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

【摘要】 We study the nonlinear Schrodinger 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 + φ_|Ψ|^αΨ+iζoΨ=0 with the same initial condition in L^q(（0,T）,B_r,2^s)for all admissible pairs（q,r）,where T ∈(0,T_max).We also show that if the dissipation coefficient ζo large enough,then,Ψ_ω is global if ω is sufficiently large and Ψ_ω converges toΨ in L^q(（0,∞）,B_r,2^s),for all admissible pairs（q,r）.In particular,our results hold for both subcritical and critical nonlinearities.更多还原

【Abstract】 We study the nonlinear Schrodinger 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 + φ_|Ψ|^αΨ+iζoΨ=0 with the same initial condition in L^q(（0,T）,B_r,2^s)for all admissible pairs（q,r）,where T ∈(0,T_max).We also show that if the dissipation coefficient ζo large enough,then,Ψ_ω is global if ω is sufficiently large and Ψ_ω converges toΨ in L^q(（0,∞）,B_r,2^s),for all admissible pairs（q,r）.In particular,our results hold for both subcritical and critical nonlinearities.更多还原