Sine-Gordon ratchets with general periodic, additive, and parametric driving forces

Abstract

We study the soliton ratchets in the damped sine-Gordon equation with periodic nonsinusoidal, additive, and parametric driving forces. By means of symmetry analysis of this system we show that the net motion of the kink is not possible if the frequencies of both forces satisfy a certain relationship. Using a collective coordinate theory with two degrees of freedom, we show that the ratchet motion of kinks appears as a consequence of a resonance between the oscillations of the momentum and the width of the kink. We show that the equations of motion that fulfill these collective coordinates follow from the corresponding symmetry properties of the original systems. As a further application of the collective coordinate technique we obtain another relationship between the frequencies of the parametric and additive drivers that suppresses the ratchetlike motion of the kink. We check all these results by means of numerical simulations of the original system and the numerical solutions of the collective coordinate equations.