Energy flow for soliton ratchets

– We study the mechanism of directed energy transport for soliton ratchets. The energy flow appears due to the progressive motion of a soliton (kink) which is an energy carrier. However, the energy current formed by internal system deformations (the total field momentum) is zero. We show that the energy flow is realized via an inhomogeneous energy exchange between the system and the external ac driving. We also discuss effects of spatial discretization and combination of ac and dc external drivings. A transport mechanism of potential relevance in various areas of physics, chemistry and biology is based on the ratchet effect [1], i.e. the generation of directed currents by zero-mean external perturbations. It is based on the breaking of relevant space-time symmetries of the underlying system evolution equations [2]. A paradigmatic model corresponds to a classical particle moving in a spatially periodic potential under the influence of zero-mean fluctuations [1], where the energy current is directly connected with the mean particle momentum and the corresponding kinetic energy of the particle. For spatially extended systems, e.g. an annular Josephson junction, which is described by a partial differential equation (PDE), the ratchet phenomenon manifests as a unidirectional motion of a collective kink excitation (soliton) [3–8]. Here the unambiguous ab initio definition of a current may become a much more complicated task, because the kink excites other modes in the system during its motion, which may contribute to an energy current as well. A well-known model in the field of soliton ratchets is the driven-damped sine-Gordon equation [9], which is also used for modelling the abovementioned annular Josephson junction [8]: φtt − φxx = −αφt − sinφ+ E(t), (1) where E(t) is a zero-mean time-periodic driving force, E(t + T ) = E(t), ∫ T 0 E(t)dt = 0. We impose the kink-bearing periodic boundary condition: φ(x+ L, t) = φ(x, t) +Q, φt(x+ L, t) = φt(x, t), (2) where Q = 2πm is the topological charge with integer m = 1, 2, . . ., and L is the system size. c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10235-7 184 EUROPHYSICS LETTERS Let us consider the easiest case m = 1, i.e. the presence of one kink in the system Q = 2π. The kink velocity V is defined, e.g., as [4, 5]