Non-sinusoidal current and current reversals in a gating ratchet

In this work, the ratchet dynamics of Brownian particles driven by an external sinusoidal (harmonic) force is investigated. The gating ratchet effect is observed when another harmonic is used to modulate the spatially symmetric potential in which the particles move. For small amplitudes of the harmonics, it is shown that the current (average velocity) of particles exhibits a sinusoidal shape as a function of a precise combination of the phases of both harmonics. By increasing the amplitudes of the harmonics beyond the small-limit regime, departures from the sinusoidal behavior are observed and current reversals can also be induced. These current reversals persist even for the overdamped dynamics of the particles.Comment: 11 pages, 9 figure

ac driven sine-Gordon solitons: dynamics and stability

The ac driven sine-Gordon equation is studied analytically and numerically, with the aim of providing a full description of how soliton solutions behave. To date, there is much controversy about when ac driven dc motion is possible. Our work shows that kink solitons exhibit dc or oscillatory motion depending on the relation between their initial velocity and the force parameters. Such motion is proven to be impossible in the presence of damping terms. For breathers, the force amplitude range for which they exist when dissipation is absent is found. All the analytical results are compared with numerical simulations, which in addition exhibit no dc motion at all for breathers, and an excellent agreement is found. In the conclusion, the generality of our results and connections to others systems for which a similar phenomenology may arise are discussed.Comment: 10 pages, latex, PostScript figures included with epsfig, to appear in European Physical Journal B, see GISC homepage at http://valbuena.fis.ucm.es/ for related wor

Exact stationary solutions of the parametrically driven and damped nonlinear Dirac equation

Two exact stationary soliton solutions are found in the parametrically driven and damped nonlinear Dirac equation. The parametric force considered is a complex ac force. The solutions appear when their frequencies are locked to half the frequency of the parametric force, and their phases satisfy certain conditions depending on the force amplitude and on the damping coe cient. Explicit expressions for the charge, the energy, and the momentum of these solutions are provided. Their stability is studied via a variational method using an ansatz with only two collective coordinates. Numerical simulations con rm that one of the solutions is stable, while the other is an unstable saddle point. Consequently, the stabilization of damped Dirac solitons can be achieved via time-periodic parametric excitations.Junta de Andalucía and Ministerio de Economía y Competitividad of Spain FIS2017-89349-PMinisterio de Ciencia, Innovación y Universidades of Spain PGC2018-093998-BI0

Reply to Comment on "Existence of Internal Modes of sine-Gordon Kinks"

In this reply to the comment by C. R. Willis, we show, by quoting his own statements, that the simulations reported in his original work with Boesch [Phys. Rev. B 42, 2290 (1990)] were done for kinks with nonzero initial velocity, in contrast to what Willis claims in his comment. We further show that his alleged proof, which assumes among other approximations that kinks are initially at rest, is not rigorous but an approximation. Moreover, there are other serious misconceptions which we discuss in our reply. As a consequence, our result that quasimodes do not exist in the sG equation [Phys. Rev. E 62, R60 (2000)] remains true.Comment: Reply to a comment by C. R. Willis on Phys. Rev. E 62, R60 (2000), to appear in Physical Review

dc motion of ac driven sine-Gordon solitons

We investigate the dependence of dc soliton motion sustained by pure ac driving in the sine-Gordon model on various parameter problems, such as frequency, initial kink velocity or relative phase. We show by means of the collective coordinate formalism that ac driving induces a net dc velocity whose modulus and direction depend on the driving phase and on the kink initial velocity. Numerical simulations of the full sine-Gordon equation confirm the correctness and accuracy of this prediction. Nontrivial cases when dc soliton motion is transformed into oscillation as well as the effects of damping are analyzed. Our results generalize previous work on the undamped problem, and clarify a long standing issue about the existence and characteristics of this phenomenon in damped problems, whose possible appearance in other systems is also discussed.CICyT (Spain) grant MAT95-0325DGES (Spain) grant PB96-011

Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism together with the Rice {\it Ansatz} to derive the equations of motion of the collective coordinates (CCs) for the perturbed sine-Gordon (sG) and $\phi^{4}$ systems. For the $N$ collective coordinates, regardless of the {\it Ansatz} used, we show that, for the nonlinear Klein-Gordon equations, this approach is equivalent to the {\it Generalized Traveling Wave Ansatz} ({\it GTWA})Comment: 9 page

Using the finite domain remnant of the continuous spectrum to examine integrability: Effect of boundary conditions

The aim of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE), by examining it in a finite but large domain. For monoparametric families of PDE/DDE’s, that are known to possess isolated integrable points, we find that very special features occur in the finite domain remnant of the continuous (“phonon”) spectrum at these “singular” points. We identify these features in the case example of a PDE and a DDE (that sustain front and pulselike solutions, respectively) for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular, integrable points and hence we propose to explore a given PDE/DDE in a finite but large domain for such traits, as a means of assessing its potential integrability.Ministerio de Ciencia y Tecnología (Spain) through Grant No. BFM2001-3878-C02Junta de Andalucıa Project No. FQM-020

On the ratchet effect

Entender el efecto trinquete inducido por la simetría temporal de fuerzas es una cuestión fundamental que ha permanecido sin resolver durante décadas. Mientras la dependencia del transporte dirigido en cada uno de los parámetros de control de trinquete ha sido investigada individual, experimental, teórica y numéricamente, no hay todavía ningún criterio general para aplicar a todo el conjunto de estos parámetros un sistema óptimo de control de dirección de transporte. También divulgamos óptimamente cómo mejorar el transporte dirigido, pues existe una fuerza universal en forma de onda que puede ser entendida como el resultado fundamental de dos competencias mecánicas: el aumento del grado de ruptura de las simetrías temporales y la disminución de la fuerza de impulso. Demostramos que esta forma de onda universal explica todos los resultados anteriores, experimental, teóricos y numéricos, en favor de una gran diversidad de sistemas, incluyendo las enzimas del motor.Understanding the ratchet effect induced by symmetry breaking of temporal forces is a fundamental issue that has remained unresolved for decades. While the dependence of the directed transport on each of the ratchet-controlling parameters has been individually investigated experimentally, theoretically, and numerically, there is still no general criterion to apply to the whole set of these parameters to optimally control directed transport in general systems. We report that, too optimally enhance directed transport, there exists a universal force waveform which can be understood as the result of two competing fundamental mechanisms: the increase of the degree of breaking of the aforementioned temporal symmetries and the decrease of the force impulse. We demonstrate that this universal waveform explains all the previous experimental, theoretical, and numerical results for a great diversity of systems, including motor enzymes.notPeerReviewe

Nonequivalence of phonon modes in the sine-Gordon equation

We study the resonances in the sine-Gordon equation driven by an ac force using a linear perturbation theory. We show that resonances take place when the driving frequency d is equal to half of the phonon modes’ frequencies as has been shown numerically in our earlier work [ N. R. Quintero, A. Sanchez, and F. G. Mertens, Phys. Rev. E 62 , R60 ( 2000)] , however, we find that the ac force is able to excite not all the phonon modes, but rather only the odd phonons ( i.e., the ones with odd eigenfunctions)Junta de Andalucia project FQM-28