Soliton Ratchets Induced by Excitation of Internal Modes
  Recently Flach et.al. used a symmetry analysis to predict the appearance of directed energy current in homogeneously spatially extended systems coupled to a heat bath in the presence of an external ac field E(t). The systems are described by nonlinear field equations. Their symmetry analysis allowed them to make the right choice of E(t) so as to obtain symmetry breaking which causes directed energy transport for systems with a nonzero topological charge. Their numerical simulations verified the existence of the directed energy current. We use a rigorous collective variable for nonlinear Klein-Gordon equations to prove the rectification of the current is due to the excitation of an internal mode, $\Gamma(t)$, which describes the oscillation of the slope of the kink. The internal mode $\Gamma(t)$ is excited by its interaction with the center of mass of the kink, X(t), which is accelerated by E(t). The external field E(t) also causes the kink to be dressed, which affects the shape of the kink and modulates the interaction of $\Gamma(t)$ with X(t). However, we show that the dressing in the absence of excitation of the internal mode $\Gamma(t)$ cannot cause a directed energy current. We derive the expressions for the dressing and numerically solve the equations of motion for $\Gamma(t)$, X(t), and the momentum, P(t), which enable us to obtain the explicit expressions for the directed energy current and the ac driven kink profile. We then show the directed energy current vanishes unless the slope $\Gamma(t)$ is a dynamical variable. 
C.R. Willis and M. Farzaneh