Soliton ratchets induced by excitation of internal modes

Recently Phys. Rev. Lett. 88, 184101 (2002)]] 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). 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. They argued that the origin of their strong rectification in the underdamped limit is due to the excitation of internal modes and their interaction with the translational kink motion. The internal mode mechanism as a cause of current rectification was also proposed by Salerno and Zolotaryuk [Phys. Rev. E. 65, 056603 (2002)]]. We use a rigorous collective variable for nonlinear Klein-Gordon equations to prove that 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, and due to a dressing of the bare kink by the ac driver. 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. 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 that the directed energy current vanishes unless the slope Gamma (t) is a dynamical variable and the kink is dressed by the ac driver.