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:
[1]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes,, Nonlinear Anal., 72 (2010), 1967. doi: 10.1016/j.na.2009.09.037. Google Scholar

[2]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A gradient-like nonautonomous evolution process,, Int. J. of Bifurcation and Chaos 20, 9 (2010), 2751. doi: 10.1142/S0218127410027337. Google Scholar

[3]

A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and stable manifolds,, J. Diff. Equations, 233 (2007), 622. doi: 10.1016/j.jde.2006.08.009. Google Scholar

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[5]

D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides,, Opt. Lett. 18 (1988), 18 (1988), 794. doi: 10.1364/OL.13.000794. Google Scholar

[6]

S. Gersgorin, Über die Abgrenzung der Eigenwerte einer Matrix,, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, (1931), 74. Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Math. Soc., (1989). Google Scholar

[8]

P. Hartman, Ordinary Differential Equations,, SIAM, (2002). doi: 10.1137/1.9780898719222. Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981). Google Scholar

[10]

Y. V. Kartashov, V. V. Konotop and V. A. Visloukh, Two-dimensional dissipative solitons supported by localized gain,, Opt. Lett., 36 (2011), 82. doi: 10.1364/OL.36.000082. Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, American Math. Soc., (2011). Google Scholar

[12]

C. K. Lam, B. A. Malomed, K. W. Chow and P. K. A. Wai, Spatial solitons supported by localized gain in nonlinear optical waveguides,, Eur. Phys. J. Special Topics, 173 (2009), 233. doi: 10.1140/epjst/e2009-01076-8. Google Scholar

[13]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623. doi: 10.1088/0951-7715/7/6/006. Google Scholar

[14]

P. Panayotaros, Continuation of normal modes in finite NLS lattices,, Phys. Lett. A, 374 (2010), 3912. doi: 10.1016/j.physleta.2010.07.022. Google Scholar

[15]

P. Panayotaros and A. Aceves, Stabilization of coherent breathers in perturbed Hamiltonian coupled oscillators,, Phys. Lett. A, 375 (2011), 3964. doi: 10.1016/j.physleta.2011.09.019. Google Scholar

[16]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Euro. Math. Soc., (2004). doi: 10.4171/003. Google Scholar

[17]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems,, Lect. Notes Math. 1907, (1907). doi: 10.1007/978-3-540-71189-6. Google Scholar

[18]

M. O. Williams, C. W. McGrath and J. N. Kutz, Light-bullet routing and control with planar waveguide arrays,, Opt. Express, 18 (2010), 11671. doi: 10.1364/OE.18.011671. Google Scholar

[19]

D. V. Zezyulin and V. V. Konotop, Nonlinear modes in finite-dimensional PT-symmetric systems,, Phys. Rev. Lett, 108 (2012). doi: 10.1103/PhysRevLett.108.213906. Google Scholar

show all references

References:
[1]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes,, Nonlinear Anal., 72 (2010), 1967. doi: 10.1016/j.na.2009.09.037. Google Scholar

[2]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A gradient-like nonautonomous evolution process,, Int. J. of Bifurcation and Chaos 20, 9 (2010), 2751. doi: 10.1142/S0218127410027337. Google Scholar

[3]

A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and stable manifolds,, J. Diff. Equations, 233 (2007), 622. doi: 10.1016/j.jde.2006.08.009. Google Scholar

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[5]

D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides,, Opt. Lett. 18 (1988), 18 (1988), 794. doi: 10.1364/OL.13.000794. Google Scholar

[6]

S. Gersgorin, Über die Abgrenzung der Eigenwerte einer Matrix,, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, (1931), 74. Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Math. Soc., (1989). Google Scholar

[8]

P. Hartman, Ordinary Differential Equations,, SIAM, (2002). doi: 10.1137/1.9780898719222. Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981). Google Scholar

[10]

Y. V. Kartashov, V. V. Konotop and V. A. Visloukh, Two-dimensional dissipative solitons supported by localized gain,, Opt. Lett., 36 (2011), 82. doi: 10.1364/OL.36.000082. Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, American Math. Soc., (2011). Google Scholar

[12]

C. K. Lam, B. A. Malomed, K. W. Chow and P. K. A. Wai, Spatial solitons supported by localized gain in nonlinear optical waveguides,, Eur. Phys. J. Special Topics, 173 (2009), 233. doi: 10.1140/epjst/e2009-01076-8. Google Scholar

[13]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623. doi: 10.1088/0951-7715/7/6/006. Google Scholar

[14]

P. Panayotaros, Continuation of normal modes in finite NLS lattices,, Phys. Lett. A, 374 (2010), 3912. doi: 10.1016/j.physleta.2010.07.022. Google Scholar

[15]

P. Panayotaros and A. Aceves, Stabilization of coherent breathers in perturbed Hamiltonian coupled oscillators,, Phys. Lett. A, 375 (2011), 3964. doi: 10.1016/j.physleta.2011.09.019. Google Scholar

[16]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Euro. Math. Soc., (2004). doi: 10.4171/003. Google Scholar

[17]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems,, Lect. Notes Math. 1907, (1907). doi: 10.1007/978-3-540-71189-6. Google Scholar

[18]

M. O. Williams, C. W. McGrath and J. N. Kutz, Light-bullet routing and control with planar waveguide arrays,, Opt. Express, 18 (2010), 11671. doi: 10.1364/OE.18.011671. Google Scholar

[19]

D. V. Zezyulin and V. V. Konotop, Nonlinear modes in finite-dimensional PT-symmetric systems,, Phys. Rev. Lett, 108 (2012). doi: 10.1103/PhysRevLett.108.213906. Google Scholar

[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]

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

[6]

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

[7]

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

[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]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]