The limit behavior of solutions for the nonlinear Schrödinger equation including nonlinear loss/gain with variable coefficient

This paper is devoted to the limit behavior as ε→0ε→0 for the solution of the Cauchy problem of the nonlinear Schrödinger equation including nonlinear loss/gain with variable coefficient: iut+Δu+λ|u|αu+iεa(t)|u|pu=0iut+Δu+λ|u|αu+iεa(t)|u|pu=0. Such an equation appears in the recent studies of Bose–Einstein condensates and optical systems. Under some conditions, we show that the solution will locally converge to the solution of the limiting equation iut+Δu+λ|u|αu=0iut+Δu+λ|u|αu=0 with the same initial data in Lγ((0,l),W1,ρ)Lγ((0,l),W1,ρ) for all admissible pairs (γ,ρ)(γ,ρ), where l∈(0,T∗)l∈(0,T∗). We also show that, if the limiting solution uu is global and has some decay property as t→∞t→∞, then uεuε is global if εε is sufficiently small and uεuε converges to uu in Lγ((0,∞),W1,ρ)Lγ((0,∞),W1,ρ), for all admissible pairs (γ,ρ)(γ,ρ). In particular, our results hold for both subcritical and critical nonlinearities.