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

doi:10.1016/S0252-9602(15)30004-7

数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (3): 567-582.doi: 10.1016/S0252-9602(15)30004-7

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

冯斌华¹, 赵敦², 孙春友²

1. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China;
2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

收稿日期:2014-01-05 修回日期:2014-04-14 出版日期:2015-05-01 发布日期:2015-05-01
通讯作者: Binhua FENG E-mail:binhuaf@163.com
基金资助:
This work was supported by the NSFC Grants 10601021 and 11475073.

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

Binhua FENG¹, Dun ZHAO², Chunyou SUN²

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.

摘要/Abstract

摘要：

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 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 +φ₀|ψ|^αψ+iζ₀ψ= 0 with the same initial condition in L^q((0, T),B_r^s,2)for all admissible pairs (q, r), where T ∈ (0,T_max). We also show that if the dissipation coefficient ζ₀ large enough, then,ψ_ω is global if ω is sufficiently large and ψ_ω converges to ψ in L^q((0,∞),B_r^s,2), for all admissible pairs (q, r). In particular, our results hold for both subcritical and critical nonlinearities.

关键词: Nonlinear Schrö, dinger equation, averaged equation, global existence, convergence

Abstract:

Key words: Nonlinear Schrö, dinger equation, averaged equation, global existence, convergence

中图分类号:

35B27

冯斌华, 赵敦, 孙春友. HOMOGENIZATION FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH PERIODIC NONLINEARITY AND DISSIPATION IN FRACTIONAL ORDER SPACES[J]. 数学物理学报(英文版), 2015, 35(3): 567-582.

Binhua FENG, Dun ZHAO, Chunyou SUN. HOMOGENIZATION FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH PERIODIC NONLINEARITY AND DISSIPATION IN FRACTIONAL ORDER SPACES[J]. Acta mathematica scientia,Series B, 2015, 35(3): 567-582.

参考文献

[1] Centurion M, Porter M A, Kevrekidis P G, Psaltis D. Nonlinearity management in optics: experiment, theory, and simulation. Phys Rev Lett, 2006, 97: 033903
[2] Pelinovsky D E, Kevrekidis P G, Frantzeskakis D J. Averaging for solitons with nonlinearity management. Phys Rev Lett, 2003, 91: 240201
[3] Beheshti S, Law K J H, Kevrekidis P G, Mason A. Averaging of nonlinearity management with dissipation. Phys Rev A, 2008, 78: 025805
[4] Abdullaev F K, Caputo J G, Kraenkel R A, Malomed B A. Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length. Phys Rev A, 2003, 67: 012605
[5] Konotop V V, Pacciani P. Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose-Einstein condensates. Phys Rev Lett, 2005, 94: 240405
[6] Atre R, Panigrahi P K, Agarwal G S. Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation. Phys Rev E, 2006, 73: 056611
[7] Porsezian K, Ganapathy R, Hasegawa A, Serkin V N. Nonautonomous Soliton Dispersion Management. IEEE J Quantum Electron, 2009, 45: 1577-1583
[8] Serkin N V, Hasegawa A. Novel soliton solutions of the nonlinear Schrödinger equation model. Phys Rev Lett, 2000, 85: 4502-4505
[9] Cazenave T, Scialom M. A Schrödinger equation with time-oscillating nonlinearity. Rev Mat Univ Complut Madrid, 2010, 23: 321-339
[10] Feng B H, Zhao D, Sun C Y. The limit behavior of solutions for the nonlinear Schrödinger equation including nonlinear loss/gain with variable coefficient. J Math Anal Appl, 2013, 405: 240-251
[11] Feng B H, Zhao D, Sun C Y. The global existence and blow-up of solutions for the nonlinear Schrödinger equation with time-dependent linear loss/gain. J Math Anal Appl, 2014, 416: 901-923
[12] Antonelli P, Saut J, Sparber C. Well-Posedness and averaging of NLS with time-periodic dispersion management. Adv Differential Equ, 2013, 18: 49-68
[13] Cazenave T. Semilinear Schrödinger equations//Courant Lecture Notes in Mathematics, 10, NYU, CIMS, AMS, 2003
[14] Fang D Y, Han Z. A Schrödinger equation with time-oscillating critical nonlinearity. Nonlinear Anal, 2011, 14: 4698-4708
[15] Carvajal X, Panthee M, Scialom M. On the critical KdV equation with time-oscillating nonlinearity. Differential and Integral Equations, 2011, 24: 541-567
[16] Panthee M, Scialom M. On the supercritical KdV equation with time-oscillating nonlinearity. NoDEA Nonlinear Differential Equations Appl, 2013, 20: 1191-1212
[17] Ohta M, Todorova G. Remarks on global existence and blow-up for damped nonlinear Schrödinger equations. Discrete Contin Dyn Syst, 2009, 23: 1313-1325
[18] Tsutsumi M. Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations. SIAM J Math Anal, 1984, 15: 357-366
[19] Goubet O. Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in R². Adv Diff Eqns, 1998, 3: 337-360
[20] Goubet O. Asymptotic smoothing effect for a weakly damped nonlinear Schrödinger equation in T². J Diff Eqns, 2000, 165: 96-122
[21] Besse Ch, Carles R, Mauser N, Stimming H P. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Disc Cont Dyn Syst Ser B, 2008, 9: 11-36
[22] Tao T. Nonlinear dispersive equations: local and global analysis. CBMS Regional Conference Series in Mathematics, AMS, 2006
[23] Tao T, Visan M, Zhang X. The nonlinear Schrödinger equation with combined power-type nonlinearities. Comm Part Diff Eqns, 2007, 32: 1281-1343
[24] Kato T. On nonlinear Schrödinger equations. Ann Inst H Poincare Phys Theor, 1987, 46: 113-129
[25] Carles R. Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J Math Anal, 2003, 35: 823-843
[26] Bergh J, Löfström J. Interpolation Spaces. New York: Springer, 1976
[27] Cazenave T, Fang D Y, Han Z. Continuous dependence for NLS in fractional order spaces. Ann Inst H Poincaré Anal Non Linéaire, 2011, 28: 135-147
[28] Keel M, Tao T. Endpoint Strichartz inequalities. Amer J Math, 1998, 120: 955-980

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

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

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