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

Multistability and localized attractors in a dissipative discrete NLS equation

Panayotis Panayotaros 1, and  Felipe Rivero 2,

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.
Keywords: Multistability, nonautonomous systems, pullback dynamics, hyperbolicity., NLS equation, attractors.
Mathematics Subject Classification: Primary: 34D45, 37D05, 37F15; Secondary: 37B5.
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), 1967-1976. 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), 2751-2760. 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), 622-653. 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, New York, 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), 794-796. 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), 74-754. Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc., Providence, 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, New York, 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), 82-84. doi: 10.1364/OL.36.000082.  Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc., Providence, 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), 233-243. 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), 1623-1643. 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), 3912-3919. 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), 3964-3972. 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, Springer, New York, 2007. 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), 11671-11682. 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), 213906. 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), 1967-1976. 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), 2751-2760. 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), 622-653. 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, New York, 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), 794-796. 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), 74-754. Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc., Providence, 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, New York, 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), 82-84. doi: 10.1364/OL.36.000082.  Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc., Providence, 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), 233-243. 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), 1623-1643. 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), 3912-3919. 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), 3964-3972. 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, Springer, New York, 2007. 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), 11671-11682. 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), 213906. doi: 10.1103/PhysRevLett.108.213906.  Google Scholar

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