# American Institute of Mathematical Sciences

June  2014, 19(4): 1137-1154. doi: 10.3934/dcdsb.2014.19.1137

## Multistability and localized attractors in a dissipative discrete NLS equation

 1 Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F. 2 Departamento de Matemática y Mecánica, I.I.M.A.S - U.N.A.M., Apdo. Postal 20-726, 01000 México D. F.

Received  June 2013 Revised  December 2013 Published  April 2014

We consider a finite discrete nonlinear Schrödinger equation with localized forcing, damping, and nonautonomous perturbations. In the autonomous case these systems are shown numerically to have multiple attracting spatially localized solutions. In the nonautonomous case we study analytically some properties of the pullback attractor of the system, assuming that the origin of the corresponding autonomous system is hyberbolic. We also see numerically the persistence of multiple localized attracting states under different types of nonautonomous perturbations.
Citation: Panayotis Panayotaros, Felipe Rivero. Multistability and localized attractors in a dissipative discrete NLS equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1137-1154. doi: 10.3934/dcdsb.2014.19.1137
##### References:

show all references

##### References:
 [1] Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 [2] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579 [3] Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011 [4] Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749 [5] Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189 [6] Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855 [7] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215 [8] Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473 [9] Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727 [10] David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96 [11] Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326 [12] Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214 [13] Luís Silva. Periodic attractors of nonautonomous flat-topped tent systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1867-1874. doi: 10.3934/dcdsb.2018243 [14] José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329 [15] Tomás Caraballo, Stefanie Sonner. Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6383-6403. doi: 10.3934/dcds.2017277 [16] Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116 [17] Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023 [18] Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991 [19] Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801 [20] Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

2018 Impact Factor: 1.008