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

This paper is devoted to the limit behavior as epsilon --> 0 for the solution of the Cauchy problem of the nonlinear Schrodinger equation including nonlinear loss/gain with variable coefficient: iu(t) + Delta u + lambda vertical bar u vertical bar(alpha) u + i epsilon alpha(t)vertical bar u vertical bar(p)u = 0. Such an equation appears in the recent studies of Bose-Einstein condensates and optical systems. tinder some conditions, we show that the solution will locally converge to the solution of the limiting equation iu(t) + Delta u + lambda vertical bar u vertical bar(alpha)u = 0 with the same initial data in L-gamma((0, l), W-1,W-rho) for all admissible pairs (gamma, rho), where l is an element of (0, T*). We also show that, if the limiting solution u is global and has some decay property as t --> infinity, then u(epsilon) is global if epsilon is sufficiently small and u(epsilon) converges to u in L-gamma ((0, infinity), W-1,W-rho), for all admissible pairs (gamma, rho). In particular, our results hold for both subcritical and critical nonlinearities. (C) 2013 Elsevier Inc. All rights reserved.