Collective coordinates theory for discrete soliton ratchets in the sine-Gordon model

A collective coordinate theory is developed for soliton ratchets in the damped discrete sine-Gordon model driven by a biharmonic force. An ansatz with two collective coordinates, namely the center and the width of the soliton, is assumed as an approximated solution of the discrete nonlinear equation. The dynamical equations of these two collective coordinates, obtained by means of the generalized travelling wave method, explain the mechanism underlying the soliton ratchet and capture qualitatively all the main features of this phenomenon. The numerical simulation of these equations accounts for the existence of a nonzero depinning threshold, the nonsinusoidal behavior of the average velocity as a function of the relative phase between the harmonics of the driver, the nonmonotonic dependence of the average velocity on the damping, and the existence of nontransporting regimes beyond the depinning threshold. In particular, it provides a good description of the intriguing and complex pattern of subspaces corresponding to different dynamical regimes in parameter space.

Figure 1

Depinning curve in the plane $(\kappa ,\epsilon )$ for $\omega =0.1,\alpha =0.1$, and $\phi =0$. Below the curve no mobile kinks exist. The area above the curve corresponds to moving kinks with nonzero average velocity. The depinning threshold computed from simulation of the discrete sG Eq. (1) (blue solid line) is very well fitted by the results obtained from numerical solutions of the CC equations (red circles).

Figure 2

Dependence of the average kink velocity on the relative phase for $\kappa =2$ (a) and $\kappa =1$ (b). The blue continuous line represents results obtained from simulations of the sG equation (1). Red circles correspond to results obtained from numerical integration of the CC equations. Other parameters are $\omega =0.1,\epsilon =0.05,\alpha =0.1$.

Figure 3

Average kink velocity versus coupling constant. Panels (a) and (b) correspond to the results from simulations of the sG system (1) and from the equations of motion for the CCs (4) and (5), respectively. Symbols refer to different dynamical regimes: periodic or quasiperiodic regular transport (black $•)$, chaotic diffusive transport (red $+)$, pinned states (blue $\times )$, and rotating states (green $*)$. The thick magenta line represents the analytical expression (15) for the average velocity in the continuous limit. The inset shows more detailed behavior around $\kappa =1.16$. Other parameters: $\omega =0.1,\epsilon =0.05,\alpha =0.1$, and $\phi =0$.

Figure 4

Evolution of the center of mass and width of the discrete sG kink for $\kappa =0.3$ (top panels) and $\kappa =1.2$ (bottom panels). The former case corresponds to a pinning state, whereas the latter constitutes an example of a nontransporting rotating state. Fourier spectra for both coordinates are depicted in the right-hand-side panels. Numerical integration of the CC equations (4) and (5) leads to a quantitatively similar outcome. Other parameters are $\omega =0.1,\epsilon =0.05,\alpha =0.1$, and $\phi =0$.

Figure 5

Top panel: Moduli of Floquet eigenvalues versus coupling constant for the pinning states of Fig. 3. Bottom panel: Moduli of Floquet eigenvalues for the rotating states stood out in the inset of Fig. 3.

Figure 6

Dynamical regimes of the kink ratchet motion for various values of the coupling constant, $\kappa$, and the driving amplitude, $\epsilon$. Panel (a) corresponds to the results from simulations of the discrete sG system (1), and panel (b) to the results from the CC equations of motion. White area: The kink remains pinned; black: periodic or quasiperiodic regular transport; red: chaotic diffusive transport; green: rotating states. Other parameters are $\omega =0.1,\alpha =0.1,\phi =0$.

Figure 7

Dependence of the normalized average kink velocity on the damping for $\kappa =2$ (a) and $\kappa =1$ (b). The blue solid line represents results from the simulations of the discrete sG equation (1) while the red circles correspond to numerical integration of the CC equations. Other parameters: $\omega =0.1,\epsilon =0.05$, and $\phi =0$.

Figure 8

Amplitudes of the series $I_8,I_7,I_3,I_1,I_5,I_{11}$, and $I_{12}$ (from top to bottom) versus coupling. The amplitudes have been evaluated at $l\left(t\right)\approx 1$ due to the small size of the fluctuations of $l\left(t\right)$.