Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation

Michael Kollmann, Hans W. Capel, Tassos Bountis

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We study an integrable discretization of the nonlinear Schrödinger equation (NLS) under the effects of damping and periodic driving, from the point of view of spatially localized solutions oscillating in time with the driver's frequency. We locate the equilibrium states of the discretized (DNLS) system in the plane of its dissipation Γ and forcing amplitude H parameters and use a shooting algorithm to construct the desired solutions ψn(t)=φn exp(it) as homoclinic orbits of a four-dimensional symplectic map in the complex φnn+1 space, for -∞<n<∞. We derive, in the Γ=0 case, closed form expressions for two fundamental such solutions having a single hump in n, ψn + , and ψn - , and determine analytically their threshold of existence in the (Γ,H) plane using Mel'nikov's theory. Then, we demonstrate numerically that above this threshold a remarkable variety of multihump structures appear, whose complexity in terms of their spatial extrema grows with increasing H. All these solutions are numerically found to be unstable in time, except for ψn - , which is seen to be stable over a certain region in the (Γ,H) plane. In the continuum limit our results are in close agreement with recent studies on the NLS equation. From a more general perspective, we view these DNLS multihump solutions as homoclinic orbits of a higher-dimensional map thereby providing a possible mechanism for explaining the occurrence of similar structures called discrete (multi-) breathers found in a wide variety of one-dimensional nonlinear lattices.

Original languageEnglish
Pages (from-to)1195-1211
Number of pages17
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume60
Issue number2 B
Publication statusPublished - 1999
Externally publishedYes

Fingerprint

Breathers
Discrete Equations
Damped
nonlinear equations
Nonlinear Equations
Homoclinic Orbit
Oscillating Solutions
Nonlinear Lattice
Shooting
Continuum Limit
Extremum
Fundamental Solution
orbits
Equilibrium State
Forcing
thresholds
Driver
Dissipation
Damping
Closed-form

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics

Cite this

Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation. / Kollmann, Michael; Capel, Hans W.; Bountis, Tassos.

In: Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Vol. 60, No. 2 B, 1999, p. 1195-1211.

Research output: Contribution to journalArticle

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