Dynamics of the Overdamped Coupled Josephson Junctions with an Interference Term

Qin, Wei; Peng, Zhi-Long

doi:10.1007/s00332-009-9040-7

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…The uniform sliding solutions are also called ponies on a merry-go-round [2], discrete rotating waves [3], single-wave-form solutions [20] or splay-phase orbits [19]. Much attention has been paid to the existence of such kind of solutions in Josephson junctions [2,3,20,21,25], coupled oscillators systems [19], discrete sine-Gordon rings [15] and the F-K model with spatially periodic boundary conditions [5,6,7,18,22,24]. In this section we employ the idea in [15,20] to study the existence of uniform sliding solutions of the F-K model with irrational mean spacing.…”

Section: Uniform Sliding Solutionsmentioning

confidence: 99%

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Dynamics of the Frenkel–Kontorova model with irrational mean spacing

Qin¹

2010

Nonlinearity

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In this paper we focus on the dynamics of the overdamped limit case of the Frenkel-Kontorova (F-K) model with irrational mean spacing. We extend the results of Baesens and MacKay on rational mean spacing case to irrational case: there exists a depinning force F d 0 such that the F-K model possesses an equilibrium with a given mean spacing if the constant driving force F ∈ [0, F d ] or it admits a uniform sliding solution provided F > F d . Moreover, the average velocity of the particles exists and is unique, and it is non-decreasing and continuous with respect to F ∈ [0, +∞). In particular, we prove that the average velocity depends continuously upon the mean spacing.

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Section: Uniform Sliding Solutionsmentioning

confidence: 99%

“…ε < 1/4(2K + 1). For time-periodic driving force F = F (t), the dynamics of the F-K model with spatially periodic boundary conditions have also been investigated recently [14,21].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of the Frenkel–Kontorova model with irrational mean spacing

Qin¹

2010

Nonlinearity

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“…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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“…For the overdamped F-K model or coupled Josephson junctions, the strong monotonicity ensures that the uniform sliding state is globally stable and unique up to phase shifts, see [5,20,21]. However, for the Hamiltonian lattices, the stability of the travelling waves is sophisticated, see [8,19].…”

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confidence: 98%