自治耦合格点非线性 SchrÖdinger 方程组的一致吸引子及熵的估计

数学物理学报 ›› 2013, Vol. 33 ›› Issue (4): 636-645.

自治耦合格点非线性 SchrÖdinger 方程组的一致吸引子及熵的估计

杨新波|赵才地|贾晓琳

温州大学数学与信息科学学院浙江温州 325035

收稿日期:2012-02-07 修回日期:2013-05-15 出版日期:2013-08-25 发布日期:2013-08-25
基金资助:
国家自然科学基金(11271290)、国家973前期预研基金(2012CB426510)、温州大学科研基金(2008YYLQ01)和温州大学研究生创新基金(31606036010129)资助

The Uniform Attractor and Entropy of the Autonomous Coupled Nonlinear SchrÖdinger Equations on Infinite Lattices

YANG Xin-Bo, ZHAO Cai-De, JIA Xiao-Lin

Department of Mathematics and Information Sciences, Wenzhou University, Zhejiang Wenzhou 325035

Received:2012-02-07 Revised:2013-05-15 Online:2013-08-25 Published:2013-08-25
Supported by:
国家自然科学基金(11271290)、国家973前期预研基金(2012CB426510)、温州大学科研基金(2008YYLQ01)和温州大学研究生创新基金(31606036010129)资助

摘要/Abstract

摘要：

讨论一类自治耦合格点非线性 SchrÖdinger 方程组解的渐近行为. 证明该格点方程组在适当意义下存在一致吸引子, 并给出一致吸引子Kolmogorov-ε 熵的上界估计.

关键词: 格点系统, 自治耦合非线性SchrÖdinger 方程组, 一致吸引子, Kolmogorov-ε 熵

Abstract:

This paper studies the asymptotic behavior of solutions for a type of autonomous coupled nonlinear SchrÖdinger equations on infinite lattices. The authors prove the existence of the uniform attractor for this lattice systems in proper sense and give an upper bound of the Kolmogorov-ε entropy of the uniform attractor.

Key words: Lattice systems, Autonomous coupled nonlinear SchrÖdinger equations, Uniform attractors, Kolmogorov-ε entropy

中图分类号:

35B40

杨新波, 赵才地, 贾晓琳. 自治耦合格点非线性 SchrÖdinger 方程组的一致吸引子及熵的估计[J]. 数学物理学报, 2013, 33(4): 636-645.

YANG Xin-Bo, ZHAO Cai-De, JIA Xiao-Lin. The Uniform Attractor and Entropy of the Autonomous Coupled Nonlinear SchrÖdinger Equations on Infinite Lattices[J]. Acta mathematica scientia,Series A, 2013, 33(4): 636-645.

参考文献

[1] Chate H, Courbage M. Lattice systems. Physica D, 1997, 103: 1--612

[2] Chow S N. Lattice Dynamical Systems. Lecture Notes in Math, 1822. Berlin: Springer, 2003

[3] Keener J P. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J Appl Math, 1987, 47: 556--572

[4] Erneux T, Nicolis G. Propagating waves in discrete bistable reaction diffusion systems. Physica D, 1993, 67: 237--244

[5] Kapval R. Discrete models for chemically reacting systems. J Math Chem, 1991, 6: 113--163

[6] Chow S N, Mallet-Paret J. Pattern formation and spatial chaos in lattice dynamical systems. IEEE Trans Circuits Syst, 1995, 42: 746--751

[7] Fabiny L, Colet P, Roy R. Coherence and phase dynamics of spatially coupled solid-state lasers. Phys Rev A, 1993, 47: 4287--4296

[8] Hillert M. A solid-solution model for inhomogeneous systems. Acta Metall, 1961, 9: 525--535

[9] Chua L O, Roska T. The CNN paradigm. IEEE Trans Circuits Systems, 1993, 40: 147--156

[10] Chua L O, Yang Y. Cellular neural networks: theory. IEEE Trans Circuits Syst, 1988, 35: 1257--1272

[11] Chow S N, Mallet-Paret J, Van Vleck E S. Pattern formation and spatial chaos in spatially discrete evolution equations. Rand Compu Dyna, 1996, 4: 109--178

[12] Bates P W, Lu K, Wang B. Attractors for lattice dynamical systems. Inter J Bifur Choas, 2001, 11(1): 143--153

[13] Wang B. Dynamics of systems on infinite lattices. J Differential Equations, 2006, 221: 224--245

[14] Zhou S. Attractors for lattice systems corresponding to evolution equations. Nonlinearity, 2002, 15: 1079--1095

[15] Zhou S. Attractors for second order lattice dynamical systems. J Differential Equations, 2002, 179: 605--624

[16] Zhou S. Attractors for first order dissipative lattice dynamical systems. Physica D, 2003, 178: 51--61

[17] Zhou S. Attractors and approximations for lattice dynamical systems. J Differential Equations, 2004, 200: 342--368

[18] Zhou S, Shi W. Attractors and dimension of dissipative lattice systems. J Differential Equations, 2006, 224: 172--204

[19] Wang B. Asymptotic behavior of non-autonomous lattice systems. J Math Anal Appl, 2007, 331: 121--136

[20] Zhao X, Zhou S. Kernel sections for processes and nonautonomous lattice systems. Disc Cont Dyna Syst (B), 2008, 9: 763--785

[21] Zhao C, Zhou S. Compact kernel sections for nonautonomous Klein-Gordon-Schr\"odinger equations on infinite lattices. J Math Anal Appl, 2007, 332: 32--56

[22] Zhou S, Zhao C. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Comm Pure Appl Math, 2007, 21: 1087--1111

[23] Bate P W, Lisei H, Lu K. Attractors for stochastic lattice dynamical systems. Stoc Dyna, 2006, 6: 1--21

[24] Lv Y, Sun J H. Dynamical behavior for stochastic lattice systems. Chaos, Solitons & Fractals, 2006, 27: 1080--1090

[25] Zhao C, Zhou S. Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications.
J Math Anal Appl, 2009, 354: 78--95

[26] Han X, Shen W, Zhou S. Random attractors for stochastic lattice dynamical systems in weighted spaces. J Differential Equations, 2011, 250: 1235--1266

[27] Wang X, Li S, Xu D. Random attractors for second-order stochastic lattice dynamical systems. Nonl Analysis (Theory, Methods \& Applications),
2010, 72: 483--494

[28] Ahmed Y. Abdallah exponential attractors for first-order lattice dynamical systems. J Math Anal Appl, 2008, 339: 217--224

[29] Ahmed Y. Abdallah uniform exponential attractors for first order non-autonomous lattice dynamical systems. J Differential Equations, 2011, 251: 1489--1504

[30] Goubet O, Kechiche W. Uniform attractor for non-autonomous nonlinear Schr\"odinger equation. Comm Pure Appl Math, 2011, 10: 639--651

[31] Chepyzhov V V, Vishik M I. Attractors for Equations of Mathematical Physics. AMS Colloquium Publications, 49. Providence, RI: AMS, 2002

[32] Lorentz G, Golistschek M, Makovoz Y. Constructive Approximation, Advanced Problem. Funct Principles of Math Sciences. Berlin: Springer-Verlag, 1996