Acta mathematica scientia,Series B ›› 2015, Vol. 35 ›› Issue (3): 567-582.doi: 10.1016/S0252-9602(15)30004-7

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

Binhua FENG1, Dun ZHAO2, Chunyou SUN2   

  1. 1. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China;
    2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • Received:2014-01-05 Revised:2014-04-14 Online:2015-05-01 Published:2015-05-01
  • Contact: Binhua FENG E-mail:binhuaf@163.com
  • Supported by:

    This work was supported by the NSFC Grants 10601021 and 11475073.

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 iψtψ +φ(ωt)|ψ|αψ+iζ(ωt)ψ= 0.Under some conditions, we show that as ω→∞, the solution ψω will locally converge to the solution of the averaged equation iψtψ +φ0|ψ|αψ+iζ0ψ= 0 with the same initial condition in Lq((0, T),Brs,2)for all admissible pairs (q, r), where T ∈ (0,Tmax). We also show that if the dissipation coefficient ζ0 large enough, then,ψω is global if ω is sufficiently large and ψω converges to ψ in Lq((0,∞),Brs,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
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