Stability of Discrete Solitons and Quasicollapse to Intrinsically Localized Modes

Laedke, E. W.; Spatschek, K. H.; Turitsyn, Sergei K.

doi:10.1103/physrevlett.73.1055

Cited by 117 publications

(110 citation statements)

References 16 publications

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“…proving linear stability of the breather [15] as well as Lyapunov stability for norm-conserving perturbations [14].…”

mentioning

confidence: 99%

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Johansson¹

2001

Phys. Rev. E

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We consider the long-time evolution of weakly perturbed discrete nonlinear Schrödinger breathers. While breather growth can occur through nonlinear interaction with one single initial linear mode, breather decay is found to require excitation of at least two independent modes. All growth and decay processes of lowest order are found to disappear for breathers larger than a threshold value.PACS numbers: 63.20. Pw, 45.05.+x, 42.65.Tg The study of intrinsically localized modes in anharmonic lattices, discrete breathers, has yielded much attention during the last decade (see e.g. [1,2]). In particular, their existence as time-periodic solutions of nonlinear lattice-equations was proven under quite general conditions [3], and numerical schemes were developed for their explicit calculation [4]. The generality of the concept of discrete breathers, which provide very efficient means to localize energy, has lead to numerous suggestions to its application in contexts where anharmonicity and discreteness are important, e.g. for describing energy and charge transport and storage in biological macromolecules [5]. Recently, discrete breathers have been experimentally observed in coupled optical waveguides [6], in charge-density wave systems [7], in magnetic systems [8], and in arrays of coupled Josephson junctions [9].Although a discrete breather under quite general conditions is linearly stable [1,3,10] (and thus no perturbations grow exponentially), there are many questions remaining concerning the long-time fate of perturbed breathers. In a previous paper [11], some of these questions were addressed considering a particular model, the discrete nonlinear Schrödinger (DNLS) equation. The interaction between stationary breathers and small perturbations corresponding to time-periodic eigensolutions to the linearized equations of motion around the breather was investigated using a multiscale perturbational approach, and it was found that the nonlinear interaction between the breather and single-mode small-amplitude perturbations could lead to breather growth through generation of radiating higher harmonics, but not to breather decay. It is the purpose of this Report to extend these results to more general perturbations of stationary DNLS breathers. In particular, we find that while the simplest growth process can be described as an inelastic scattering process with a one-frequency incoming wave and an additional outgoing higher-harmonic wave, the description of a decay process requires at least two incoming modes yielding outgoing modes with frequencies being linear combinations of the original ones.In order to make this Report self-contained, we first recapitulate the main formalism from [11] (to which the reader is referred for details; see also [12] for a similar approach in continuous NLS models). With canonical conjugated variables {iψ n }, {ψ * n }, the DNLS equation can be derived from the Hamiltonian H ({iψ n } , {ψ * n }) = n

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“…proving linear stability of the breather [15] as well as Lyapunov stability for norm-conserving perturbations [14].…”

mentioning

confidence: 99%

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Johansson¹

2001

Phys. Rev. E

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“…The Vakhitov-Kolokolov criterion was shown also to hold for discrete space-time solitons that exist in Kerr nonlinear media [22,23]. Moreover, the unstable odd cubic continuous-discrete solitons can display collapse-type instabilities, a reminiscent feature of the two-dimensional stationary solutions of nonlinear Schrödinger equation, while the unstable quadratic discrete space-time odd solitons do not display this type of instability [24].…”

Section: Stability Analysismentioning

confidence: 99%

“…The existence of STS's in quadratic nonlinear materials was theoretically predicted [17] and thereafter experimentally realized in a two-dimensional geometry involving one temporal and one spatial coordinate [18]. The existence and properties of continuous-discrete spatiotemporal solitons has been extensively investigated in cubic nonlinear media and stable odd solitons have been shown to exist [19,20,21,22,23]. It was shown that the cubic weakly-coupled waveguide arrays act as collapse compressors [19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal discrete multicolor solitons

Kartashov

Crasovan

et al. 2004

Phys. Rev. E

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We have found various families of two-dimensional spatiotemporal solitons in quadratically nonlinear waveguide arrays. The families of unstaggered odd, even and twisted stationary solutions are thoroughly characterized and their stability against perturbations is investigated. We show that the twisted and even solutions display instability, while most of the odd solitons show remarkable stability upon evolution.

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“…1. It has previously been shown [20][21][22] that the linear stability of the stationary states in the discrete case is determined by the Vakhitov-Kolokolov criterion, 23 yielding linear stability whenever dN/d⌳Ͼ0. This, together with the solid curve in Fig.…”

Section: ͑6͒mentioning

confidence: 99%

Stationary states of the two-dimensional nonlinear Schrödinger model with disorder

et al. 1998

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Solitonlike excitations in the presence of disorder in the two-dimensional cubic nonlinear Schrödinger equation are analyzed. The continuum as well as the discrete problem are analyzed. In the continuum model, otherwise unstable excitations are stabilized in the presence of disorder. In the discrete model, the disorder is found to leave the narrow excitations unaffected. Our results suggest that the disorder provides a possibility to control the spatial extent of the stable excitations in the continuum system. ͓S0163-1829͑98͒06429-7͔

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Stability of Discrete Solitons and Quasicollapse to Intrinsically Localized Modes

Cited by 117 publications

References 16 publications

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Spatiotemporal discrete multicolor solitons

Stationary states of the two-dimensional nonlinear Schrödinger model with disorder

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