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]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[2]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[3]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004

[4]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172

[5]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012

[6]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[7]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[8]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

[9]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364

[10]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[11]

Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020402

[12]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[13]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278

[14]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[15]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[16]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018

[17]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021023

[18]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[19]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039

[20]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]