The alternating-current-driven motion of dislocations in a weakly damped Frenkel - Kontorova lattice

Abstract

By means of numerical simulations, we demonstrate that ac field can support stably moving collective nonlinear excitations in the form of dislocations (topological solitons, or kinks) in the Frenkel-Kontorova (FK) lattice with weak friction, which was qualitatively predicted by Bonilla and Malomed [Phys. Rev. B 43, 11539 (1991)]. Direct generation of the moving dislocations turns out to be virtually impossible; however, they can be generated initially in the lattice subject to an auxiliary spatial modulation of the on-site potential strength. Gradually relaxing the modulation, we are able to get the stable moving dislocations in the uniform FK lattice with the periodic boundary conditions, provided that the driving frequency is close to the gap frequency of the linear excitations in the uniform lattice. The excitations that can be generated this way have a large and noninteger index of commensurability with the lattice (so suggesting that the actual value of the commensurability index is irrational). The simulations reveal two dif-1 ferent types of the moving dislocations: broad ones, that extend, roughly, to half the full length of the periodic lattice (in that sense, they cannot be called solitons), and localized soliton-like dislocations, that can be found in an excited state, demonstrating strong persistent internal vibrations. The minimum (threshold) amplitude of the driving force necessary to support the traveling excitation is found as a function of the friction coefficient. Its extrapolation suggests that the threshold does not vanish at the zero friction, which may be explained by radiation losses. The moving dislocation can be observed experimentally in an array of coupled small Josephson junctions in the form of an inverse Josephson effect, i.e., a dc-voltage response to the uniformly applied ac bias current.

Progressive motion of an ac-driven kink in an annular damped system

“…The results of the simulations are displayed below in the form u = u(γ ac ), using numerically found dependencies V dc (γ ac ) and the relation (8). This way of the presentation of results is the most appropriate one if the objective is to look for dynamical regimes with a nonzero average velocity of the ac-driven kink.…”

“…Search for such regimes in FK models turned out to be more difficult, but simulations have finally demonstrated that the soliton in the form of a dislocation may be driven with a nonzero mean velocity by the ac force [8].…”

Bound fluxon pair in one-dimensional SQUID array

“…One such phenomenon is the stable progressive motion of kinks with homogeneous pure AC driving [8][9][10][11]. In a discrete system, kinks experience a periodic potential, namely the Peierls-Nabarro (PN) potential, whose period is the lattice constant of the system, which here is d. Therefore, stable propagation can be maintained only when a kind of mode locking occurs between the external driving and the oscillation of the propagating kinks in the PN potential.…”

“…In a discrete system, kinks experience a periodic potential, namely the Peierls-Nabarro (PN) potential, whose period is the lattice constant of the system, which here is d. Therefore, stable propagation can be maintained only when a kind of mode locking occurs between the external driving and the oscillation of the propagating kinks in the PN potential. Thus, the mean velocity is predicted to be v ¼ m e d Á m=M, where m and M are the super-and subharmonic resonance orders [11]. 3) is considered, the fluxon pair would experience asymmetric PN potential, and the direction of progress would be fixed.…”

Stability of Mode-Locked Kinks in the AC Driven and Damped Sine-Gordon Lattice

“…The ac-driven DSG lattice has two independent sources of non-integrability: the external drive (bias) and the discreteness. Interplay of these two sources has led to a number of interesting effects: mode-locking to the frequency of the external drive [13] and kink mobility [14,15] (including its experimental detection in periodically modulated Josephson junctions [16]), various regimes of the dynamical chaos [13,18], biharmonically driven discrete kink ratchet [17,18] to name a few. However, these studies have been performed mostly in the adiabatic, subband (the driving frequency lies in the gap of the linear spectrum) or resonant (the driving frequency lies in the linear band) cases.…”