Soliton complexity in the damped-driven nonlinear Schrödinger equation: Stationary to periodic to quasiperiodic complexes

Barashenkov, I. V.; Zemlyanaya, E. V.

doi:10.1103/physreve.83.056610

Cited by 31 publications

(19 citation statements)

References 65 publications

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“…Hopf bifurcations and the corresponding periodic solitons were reported and discussed therein. The connection between the results presented in this work, which correspond to weakly coupled lattices, and those of [24,25,26] are proposed for future study.…”

Section: Resultsmentioning

confidence: 73%

“…Following [24,25,26], we assume that both the damping coefficient α and the driving strength γ are positive. For the coupling constant ε, we also set it to be positive (the case ε < 0 can be obtained accordingly by the so-called staggering transformation u n → (−1) n u n and Λ → (Λ − 4ε)).…”

Section: Analytical Formulationmentioning

confidence: 99%

“…The lower and upper horizontal dotted lines correspond to Eqs. (31) and(26), respectively (see text).…”

mentioning

confidence: 99%

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Solitons in a Parametrically Driven Damped Discrete Nonlinear Schrödinger Equation

Syafwan

Susanto

Cox

2012

Progress in Optical Science and Photonics

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We consider a parametrically driven damped discrete nonlinear Schrödinger (PDDNLS) equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental discrete bright solitons. We show that there are two types of onsite discrete soliton, namely onsite type I and II. We also show that there are four types of intersite discrete soliton, called intersite type I, II, III, and IV, where the last two types are essentially the same, due to symmetry. Onsite and intersite type I solitons, which can be unstable in the case of no dissipation, are found to be stabilized by the damping, whereas the other types are always unstable. Our further analysis demonstrates that saddlenode and pitchfork (symmetry-breaking) bifurcations can occur. More interestingly, the onsite type I, intersite type I, and intersite type III-IV admit Hopf bifurcations from which emerge periodic solitons (limit cycles). The continuation of the limit cycles as well as the stability of the periodic solitons are computed through the numerical continuation software Matcont. We observe subcritical Hopf bifurcations along the existence curve of the onsite type I and intersite type III-IV. Along the existence curve of the intersite type I we observe both supercritical and subcritical Hopf bifurcations.

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Section: Resultsmentioning

confidence: 73%

Section: Analytical Formulationmentioning

confidence: 99%

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Solitons in a Parametrically Driven Damped Discrete Nonlinear Schrödinger Equation

Syafwan

Susanto

Cox

2012

Progress in Optical Science and Photonics

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“…Physical systems exhibiting parity-time (PT) symmetry have been the subject of much investigation in recent years and are now extensively considered in diverse areas of physics, namely, quantum field theories, non-Hermitian Anderson models, and complex Lie algebras, just to name a few [1][2][3][4][5][6][7][8][9][10]. We know that even a single PT cell can exhibit unconventional features; it follows that one may wish to investigate what new behaviour and properties can be expected from PT symmetric lattices [11,12].…”

Section: Introductionmentioning

confidence: 99%

On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions

Masemola

Kara

2014

Journal of Applied Mathematics

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An analysis of a PT symmetric coupler with “gain in one waveguide and loss in another” is made; a transformation in the PT system and some assumptions results in a scalar cubic Schrödinger equation. We investigate the relationship between the conservation laws and Lie symmetries and investigate a Lagrangian, corresponding Noether symmetries, conserved vectors, and exact solutions via “double reductions.”

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“…Thus the two-pulse solution is unstable for 1−p C < 0 and stable for 1−p C > 0 and C = C − , and this is opposite to the stability of the single-pulse solution, at least if we do not consider the possibility of the oscillatory instability. Note that multipulse solitons were studied formerly for the parametrically driven [5,6] and ac-driven [4] damped nonlinear Schrödinger equation.…”

Section: Two-pulse Solutionmentioning

confidence: 99%

Low-Dimensional Description of Pulses under the Action of Global Feedback Control

Kanevsky

Nepomnyashchy

2012

Math. Model. Nat. Phenom.

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Abstract. The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem.

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Soliton complexity in the damped-driven nonlinear Schrödinger equation: Stationary to periodic to quasiperiodic complexes

Abstract: Stationary and oscillatory bound states, or complexes, of the damped-driven solitons are numerically path-followed in the parameter space. We compile a chart of the two-soliton attractors, complementing the one-soliton attractor chart.

Cited by 31 publications

References 65 publications

Solitons in a Parametrically Driven Damped Discrete Nonlinear Schrödinger Equation

Solitons in a Parametrically Driven Damped Discrete Nonlinear Schrödinger Equation

On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions

Low-Dimensional Description of Pulses under the Action of Global Feedback Control

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