Dynamics and stability of localized modes in nonlinear media with point defects

Bogdan, M. M.; Kovalev, A. S.; Gerasimchuk, I. V.

doi:10.1063/1.593346

Cited by 56 publications

(22 citation statements)

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“…These soliton profiles are similar to those discussed in the long-wave approximation in Refs. [14,15,22]. For α = 0 (homogeneous case), the circles denote the discretization of the continuum profile in accordance with Eq.…”

Section: Localized Modes and Their Stabilitymentioning

confidence: 99%

“…For the continuum problems, this interplay is known to lead to the existence of symmetric and asymmetric impurity modes and their interesting stability properties (see Refs. [14,15] and references therein). Here we concentrate on the study of the discrete systems such as the DNLS model.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, in the continuum approximation, a two-hump localized mode centered at the defect [22] is known to be unstable with respect to an exponential growth of antisymmetric linear perturbations [14,15]. The abovementioned instability leads to the motion of the localized mode away from the defect [15], and for small distances ξ between the mode center and the defect, the effective interaction energy can be presented in the form H int ≃ H 0 − Aξ 2 , where A is defined by the defect parameters.…”

Section: Introductionmentioning

confidence: 99%

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Instabilities and bifurcations of nonlinear impurity modes

2003

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We study the structure and stability of nonlinear impurity modes in the discrete nonlinear Schrödinger equation with a single on-site nonlinear impurity emphasizing the effects of interplay between discreteness, nonlinearity and disorder. We show how the interaction of a nonlinear localized mode (a discrete soliton or discrete breather) with a repulsive impurity generates a family of stationary states near the impurity site, as well as examine both theoretical and numerical criteria for the transition between different localized states via a cascade of bifurcations.

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Section: Localized Modes and Their Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Instabilities and bifurcations of nonlinear impurity modes

2003

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“…Another remark is that the unpinning instability is not connected with overdriving the chain; it occurs already in the undriven NLS [13]. q > 0: Numerical stability analysis.…”

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confidence: 99%

“…These two facts indicate that Q > 0-impurities should attract and trap solitons (cf. [13]). In the Q < 0 case, conversely, distant solitons should be repelled while an initially pinned soliton is expected to unpin and move away from the impurity regaining its cusp-free shape.…”

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confidence: 99%

Taming Spatiotemporal Chaos by Impurities in the Parametrically Driven Damped Nonlinear Schrodinger Equation

ALEXEEVA¹,

BARASHENKOV²,

TSIRONIS³

2001

JNMP

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Solitons of the parametrically driven, damped nonlinear Schrödinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the soliton on an "attractive" inhomogeneity expands its stability region whereas "repulsive" impurities produce an effective partitioning of the interval. We also show that attractive impurities may spontaneously nucleate solitons which subsequently remain pinned on these defects. A brief account of these results has appeared in patt-sol/9906001, where the interested reader can also find multicolor versions of the figures.Motivation. The ability to synchronize populations of coupled nonlinear oscillators would afford enormous technological benefits. A textbook example is provided by chains of Josephson junctions. A single junction can serve as an unparalleled source of ultrahighfrequency voltage oscillations; however, its industrial utilization was hindered by anomalously low power outputs. A natural way out would be to assemble a large array of coupled identical junctions, in anticipation that the coupling would force them to pulsate in unison. However -even if individual oscillators are nonchaotic -the synchronized regime may be unstable and evolve into a highly incoherent state, usually referred to as the spatio-temporal chaos.In an exciting twist of events, recent numerical simulations revealed that the introduction of slight uncorrelated differences between the oscillators may result in a significant improvement of the synchronization of the array [1,2,3]. The disorder was seen to suppress the chaos! In an attempt to gain a deeper insight into the nature of this counter-intuitive phenomenon, a numerical study of the effect of a single impurity on an otherwise homogeneous array was carried out [4]. Surprisingly, a single impurity was found to be sufficient to "tame" the chaotic behaviour and produce simple spatiotemporal patterns in very long chains.

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Localized states in the crystal characterized by the switching between self-focusing and defocusing Kerr type nonlinearities and the surface interaction

Савотченко

2021

Opt Quant Electron

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Dynamics and stability of localized modes in nonlinear media with point defects

Cited by 56 publications

References 0 publications

Instabilities and bifurcations of nonlinear impurity modes

Instabilities and bifurcations of nonlinear impurity modes

Taming Spatiotemporal Chaos by Impurities in the Parametrically Driven Damped Nonlinear Schrodinger Equation

Localized states in the crystal characterized by the switching between self-focusing and defocusing Kerr type nonlinearities and the surface interaction

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