Uniform sliding states in the undamped Frenkel–Kontorova model

Qin, Wei

doi:10.1016/j.jde.2010.07.028

Cited by 7 publications

(7 citation statements)

References 32 publications

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php diatomic case. Both topological [1,6,15,22] and variational [6,21] methods are used to investigate the traveling waves of the monatomic FK model. Since the on-site potential of the FK model is periodic and hence bounded, the approach for asymptotic quadratic potential in this paper seems workable for the diatomic FK model.…”

Section: Discussionmentioning

confidence: 99%

“…Pankov and Rothos [20] investigated the existence of traveling waves for FPU chains with saturable nonlinearities under some nonresonance condition. Following the idea in [21], in which the periodic traveling waves for the monatomic Frenkel-Kontorova model were studied, we apply the saddle point theorem to discuss the existence of wave trains with two different waveform functions for diatomic FPU chains with asymptotic quadratic potential.…”

Section: Introductionmentioning

confidence: 99%

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Wave Propagation in Diatomic Lattices

Qin¹

2015

SIAM J. Math. Anal.

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We study periodic traveling waves (wave trains) in diatomic Fermi-Pasta-Ulam chains (FPU). By applying the minimax principle, we demonstrate the existence of two different periodic waveform functions corresponding, respectively, to light and heavy particles. Our approach applies to the FPU β-model for each wavenumber and each frequency, and to FPU chains with asymptotic quadratic potential for wavenumbers and frequencies satisfying the nonresonance condition. As an application to monatomic lattices, we show for the monatomic soft FPU β-model the existence of supersonic wave trains with two different waveform functions for adjacent particles, contrary to the nonexistence of supersonic wave trains with only one waveform function.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wave Propagation in Diatomic Lattices

Qin¹

2015

SIAM J. Math. Anal.

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“…If γ = F = 0, i.e., the particles system (1.1) is Hamiltonian, then we can use the variational method to study system (3.1) under some conditions on interaction potentials [30]. If the interaction potential in (1.1) is harmonic, then by reducing to a fixed point theorem, we applied the Schauder fixed point theorem to discuss in [31] the uniform sliding states for the overdamped limit case (1.5).…”

Section: Continuous Dependencementioning

confidence: 99%

“…The analogue of uniform sliding states with periodic boundary conditions for Josephson junction systems are called ponies on a merry-go-round [2], discrete rotating waves [3], single-wave-form solutions [29], or splay-phase solutions [28] in the literature. Various approaches have been developed to study the existence, stability, and other properties of such kind of solutions, such as degree theory [2,16], the Schauder fixed point theorem [20,29], variational methods [16,30], the Lefschetz trace formula [28], and monotone dynamical systems theory [6,7,27,32].…”

Section: Introductionmentioning

confidence: 99%

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Qin

2011

Commun. Math. Phys.

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In this paper we are mainly concerned with existence and modulation of uniform sliding states for particle chains with damping γ and external driving force F. If the on-site potential vanishes, then for each F > 0 there exist trivial uniform sliding states x n (t) = nω + νt + α for which the particles are uniformly spaced with spacing ω > 0, the sliding velocity of each particle is ν = F/γ , and the phase α is arbitrary. If the particle chain with convex interaction potential is placed in a periodic on-site potential, we show under some conditions the existence of modulated uniform sliding states of the formwhere the modulation function u is periodic and unique up to phase. The conditions are that the system is overdamped and the driving force F exceeds some critical value F d (ω) ≥ 0 depending on mean spacing ω. If F ∈ [0, F d (ω)], the system possesses a set of rotationally ordered equilibrium states for irrational ω, which can be described by a non-decreasing hull function, just as the case γ = F = 0, where Aubry-Mather theory applies to ground states. Meanwhile, we prove that F d (ω) = 0, which was argued physically much earlier, if the hull function of ground states with irrational rotation number ω for F = 0 is continuous.

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“…There is a wealth of studies of Frenkel-Kontorova models. We refer the reader to the monograph by Braun [4] and only mention more recent results for sliding states by Qin for a forced Frenkel-Kontorova chain, both with and without damping [5,6] and periodic travelling waves (wave trains) by Fečkan and Rothos [7].…”

Section: Introductionmentioning

confidence: 99%

Travelling heteroclinic waves in a Frenkel–Kontorova chain with anharmonic on-site potential

Buffoni

Schwetlick

Zimmer

2019

Journal de Mathématiques Pures et Appliquées

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The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by a chain of atoms, where neighbouring atoms interact through a linear spring and are exposed to a smooth periodic on-site potential. A dislocation moving with constant speed corresponds to a heteroclinic travelling wave, making a transition from one well of the on-site potential to another. The ensuing system is nonlocal, nonlinear and nonconvex. We present an existence result for a class of smooth nonconvex on-site potentials. Previous results in mathematics and mechanics have been limited to on-site potentials with harmonic wells. To overcome this restriction, we propose a novel approach: we first develop a global centre manifold theory for anharmonic wave trains, then parametrise the centre manifold to obtain asymptotically correct approximations to the solution sought, and finally obtain the heteroclinic wave via a fixed point argument.

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Uniform sliding states in the undamped Frenkel–Kontorova model

Abstract: The uniform sliding states of the undamped Frenkel-Kontorova model are reduced to the periodic solutions of an advance-delay differential equation, the existence of which is investigated in this paper by employing variational methods.

Cited by 7 publications

References 32 publications

Wave Propagation in Diatomic Lattices

Wave Propagation in Diatomic Lattices

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

Travelling heteroclinic waves in a Frenkel–Kontorova chain with anharmonic on-site potential

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