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Recently Flach et.al. used a symmetry analysis to predict the appearance ofdirected energy current in homogeneously spatially extended systems coupled toa heat bath in the presence of an external ac field E(t). The systems aredescribed by nonlinear field equations. Their symmetry analysis allowed them tomake the right choice of E(t) so as to obtain symmetry breaking which causesdirected energy transport for systems with a nonzero topological charge. Theirnumerical simulations verified the existence of the directed energy current. Weuse a rigorous collective variable for nonlinear Klein-Gordon equations toprove the rectification of the current is due to the excitation of an internalmode, $\Gamma(t)$, which describes the oscillation of the slope of the kink.The internal mode $\Gamma(t)$ is excited by its interaction with the center ofmass 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 andmodulates the interaction of $\Gamma(t)$ with X(t). However, we show that thedressing in the absence of excitation of the internal mode $\Gamma(t)$ cannotcause a directed energy current. We derive the expressions for the dressing andnumerically solve the equations of motion for $\Gamma(t)$, X(t), and themomentum, P(t), which enable us to obtain the explicit expressions for thedirected energy current and the ac driven kink profile. We then show thedirected energy current vanishes unless the slope $\Gamma(t)$ is a dynamicalvariable.Comment: 6 pages, 5 figure

Soliton ratchets induced by excitation of internal modes