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August 2017

EPL, 119 (2017) 40003 www.epljournal.org

doi: 10.1209/0295-5075/119/40003

π-kink propagation in the damped Frenkel-Kontorova model

K. Alfaro-Bittner1,M.G.Clerc

2,M.A.Garc

´

ıa- ˜

Nustes1and R. G. Rojas1

1Instituto de F´ısica, Pontiﬁcia Universidad Cat´olica de Valpara´ıso - Casilla 4059, Valpara´ıso, Chile

2Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile

Casilla 487-3, Santiago, Chile

received 2 June 2017; accepted in ﬁnal form 3 October 2017

published online 6 November 2017

PAC S 05.45.-a – Nonlinear dynamics and chaos

PAC S 05.45.Xt – Synchronization; coupled oscillators

Abstract – Coupled dissipative nonlinear oscillators exhibit complex spatiotemporal dynamics.

Frenkel-Kontorova is a prototype model of coupled nonlinear oscillators, which exhibits coexistence

between stable and unstable state. This model accounts for several physical systems such as the

movement of atoms in condensed matter and magnetic chains, dynamics of coupled pendulums,

and phase dynamics between superconductors. Here, we investigate kinks propagation into an

unstable state in the Frenkel-Kontorova model with dissipation. We show that unlike point-like

particles π-kinks spread in a pulsating manner. Using numerical simulations, we have characterized

the shape of the π-kink oscillation. Diﬀerent parts of the front propagate with the same mean

speed, oscillating with the same frequency but diﬀerent amplitude. The asymptotic behavior

of this propagation allows us to determine the minimum mean speed of fronts analytically as a

function of the coupling constant. A generalization of the Peierls-Nabarro potential is introduced

to obtain an eﬀective continuous description of the system. Numerical simulations show quite fair

agreement between the Frenkel-Kontorova model and the proposed continuous description.

Copyright c

EPLA, 2017

Introduction. – Nonlinear oscillators such as the

pendulum have played a primary role in the understand-

ing of complex dynamics since the dawn of modern sci-

ence [1,2]. Even a simple two-oscillators coupled system

shows interesting behavior such as synchronization [3].

A chain of coupled oscillators to nearest neighbors also

can present a rich spatiotemporal dynamics [3–5], such

as phase turbulence [4], synchronization [3], defects tur-

bulence [6], random occurrence of coherence events [7],

defect-mediated turbulence [8], spatiotemporal intermit-

tency [9], quasiperiodicity in extended system [10] and

coexisting of coherent and incoherent behavior, known

as chimera states [11]. A prototype model of coupled

nonlinear oscillators to nearest neighbors is the Frenkel-

Kontorova model [12]. Figure 1 illustrates a chain of cou-

pled pendulums. In the context of condensed matter, it is

the simplest model that describes the dynamics of a chain

of particles interacting with the nearest neighbors un-

der the inﬂuence of an external periodic potential [12,13].

It has been used to describe several nonlinear phenomena

such as solitons, kinks, breathers, and glass-like behavior.

Likewise, this model has been used to describe cluster of

atoms in DNA-like chain, spin in magnetic chain, ﬂuxon

in coupled Josephson junctions and plastic deformations

in metals (see textbook [12] and references therein). The

Frenkel-Kontorova model exhibits coexistence between a

stable extended state and an unstable one. The solution

that connects both is called the π-kink [14], which corre-

sponds to a topological particle-type solution. As a mat-

ter of fact, a rich domain dynamics can emerge between

π-kink solutions. In dissipative systems, π-kinks are also

known as front solutions, or wavefronts [15–17], depending

on the physical context where they are considered. Front

dynamics occurs in a variety of systems ranging from bi-

ology to physics [18,19]. Interfaces between metastable

states can also appear in the form of propagating fronts,

leading to a rich spatiotemporal dynamics [20,21]. Most

of the theoretical studies of fronts propagation have been

achieved considering the continuous limit [15,18,21]. At

this limit, the fronts propagate as a rigid solid with a speed

determined by the initial conditions. However, recently, in

coupled micropillar laser with saturable absorber has been

observed the propagation of self-pulsating states [22,23].

Hence, the nonlinear waves propagation in these optical

oscillators is not correctly well contained in the continu-

ous description.

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