Kink ratchet induced by a time-dependent symmetric field potential

The ratchet effect of a sine-Gordon kink is investigated in the absence of any external force while the symmetry of the field potential at every time instant is maintained. The directed motion appears by a time shift of the sine-Gordon potential through a time-dependent additional phase. A symmetry analysis provides the necessary conditions for the existence of net motion. It is also shown analytically, by using a collective coordinate theory, that the novel physical mechanism responsible for the appearance of the ratchet effect is the coupled dynamics of the kink width with the background field. Biharmonic and dichotomic periodic variations of the additional phase of the sine-Gordon potential are considered. The predictions established by the symmetry analysis and the collective coordinate theory are verified by means of numerical simulations. Inversion and maximization of the resulting current as a function of the system parameters are investigated.

Figure 1

(a) Kink velocity versus the amplitude parameter $\theta$ for fixed $\delta =0.8$. The circles are the results obtained by simulation of the sine-Gordon equation (2, 3), the solid line represents the average velocity obtained by using the collective coordinate Eq. (21), and the dashed line corresponds to its linear approximation (25). (b) Kink velocity versus the phase difference $\delta$ for fixed $\theta =1$. In both panels, $\beta =0.1$ and $\omega =0.1$.

Figure 2

Kink velocity versus frequency $\omega$ for fixed $\theta =0.1, \delta =0.8$, and $\beta =0.2$. The circles are the results obtained by simulation of the sine-Gordon equation (2, 3), the solid line represents the average velocity obtained by using the collective coordinate Eq. (21), and the dashed line corresponds to its linear approximation (25).

Figure 3

Time evolution of the kink center with a two-state potential given by (3) and (26). No ratchet effect is observed when $\eta \left(t\right)$ satisfies the time-shift symmetry (11) (open circles, $\mathrm{\Delta }\tau =0$). Net kink motion appears breaking that symmetry by setting $\mathrm{\Delta }\tau =-2$ (full circles). The remaining parameters are $\theta =0.8, \omega =\pi /3$, and $\beta =0.8$.

Figure 4

Comparison of the average kink velocity (circles) with the average velocity obtained using the collective coordinate approximation (solid line) for $\beta =0.05, \mathrm{\Delta }\tau =-2$, and $\omega =0.1$.

Figure 5

Kink velocity versus the translation parameter $\theta$ for fixed $\mathrm{\Delta }\tau =-2$ and $\omega =\pi /3$. The open circles correspond to a damping coefficient $\beta =1$, while full circles correspond to $\beta =0.8$.

Figure 6

Kink velocity versus $\mathrm{\Delta }\tau /T$ for fixed $T$ and $\theta$. Open squares: $T=4$. Full squares: $T=6$. Open circles: $T=10$. Full circles: $T=14$. In all cases, $\theta =0.8$ and $\beta =0.8$.

Figure 7

Kink velocity versus $\omega =2\pi /T$ for fixed $\mathrm{\Delta }\tau$. Full circles correspond to $\mathrm{\Delta }\tau =-2$, while open circles correspond to $\mathrm{\Delta }\tau =-0.5$. The remaining parameters are $\theta =0.8$ and $\beta =0.8$.