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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:405

Article

Feng, Binhua¹  Zhao, Dun¹  Sun, Chunyou¹ 

[1] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China

关键词: Nonlinear Schrodinger equation;    Limit behavior of solutions;    Variable coefficient loss/gain;    Convergence;    Global existence;   

DOI  :  10.1016/j.jmaa.2013.04.001

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

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