Sine-Gordon kinks on a discrete lattice. II. Static properties

Stancioff, P.; Willis, C. R.; El‐Batanouny, M.; Burdick, S.

doi:10.1103/physrevb.33.1912

Cited by 51 publications

(11 citation statements)

References 14 publications

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“…Thus a Klein-Gordon kink when boosted can only radiate some energy away and finally stop to become a static kink solution! In order to account for these effects a collective coordinate approach was developed [195], [165], [28], [194], [29], [193], [57], [30]. Within this approach one performs a canonical transformation to new coordinates some of whom are collective (nonlocal in the old coordinates).…”

Section: Movabilitymentioning

confidence: 99%

Discrete breathers

1998

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Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices -quantum breathers.Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.

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Section: Movabilitymentioning

confidence: 99%

Discrete breathers

1998

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“…Of particular relevance to our work are the more recent articles of Boesch, Willis and coworkers [59,54,8,9,11,10]. In this work the discrete sine-Gordon equation is solved with kink-like initial data on the lattice.…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that there are two static kink solutions, a high energy one centered on a lattice site and a low energy one centered between two consecutive lattice sites [48,27,15,59,54,8,9,11,7,34,30,31]. The low energy kink corresponds to the minimizer of the static Hamiltonian energy, h[u], displayed in (2.10).…”

Section: The Sine-gordon Equationmentioning

confidence: 99%

Dynamics of lattice kinks

Kevrekidis

Weinstein

2000

Physica D: Nonlinear Phenomena

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We consider a class of Hamiltonian nonlinear wave equations governing a field defined on a spatially discrete one dimensional lattice, with discreteness parameter, d = h −1 , where h > 0 is the lattice spacing. The specific cases we consider in detail are the discrete sine-Gordon (SG) and discrete φ 4 models. For finite d and in the continuum limit (d → ∞) these equations have static kink-like (heteroclinic) states which are stable. In contrast to the continuum case, due to the breaking of Lorentz invariance, discrete kinks cannot be "Lorentz boosted" to obtain traveling discrete kinks. Peyrard and Kruskal pioneered the study of how a kink, initially propagating in the lattice dynamically adjusts in the absence of an available family of traveling kinks. We study in detail the final stages of the discrete kink's evolution during which it is pinned to a specified lattice site (equilibrium position in the Peierls-Nabarro barrier). We find: (i) for d sufficiently large (sufficiently small lattice spacing), the state of the system approaches an asymptotically stable ground state static kink (centered between lattice sites).(ii) for d sufficiently small d < d * the static kink bifurcates to one or more time periodic states. For the discrete φ 4 we have: wobbling kinks which have the same spatial symmetry as the static kink as well as "g-wobblers" and "e-wobblers", which have different spatial symmetry. In the discrete sine-Gordon case, the "ewobbler" has the spatial symmetry of the kink whereas the "g-wobbler" has the opposite one. These time-periodic states may be regarded as a class of discrete breather / topological defect states; they are spatially localized and time periodic oscillations mounted on a static kink background.The large time limit of solutions with initial data near a kink is marked by damped oscillation about one of these two types of asymptotic states. In case (i) we compute the characteristics of the damped oscillation (frequency and d-dependent rate of decay). In case (ii) we prove the existence of, and give analytical and numerical evidence for the asymptotic stability of wobbling solutions. * Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA † Mathematical Sciences Research, Bell Laboratories -Lucent Technologies, and Department of Mathematics, University of Michigan -Ann Arbor, 600 Mountain Avenue, Murray Hill, NJ 07974-0636 USA 1The mechanism for decay is the radiation of excess energy, stored in internal modes, away from the kink core to infinity. This process is studied in detail using general techniques of scattering theory and normal forms. In particular, we derive a dispersive normal form, from which one can anticipate the character of the dynamics. The methods we use are very general and are appropriate for the study of dynamical systems which may be viewed as a system of discrete oscillators (e.g. kink together with its internal modes) coupled to a field (e.g. dispersive radiation or phonons). The approach is based on and extends an approach...

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“…Fractals can give rise to very new and striking phenomena, like superlocalization (Levy and Souillard, 1987) or anomalous diffusion (Gefen et al, 1983; see also the review by Havlin and Ben-Avraham, 1987). Besides, the recent introduction of multifractals (see the review article by Stanley and Meakin, 1988, and references therein) is opening a lot of new fields to research and providing new tools to classify objects and process. We are not going to talk about quasicrystals and fractals in the rest of this the paper, because, to our knowledge, nonlinear problems have not been studied when perturbed by or happening in such substrates, or potentials, or whatever; we just pointed them out as the first open question we mention, a very fascinating subject that will certainly be important in the near future.…”

Section: Models Of Disordermentioning

confidence: 99%

Nonlinear Wave Propagation in Disordered Media

Sánchez

Vázquez

1991

Int. J. Mod. Phys. B

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We briefly review the state-of-the-art of research on nonlinear wave propagation in disordered media. The paper is intended to provide the non-specialist reader with a flavor of this active field of physics. Firstly, a general introduction to the subject is made. We describe the basic models and the ways to study disorder in connection with them. Secondly, analytical and numerical techniques suitable for this purpose are outlined. We summarize their features and comment on their respective advantages, drawbacks and applicability conditions. Thirdly, the Nonlinear Klein-Gordon and Schrbdinger equations are chosen as specific examples. We collect a number of results that are representative of the phenomena arising from the competition between nonlinearity and disorder. The review is concluded with some remarks on open questions, main current trends and possible further developments.

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Sine-Gordon kinks on a discrete lattice. II. Static properties

Cited by 51 publications

References 14 publications

Discrete breathers

Discrete breathers

Dynamics of lattice kinks

Nonlinear Wave Propagation in Disordered Media

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