),B_r^s,2)for all admissible pairs (q, r), where T ∈ (0,T_max). We also show that if the dissipation coefficient ζ₀ large enough, then,ψ_ω is global if ω is sufficiently large and ψ_ω converges to ψ in L^q((0,∞),B_r^s,2), for all admissible pairs (q, r). In particular, our results hold for both subcritical and critical nonlinearities.

Key words: Nonlinear Schrö, dinger equation, averaged equation, global existence, convergence

CLC Number:

35B27

Binhua FENG, Dun ZHAO, Chunyou SUN. HOMOGENIZATION FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH PERIODIC NONLINEARITY AND DISSIPATION IN FRACTIONAL ORDER SPACES[J].Acta mathematica scientia,Series B, 2015, 35(3): 567-582.

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