Exact determination of the Peierls-Nabarro frequency

Boesch, R.; Willis, C. R.

doi:10.1103/physrevb.39.361

Cited by 56 publications

(21 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If these length scales are very different from each other the perturbed dynamics can support solitonlike or breatherlike excitations. The motion of these excitations can be described by a collective variable approach [3,4]. On the other hand, if the length scales are comparable localized excitations break up or dissipate into radiation even for relatively small strength of the perturbation.…”

Section: Theoretical Division and Center For Nonlinear Studies Los Amentioning

confidence: 99%

Sine-Gordon kink-antikink generation on spatially periodic potentials

et al. 1992

View full text Add to dashboard Cite

A spatially periodic perturbation can lead to a breakup of large-amplitude sine-Gordon breathers into kink and anti kink solutions, each oscillating around a minimum of the perturbing potential. This behavior can be understood by studying the effective potential experienced by the breather (bound kink-antikink) or the (free) kink-antikink solution as long as kink and antikink are sufficiently far apart. The resulting kinks and antikinks move independently and nearly radiationlessly in the presence of the perturbation and can travel arbitrarily far for sufficiently large initial kinetic energy. Upon interacting with each other they are strongly affected by the perturbation, lose energy by radiating, and can end in a bound state having the character of a distorted breather.PACS numbers: 03.40. Kf, 61.70.Ga Adding spatial disorder to completely integrable nonlinear dynamics like the one governed by the nonlinear Schrodinger (NLS) equation or the sine-Gordon (SG) equation leads to a variety of novel effects having practical relevance [1]. Competition of the length scales introduced by the perturbation and by the nonlinearity is one example we have studied recently [2]. If these length scales are very different from each other the perturbed dynamics can support solitonlike or breatherlike excitations. The motion of these excitations can be described by a collective variable approach [3,4]. On the other hand, if the length scales are comparable localized excitations break up or dissipate into radiation even for relatively small strength of the perturbation. This behavior has been observed in the case of the SG equation[2] as well as for the NLS equation [5].Another class of effects induced by spatially periodic perturbations can be observed in dynamics which allow for topological solitons like the SG equation. Although very many perturbed SG problems have been considered in the literature (see, e.g., [6,7] for reviews), to our best knowledge, only Mkrtchyan and Shmidt [8] and Malomed and Tribelsky [9] have studied the motion of SG kinks under the influence of a cosine potential. In this case kinks (and antikinks) seem to move radiationlessly below a certain velocity threshold. The question arises whether kinks and anti kinks also interact radiationlessly under these circumstances. Finally it is interesting to see under which conditions a breather can break up into a kinkantikink (f{-f{) pair under the perturbation [2].In this Rapid Communication we address the question of how large-amplitude SG breathers behave under the influence of spatially periodic potentials. These potentials include the effects of discreteness as a particular case; they also can be relevant to understand the interaction of domain wall excitations in discrete solid-state and materials science problems governed by SG-like equations of motion, for example, dislocations in a crystal, walls in ferroelectrics or ferromagnets, or discommensurations in superlattices. A breather can be interpreted as a bound f{-f{ state (see a detailed account of the ...

show abstract

Section: Theoretical Division and Center For Nonlinear Studies Los Amentioning

confidence: 99%

Sine-Gordon kink-antikink generation on spatially periodic potentials

et al. 1992

View full text Add to dashboard Cite

show abstract

“…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%

Dynamics of lattice kinks

Kevrekidis

Weinstein

2000

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

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...

show abstract

“…A different procedure, which has the advantage of its easy physical interpretation, is the already mentioned one of the collective coordinates (see [14] for a very recent account on these techniques). We will show that they indeed provide some useful information, without overly involved algebra.…”

mentioning

confidence: 99%

Sine-Gordon breathers on spatially periodic potentials

et al. 1992

View full text Add to dashboard Cite

We have carried out an extensive simulation program to study the behavior of sine-Gordon breathers initially at rest in the presence of perturbations that are periodic in space and constant in time. We report here a number of different observed phenomena and the range of the relevant parameters (the ratio of the breather width to the perturbation wavelength and the perturbation magnitude) for which each one of them occurs. We also propose some qualitative explanations valid for certain regimes. PACS number(s): 03.40.KfThe study of nonlinear disordered systems has recently become the subject of a great deal of research [1]. A model that has been widely used as an approximate description of quasi-one-dimensional physical phenomena is the sine-Gordoll (SG) equation (see, for instance, [2] and references therein). The effects of a large variety of perturbations on the properties of that equation have been investigated (see [3,4] for recent reviews). However, starting from the seminal work by Fogel et al. [5], only a few papers have been devoted to disordered systems, corresponding to the addition of certain properly chosen inhomogeneous terms to the original equation. In particular, a simple form of inhomogeneity, which to some extent is amenable to analytical work, is a spatially periodic, external potential. The propagation of SG kinks on such a potential has been previously studied by Mkrtchyan and Shmidt [6] and Malomed and Tribelsky [7], and the same.problem has been analyzed as the limiting case of a spatial lattice of impurities when this lattice becomes very dense [8].. In this work we concern ourselves with the influence of these periodic potentials on the SG breathers. Some preliminary work has been done by Pascual [9]. We introduce this issue as an excellent example of the competition between two characteristic length scales, which is an ubiquitous phenomenon in real physical systems: In this case, these length scales are the breather width (AB) and the perturbation period (Ap). The aim of this work is twofold. First, our immediate goal is to learn whether the particle picture of solitons, often very useful, holds or not, and, if not, when and how it breaks down due to this perturbation; second, our ultimate purpose is to apply our results to nonlinear wave propagation in potentials which are stochastic in space (this problem has been considered in [10]), viewing the periodic potential as a particular component or "color" of the noisy one. As is explained below, understanding one color is a fundamental input to treating propagation in a general color distribution. On the other hand, we are also interested in comparing the phenomena we are describing here to those caused by the same kind of potential on the nonlinear Schrodinger (NLS) equation, often viewed as a weak 45 nonlinearity limit of the SG equation. Our investigations on this related system are planned to be reported elsewhere [11].To approach the above issues, we have carried out a number of numerical simulations spanning large intervals of the p...

show abstract

Exact determination of the Peierls-Nabarro frequency

Cited by 56 publications

References 11 publications

Sine-Gordon kink-antikink generation on spatially periodic potentials

Sine-Gordon kink-antikink generation on spatially periodic potentials

Dynamics of lattice kinks

Sine-Gordon breathers on spatially periodic potentials

Contact Info

Product

Resources

About