The mechanism underlying the soliton ratchet, both in absence and in presence of noise, is investigated. We show the existence of an asymmetric internal mode on the soliton profile that couples, through the damping in the system, to the soliton translational mode. Effective soliton transport is achieved when the internal mode and the external force are phase locked. We use as a working model a generalized double sine-Gordon equation. The phenomenon is expected to be valid for generic soliton systems. DOI: 10.1103/PhysRevE.65.025602 PACS number͑s͒: 05.45.Yv, 05.60.Cd, 63.20.Pw One of the phenomena which is presently attracting interest both in physics and in biology is the so-called ratchet effect ͓1͔. In simple terms, a ratchet system can be described as a periodically forced Brownian particle moving in an asymmetric potential in the presence of damping and periodic driving. The periodic forcing keeps the system out of equilibrium so that the thermal energy can assist the conversion of the ac driver into effective work ͑direct motion of the particle͒ without any conflict with the second law of thermodynamics. This phenomenon has been found in several physical ͓2͔ and biological ͓3͔ systems and is presently considered as a possible mechanism by which biological motors perform their functions ͓4͔. For ode systems with damping, additive forcing, and noise, the ratchet effect can be viewed as a phase-locking phenomenon between the motion of the particle in the periodic potential and the external driver ͓5͔. Ratchet dynamics have also been observed in more complicated systems such as overdamped 4 models ͓6͔, chains of coupled particles with degenerate on-site potentials ͓7͔, long Josephson junctions with modulated widths ͓8͔, and inhomogeneous parallel Josephson arrays ͓9͔, three-dimensional helical models ͓10͔, etc. These are infinite dimensional systems described by continuous or discrete equations of soliton type, with asymmetric potentials, damping, and periodic forcing, in which the ratchet phenomenon manifests as unidirectional motion of the soliton ͑soliton ratchet͒. For overdamped systems one can reduce the soliton ratchet to the usual singleparticle ratchet by using a collective coordinate approach for the center-of-mass of the soliton ͓6͔. For underdamped or moderately damped systems, however, this could be inappropriate, since the radiation field present in the system can play an important role for the generation of the phenomenon.The aim of this article is to investigate the mechanism underlying soliton ratchets both in the absence and in the presence of noise. To this end we use an asymmetric double sine-Gordon equation as a working model for studying the effect ͑the phenomena, however, will not depend on the particular model used͒. We show that the asymmetry of the potential induces a spatially asymmetric internal mode on the soliton profile which can be excited by the periodic force. In the presence of damping, this mode can exchange energy with the translational mode so that the soliton can have a...

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We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation ͑NLSE͒ with the following perturbations: nonparametric spatiotemporal driving of the form f͑x , t͒ = a exp͓iK͑t͒x͔, damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordinate-theory which yields a set of ordinary differential equations ͑ODEs͒ for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force f͑x͒. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of f͑x͒. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum P͑t͒ and the soliton velocity V͑t͒: This is a parameter representation of a curve P͑V͒ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.

We study whether or not sine-Gordon kinks exhibit internal modes or ''quasimodes.'' By considering the response of the kinks to ac forces and initial distortions, we show that neither intrinsic internal modes nor ''quasimodes'' exist in contrast to previous reports. However, we do identify a different kind of internal mode bifurcating from the bottom edge of the phonon band which arises from the discretization of the system in the numerical simulations, thus confirming recent predictions.

We investigate the nonparametric, pure ac driven dynamics of nonlinear Klein-Gordon solitary waves having an internal mode of frequency V i . We show that the strongest resonance arises when the driving frequency d V i ͞2, whereas when d V i the resonance is weaker, disappearing for nonzero damping. At resonance, the dynamics of the kink center of mass becomes chaotic. As we identify the resonance mechanism as an indirect coupling to the internal mode due to its symmetry, we expect similar results for other systems. PACS numbers: 05.45.Yv, 02.30.Jr, 63.20.Pw An important paradigm established over the last two decades is that solitary waves or solitons behave very much like point particles when subjected to (a large class of) external forces and perturbations [1][2][3]. However, many solitary waves possess one (sometimes more than one) internal or shape mode [4,5], and in that case the particle picture of their dynamics may be oversimplified: Indeed, the internal mode can temporarily store energy and release it at a later stage, giving rise to resonance phenomena in solitary wave collisions [5] or in solitary wave interactions with inhomogeneities [6]. As internal modes are quite common in nonlinear systems, either intrinsically or as a result of small perturbations [7], the question of their influence on the dynamics of solitary waves is a very general and relevant one.One aspect of solitary wave dynamics that has proven itself difficult to understand is that of topological solitary waves or kinks subjected to pure, i.e., nonparametric ac driving. Thus, only recently [8] the ac driven dynamics of sine-Gordon kinks (that do not possess an internal mode) has been definitely clarified. Naively, the only new phenomenon one expects when a nonparametric external driving acts on solitons with internal modes is a resonance when its frequency, D, matches that of an internal mode, V i . The aim of this Letter is to show that, in fact, the actual scenario is most unexpected and highly nontrivial. As we will see below, a strong, anomalous resonance arises when d V i ͞2, whereas the normal resonance at d V i is definitely weaker, only possible at exactly zero damping, and even then it can be suppressed by appropriate choices of other parameters. We expect this result to be generic, because our analytical approach allows us to identify the mechanism for such a peculiar phenomenon: The ac force does not act directly on the internal mode (because of symmetry reasons), but rather, they interact indirectly via the translational motion which couples to the internal mode. Our predictions are fully confirmed by numerical simulations, which in addition show the implications of these resonances for the kink dynamics.As a specific example of a kink with internal mode, we take the well known [1] f 4 equation, which, when driven with an ac force f͑t͒ e sin͑dt 1 d 0 ͒, readswhere U͑f͒ ͑f 2 2 1͒ 2 ͞4 and b is a damping coefficient. Previous related works on this system are [9], where resonances in the presence of an external (time indep...