格点量子Zakharov 方程组紧致核截面的存在性与熵的估计

数学物理学报 ›› 2014, Vol. 34 ›› Issue (5): 1203-1218.

格点量子Zakharov 方程组紧致核截面的存在性与熵的估计

梁芸芸¹, 李楚进², 赵才地^1**

1.温州大学数学与信息科学学院 |浙江温州 325035;
2.华中科技大学数学与统计学院武汉 430071

收稿日期:2013-06-14 修回日期:2013-12-06 出版日期:2014-10-25 发布日期:2014-10-25
通讯作者: 赵才地,lichujin@hust.edu.cn E-mail:zhaocaidi2013@163.com
基金资助:
国家自然科学基金(11271290)资助

Compact Kernel Sections and Kolmogorov Entropy for the Lattice Zakharov Equations with a Quantum Correction

LIANG Yun-Yun¹, LI Chu-Jin², ZHAO Cai-Di^1**

1.Department of Mathematics and Information Science, Wenzhou University, Zhejiang |Wenzhou 325035;
2.School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan |430071

Received:2013-06-14 Revised:2013-12-06 Online:2014-10-25 Published:2014-10-25
Contact: ZHAO Cai-Di,lichujin@hust.edu.cn E-mail:zhaocaidi2013@163.com
Supported by:
国家自然科学基金(11271290)资助

摘要/Abstract

摘要：

讨论一类非自治带量子纠正的修正 Zakharov 方程组在无穷格点上解的渐近行为. 应用截断函数的技巧证明该格点方程组
紧致核截面的存在性, 并给出核截面的Kolmogorov-$\varepsilon$ 熵的上界估计.

关键词: 格点系统, 带量子纠正的修正Zakharov 方程组, 核截面, Kolmogorov-&epsilon, 熵

Abstract:

This work discusses the asymptotic behavior of solutions of the modified Zakharov equations with a quantum correction
on infinite lattices. The authors use the technique of truncation function to derive the existence of compact kernel sections and give an upper bound for the Kolmogorov ε-entropy of the obtained compact kernel sections.

Key words: Lattice dynamical systems, Modified Zakharov equations with a quantum correction, Kernel sections, Kolmogorov ε-entropy

中图分类号:

35B40

梁芸芸, 李楚进, 赵才地. 格点量子Zakharov 方程组紧致核截面的存在性与熵的估计[J]. 数学物理学报, 2014, 34(5): 1203-1218.

LIANG Yun-Yun, LI Chu-Jin, HAO Cai-Di. Compact Kernel Sections and Kolmogorov Entropy for the Lattice Zakharov Equations with a Quantum Correction[J]. Acta mathematica scientia,Series A, 2014, 34(5): 1203-1218.

参考文献

[1] Hale J. Asymptotic Behavior of Dissipative Systems. Province, RI: AMS, 1988

[2] Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics.2nd ed.Berlin: Springer, 1997

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

[4] Sell G, You Y. Dynamics of Evolutionary Equations. New York: Springer, 2002

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

[6] Chow S N. Lattice dynamical systems//Dynamical Systems, Lecture Notes in Math, 1822. Berlin: Springer, 2003, 1-102

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

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

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

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

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

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

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

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

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

[16] Zhou S, Han X. Pullback exponential attractors for non-autonomous lattice systems. J Dyna Differential Equations,
2012, 24(3): 601--631

[17] Carrol T L, Pecora L M. Synchronization in chaotic systems. Phys Rev Lett, 1990, 64: 821--824

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

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

[20] 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

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

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

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

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

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

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

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

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

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

[30] 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

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

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

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

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

[35] Zakharov V E. Collapse of Langmuir waves. Zh Eksp Teog Fiz, 1972, 62: 1745--1751

[36] Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor Equations. Vienna: Springer, 1990

[37] Jung Y D. Quantum-mechanical effects on electron-electron scattering in dense high-temperature plasmas. Phys Plasmas, 2001, 8: 3842--3844

[38] Kremp D, Bornath Th, Bonitz M, Schlanges M. Quantum kinetic theory of plasmas in strong laser fields. Phys Rev E, 1990, 60: 4725--4732

[39] Garcia L G, Haas F, De Oliveira L P L, Goedert J. Modified Zakharov equations for plasmas with a quantum correcion.
Phys Plasma, 2005, 12: 012302-8

[40] Flahaut I. Attractors for the dissipative Zakharov system. Nonlinear Anal, 1991, 16: 599--633

[41] Fang S, Guo C, Guo B. Exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction. Acta Math Sci, 2012, 32B(3): 1073--1082

[42] Fang S M, Jin L Y, Guo B L. Existence of weak solution for quantum Zakharov equations for plasmas mode. Appl Math Mech, 2011, 32: 1339--1344

[43] Guo C, Fang S, Guo B. Long time behavior of the solutions for the dissipative modified Zakharov equations for plasmas with a quantum correction. J Math Anal Appl, 2013, 403: 183--192

[44] Yin F, Zhou S, Ou Yang Z, Xiao C. Attractor for lattice system of dissipative Zakharov equation. Acta Math Sinica, 2009, 25: 321--342

[45] 郭艳凤, 郭柏灵, 李栋龙. 耗散的量子Zakharov方程解的渐进性行为. 应用数学和力学, 2012, 33(4): 486--499

[46] Zhao C, Zhou S. Compact Kernel sections for nonautonomous Klein-Gordon-Schrodinger equations. J Math Anal Appl, 2007, 332: 32--56

[47] Yang X, Zhao C, Cao Juan. Dynamics of the discrete coupled nonlinear Schrodinger-Boussinesq equations. Math Comp Appl, 2013, 219: 8508--8524