# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022056
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## Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise

 College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Shengfan Zhou

Celebrating the 80th Birthday of Professor Jibin Li

Received  December 2021 Revised  January 2022 Early access March 2022

Fund Project: The second author is supported by NSF of China grant 11871437

We mainly consider the existence of random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) defined on the space of infinite sequences with complex-valued components. Then we transfer the considered stochastic Schrödinger lattice system into a random system with random parameters and quasi-periodic forces without white noise through Ornstein-Uhlenbeck process, whose solutions generate a jointly continuous NRDS. Thirdly, we prove the existence of a uniform absorbing random set for this NRDS. Fourthly, we construct a bounded closed random set and verify the Lipschitz continuity of the NRDS on this random set, and next we decompose the difference between two solutions into a sum of two parts including some random variables with bounded expectation. Finally, we obtain the existence of a random uniform exponential attractor for the considered system.

Citation: Sijia Zhang, Shengfan Zhou. Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022056
##### References:
 [1] A. Y. Abdallah, Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7. [3] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [4] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031. [5] P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [6] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [7] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., 2002. [8] H. Cui and J. A. Langa, Uniform attractors for nonautonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018. [9] H. Cui and S. Zhou, Random attractor for Schrödinger lattice system with multiplicative white noise, J. Zhejiang Normal University (Nat. Sci.), 40 (2017), 17-23.  doi: 10.16218/j.issn.1001-5051.2017.01.003. [10] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Res. Appl. Math., 37 (1994). [11] X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860. [12] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018. [13] Z. Han and S. Zhou, Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh-Nagumo lattice systems with quasi-periodic forces and multiplicative noise, Stoch. Dynam., 20 (2020), 2050036, 38 pp. doi: 10.1142/S0219493720500367. [14] X. Jiang, S. Zhou and Z. Han, Random attractor for Schrö dinger lattice system with multiplicative white noise, J. Zhejiang Normal University (Nat. Sci.), 43 (2020), 251-258.  doi: 10.16218/j.issn.1001-5051.2020.03.002. [15] N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123.  doi: 10.1016/j.jde.2005.06.002. [16] G. P. Kevrekidis, O. K. Rasmussen and R. A. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Inter. J. Modern Phys. B, 15 (2001), 2833-2900.  doi: 10.1142/S0217979201007105. [17] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [18] B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070. [19] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015. [20] B. Wang and R. Wang, Asymptotic behavior of stochastic Schr ödinger lattice systems driven by nonlinear noise, Stoch. Anal. Appl., 38 (2020), 213-237.  doi: 10.1080/07362994.2019.1679646. [21] Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with multiplicative white noise, Disc. Conti. Dyna. Syst., 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210. [22] C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.  doi: 10.1016/j.jmaa.2008.12.036. [23] S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044. [24] S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces, Nonlinear Anal., 78 (2013), 141-155.  doi: 10.1016/j.na.2012.10.001. [25] S. Zhou, M. Zhao and H. Tan, Pullback and uniform exponential attractor for non-autonomous Schrödinger lattice equation, Acta Math. Appl. Sin., 42 (2019), 145-161.

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##### References:
 [1] A. Y. Abdallah, Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7. [3] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [4] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031. [5] P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [6] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020. [7] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., 2002. [8] H. Cui and J. A. Langa, Uniform attractors for nonautonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018. [9] H. Cui and S. Zhou, Random attractor for Schrödinger lattice system with multiplicative white noise, J. Zhejiang Normal University (Nat. Sci.), 40 (2017), 17-23.  doi: 10.16218/j.issn.1001-5051.2017.01.003. [10] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Res. Appl. Math., 37 (1994). [11] X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860. [12] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018. [13] Z. Han and S. Zhou, Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh-Nagumo lattice systems with quasi-periodic forces and multiplicative noise, Stoch. Dynam., 20 (2020), 2050036, 38 pp. doi: 10.1142/S0219493720500367. [14] X. Jiang, S. Zhou and Z. Han, Random attractor for Schrö dinger lattice system with multiplicative white noise, J. Zhejiang Normal University (Nat. Sci.), 43 (2020), 251-258.  doi: 10.16218/j.issn.1001-5051.2020.03.002. [15] N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123.  doi: 10.1016/j.jde.2005.06.002. [16] G. P. Kevrekidis, O. K. Rasmussen and R. A. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Inter. J. Modern Phys. B, 15 (2001), 2833-2900.  doi: 10.1142/S0217979201007105. [17] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [18] B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070. [19] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015. [20] B. Wang and R. Wang, Asymptotic behavior of stochastic Schr ödinger lattice systems driven by nonlinear noise, Stoch. Anal. Appl., 38 (2020), 213-237.  doi: 10.1080/07362994.2019.1679646. [21] Z. Wang and S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with multiplicative white noise, Disc. Conti. Dyna. Syst., 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210. [22] C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.  doi: 10.1016/j.jmaa.2008.12.036. [23] S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044. [24] S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces, Nonlinear Anal., 78 (2013), 141-155.  doi: 10.1016/j.na.2012.10.001. [25] S. Zhou, M. Zhao and H. Tan, Pullback and uniform exponential attractor for non-autonomous Schrödinger lattice equation, Acta Math. Appl. Sin., 42 (2019), 145-161.
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