Journal ArticleDOI
Kink’s internal modes in the Frenkel-Kontorova model
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In this paper, the authors consider a generalized Frenkel-Kontorova model, describing the dynamics of a chain of particles in a periodic substrate potential, and analyze the effect of discreteness on the existence and properties of internal modes of kinks, topological excitations of the chain.Abstract:
We consider a generalized Frenkel-Kontorova model, describing the dynamics of a chain of particles in a periodic substrate potential, and analyze the effect of discreteness on the existence and properties of internal ~or shape! modes of kinks, topological excitations of the chain. In particular, we show that kink’s internal modes can appear not only below but also above the phonon spectrum band and, in the latter case, the localized mode describes out-of-phase oscillations of the kink’s shape. For the sinusoidal on-site potential, when the model is described by the discrete sine-Gordon equation, we reveal, in sharp contrast with the continuum limit, the existence of the kink’s internal mode in a narrow region of the discreteness parameter. We apply two different analytical techniques to describe the cases of weak and strong coupling between particles, explaining qualitatively and even quantitatively the main features of the kink oscillations observed in numerical simulations. We also discuss the effect of nonlinearity on the existence and properties of kink’s internal modes and show, in particular, that a nonlinearity-induced frequency shift of the lattice vibrations can lead to the creation of the nonlinear kink’s internal modes, which, however, slowly decay due to a generation of radiation through higher-order harmonics. @S1063-651X~97!11010-8#read more
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Journal ArticleDOI
Nonlinear dynamics of the Frenkel-Kontorova model
Oleg M. Braun,Yuri S. Kivshar +1 more
TL;DR: An overview of the dynamics of one of the fundamental models of low-dimensional nonlinear physics, the Frenkel-Kontorova (FK) model, is presented in this article.
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On a class of discretizations of Hamiltonian nonlinear partial differential equations
TL;DR: In this article, a new class of discretizations of partial differential equations (PDEs) that preserve a (momentum-like) integral of the motion is presented, which results in an effective translational invariance for the dynamical problem and the absence of a Peierls-Nabarro barrier.
Journal ArticleDOI
Dynamics of lattice kinks
TL;DR: In this paper, the authors 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.
Journal ArticleDOI
Quantum coherence of discrete kink solitons in ion traps.
TL;DR: It is shown that long-lived solitonlike configurations are manifested as deformations of the zigzag structure in the linear Paul trap, and are topologically protected in a circular trap with an odd number of ions, suggesting that ion traps can be used to test quantum-mechanical effects with solitons.
Journal ArticleDOI
Bound states of two-dimensional solitons in the discrete nonlinear Schrödinger equation
TL;DR: In this article, the existence and stability of bound states of solitary excitations in the two-dimensional discrete nonlinear Schrodinger lattice are considered, and the interaction of S = 0 solitons is investigated.