### 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 ψ_{n}(t)=φ_{n} exp(it) as homoclinic orbits of a four-dimensional symplectic map in the complex φ_{n},φ_{n+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, ψ_{n}
^{+} , and ψ_{n}
^{-} , 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 ψ_{n}
^{-} , 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 language | English |
---|---|

Pages (from-to) | 1195-1211 |

Number of pages | 17 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 60 |

Issue number | 2 B |

Publication status | Published - 1999 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*60*(2 B), 1195-1211.

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

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 60, no. 2 B, pp. 1195-1211.

}

TY - JOUR

T1 - Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation

AU - Kollmann, Michael

AU - Capel, Hans W.

AU - Bountis, Tassos

PY - 1999

Y1 - 1999

N2 - 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 φn,φn+1 space, for -∞n + , 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.

AB - 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 φn,φn+1 space, for -∞n + , 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.

UR - http://www.scopus.com/inward/record.url?scp=0000702171&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000702171&partnerID=8YFLogxK

M3 - Article

VL - 60

SP - 1195

EP - 1211

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 2 B

ER -