Double Sine-Gordon ratchet induced by excitation of an internal mode

In a recent paper [Phys. Rev. E 69, 056612 (2004)] we showed the symmetry analysis of Flach [Phys. Rev. Lett. 88, 184101 (2002)] which predicted the appearance of directed energy current in homogeneously spatially extended systems described by nonlinear field equations coupled to a heat bath in the presence of a correct choice for the time dependence of an external ac field, E(t) , was due to the excitation of an internal mode. Flach applied their analysis to the sine-Gordon (SG) equation and verified the symmetry breaking numerically. In the SG case we showed the internal mode coupled to the center of the mass variable, X(t) , that caused the symmetry breaking was Γ(t) the slope of the kink. We also found that the phonon dressing of the SG kink by the ac driver, χ(t) , was necessary for the occurrence of a directed energy current in the SG equation. We show in the case of the double sine-Gordon (DSG) equation that the excitation of the internal mode, R(t) (where R(t) is the separation of the two subkinks that make up the DSG soliton), combined with the phonon dressing of the DSG soliton also causes a directed energy current.