Solitons on lattices

Duncan, Dugald B.; Eilbeck, J. C.; Feddersen, Henrik; Wattis, Jonathan A. D.

doi:10.1016/0167-2789(93)90020-2

Cited by 117 publications

(104 citation statements)

References 20 publications

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“…To find the whole class of soliton solutions as well as to calculate those values of velocity s at which such solutions exist, we use the pseudospectral method developed by Eilbeck and Flesch 8 and Duncan et al 9 Since we look for traveling-wave solutions, r n ( )ϭr(z), zϭnaϪs , the set of Eqs. ͑A1͒ is reduced to…”

Section: Discussionmentioning

confidence: 99%

“…We start in this paper with the 1D case and afterwards pass to the 2D lattice. Because the lattice solitons are essentially discrete objects and the techniques of the continuum analysis are not valid anymore, we find the soliton solutions by applying the pseudospectral method suggested and developed previously by Eilbeck and Flesch 8 and Duncan et al 9 This method allows us to find numerically the traveling-wave solutions for all admissible velocities, the profile of which can be obtained with any given accuracy. In dependence on the direction of solitary wave propagation, we show that in the 2D lattice the solitary plane wave has either a finite or infinite interval of supersonic velocities.…”

Section: Introductionmentioning

confidence: 99%

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Solitary plane waves in an isotropic hexagonal lattice

1998

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Solitary plane-wave solutions in a two-dimensional hexagonal lattice which can propagate in different directions on the plane are found by using the pseudospectral method. The main point of our studies is that the lattice model is isotropic and we show that the sound velocity is the same for different directions of wave propagation. The pseudospectral method allows us to obtain solitary wave solutions with very narrow profile, the thickness of which may contain a few atoms or even less than one lattice spacing ͑i.e., essentially discrete solutions͒. Since these nonlinear waves are quite narrow, details of lattice microstructure appear to be important for their motion. Particularly, the regime of their propagation qualitatively depends on whether or not the direction of their motion occurs along the lattice bonds. Two types of solitary plane waves are found and studied. The stability of these solitary waves is investigated numerically by their interactions with vacancies and lattice edges. Propagation of solitary plane waves through finite lattice domains with isotopic disorder is extensively studied. Comparison of these results with the soliton propagation in one-dimensional lattices with mass impurities is presented. ͓S0163-1829͑98͒00522-0͔

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solitary plane waves in an isotropic hexagonal lattice

1998

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“…͑11͒ and ͑12͒. Note that there is another effective method for this purpose which was discovered by Eilbeck and Flesch 19 and further developed 20 for variety of nonlinear dynamical systems, including also the 1D FK model. When a ͑two-component͒ kink profile has been found by the minimization method, then it can be chosen as an initial condition for numerical simulations of Eqs.…”

Section: A Numerical Methodsmentioning

confidence: 99%

Zig-zag version of the Frenkel-Kontorova model

1996

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We study a generalization of the Frenkel-Kontorova model which describes a zig-zag chain of particles coupled by both the first-and second-neighbor harmonic forces and subjected to a planar substrate with a commensurate potential relief. The particles are supposed to have two degrees of freedom: longitudinal and transverse displacements. Two types of two-component kink solutions corresponding to defects with topological charges QϭϮ1,Ϯ2 have been treated. The topological defects with positive charge ͑excess of one or two particles in the chain͒ are shown to be immobile while the negative defects ͑vacancies of one or two particles͒ have been proved at the same parameter values to be mobile objects. In our studies we apply a minimization scheme which has been proved to be an effective numerical method for seeking solitary wave solutions in molecular systems of large complexity. The dynamics of both these types of defects has also been investigated.

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“…Note that when investigating only wide kinks, we may omit interesting effects due to discreetness. 20,21 In order to treat highly discrete solutions, we should use more complicated numerical techniques such as the pseudospectral method suggested by Eilbeck and Flesch 22 and further developed by Duncan et al 23 …”

Section: ͑14͒mentioning

confidence: 99%