Multistability and localized attractors in a dissipative discrete NLS equation

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	S. Gersgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, (1931), 74-754.
[7]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc., Providence, 1989.
[8]	P. Hartman, Ordinary Differential Equations, SIAM, 2002. doi: 10.1137/1.9780898719222.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.
[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.
[11]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc., Providence, 2011.
[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.
[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.
[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.
[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.
[16]	Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Euro. Math. Soc., 2004. doi: 10.4171/003.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[6]	S. Gersgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, (1931), 74-754.
[7]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Math. Soc., Providence, 1989.
[8]	P. Hartman, Ordinary Differential Equations, SIAM, 2002. doi: 10.1137/1.9780898719222.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.
[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.
[11]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc., Providence, 2011.
[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.
[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.
[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.
[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.
[16]	Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Euro. Math. Soc., 2004. doi: 10.4171/003.
[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.
[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.
[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.

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