摘要
主要考虑非自治薛定谔格点系统的拉回指数吸引子和一致指数吸引子的存在性以及它们的分形维数.首先,证明具时变耦合系数的薛定谔格点系统在依时间外力作用下的拉回指数吸引子的存在性;然后,证明拟周期外力驱动下的非自治薛定谔格点系统的一致指数吸引子的存在性。
In this paper, we mainly study the existence and fractal dimension of a pullback exponential attractor and a uniform exponential attractor for non-autonomous Schr?dinger lattice system. Firstly, we prove the existence of a pullback exponential attractor for the stochastic Schr?dinger lattice system with time-dependent coupled coefficients and forces.Then we prove the existence of a uniform exponential attractor for non-autonomous Schr?dinger lattice system driven by quasi-periodic external forces.
引文
[1]Chate H, Courbage M. Lattice systems. Physica D, 1997, 10:1-612
[2]Chow S. Lattice Dynamical Systems. In:Dynamical Systems, Springer-Verlag, 2003
[3]Bates P, Lisei H, Lu K N. Attractors for stochastic lattice dynamical systems. Stoch. Dyn., 2006,6(1):1-21
[4]Chen T, Zhou S F, Zhao C D. Attractors for discrete nonlinear Schr?dinger equation with delay. Acta Math. Appl. Sin. Engl. Ser., 2010, 26(4):633-642
[5]Han X Y, Shen W X, Zhou S F. Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differential Equations, 2011, 250(3):1235-1266
[6]Karachalios N, Yannacopoulos A. Global existence and compact attractors for the discrete nonlinear Schrodinger equation. J. Differential Equations, 2005, 217(1):88-123
[7]Zhao X Q, Zhou S F. Kernel sections for processes and nonautonomous lattice systems. Discrete Contin. Dyn. Syst. Ser. B, 2008, 9(3-4):763-785
[8]Zhou S F, Zhao C D, Liao X Y. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Commun. Pure Appl. Anal., 2007, 6(4):1087-1111
[9]Eden A, Foias C, Nicolaenko B, Temam R. Exponential attractors for dissipative evolution equations.Res. Appl. Math., 1996
[10]Fan X M, Yang H. Exponential attractor and its fractal dimension for a second order lattice dynamical system. J. Math. Anal. Appl., 2010, 367(2):350-359
[11]Abdallah A. Uniform exponential attractors for first order non-autonomous lattice dynamical systems.J. Differential Equations, 2011, 251(6):1489-1504
[12]Abdallah A. Uniform exponential attractors for non-autonomous Klein-Gordon-Schrodinger lattice systems in weighted spaces. Nonlinear Anal., 2015, 127:279-297
[13]Li X J, Wei K J, Zhang H Y. Exponential attractors for lattice dynamical systems in weighted spaces.Acta Appl. Math., 2011, 114(3):157-172
[14]赵才地,周盛凡.格点系统存在指数吸引子的充分条件及应用.数学学报,2010, 53(2):233-242(Zhao C D, Zhou S F. Sufficient Conditions for the Existence of Exponential Attractors for Lattice Systems and Applications. Acta Mathematica Sinica, 2010, 53(2):233-242)
[15]Zhou S F, Han X Y. Pullback exponential attractors for non-autonomous lattice systems. J. Dynam.Differential Equations, 2012, 24(3):601-631
[16]Zhou S F, Han X Y. Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces. Nonlinear Anal., 2013, 78(1):141-155
[17]周盛凡,谭慧荣.非线性薛定谔格点方程的指数吸引子.淅江师范大学学报:自然科学版,2015, 38(4):361-365(Zhou S F, Tan H R. Exponential attractor for nonlinear Schrodinger lattice equation. Journal of Zhejiang Normal University(Nat. Sci.), 2015, 38(4):361-365)