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Discrete soliton ratchets driven by biharmonic fields Yaroslav Zolotaryuk† and Mario Salerno‡ † Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine Kyiv 03143, Ukraine ‡ Dipartamento di Fisica “E. R. Caianiello” and Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia (CNISM), Universit´a di Salerno, I-84081 Baronissi, Salerno, Italy (February 4, 2008) Directed motion of topological solitons (kinksor antikinks) in thedampedand AC-drivendiscrete sine-Gordon system is investigated. We show that if the driving field breaks certain time-space symmetries, the soliton can perform unidirectional motion. The phenomenon resembles the well 6 known effects of ratchet transport and nonlinear harmonic mixing. Direction of the motion and its 0 velocitydependsontheshapeoftheACdrive. Necessaryconditionsfortheoccurrenceoftheeffect 0 are formulated. In comparison with thepreviously studied continuumcase, thediscrete case shows 2 a number of new features: non-zero depinning threshold for the driving amplitude, locking to the n rationalfractionsofthedrivingfrequency,anddiffusiveratchetmotioninthecaseofweakintersite a coupling. J 0 Pacs: 05.45.-a,07.90.+c, 89.20.-a demonstrated by using asymmetric magnetic fields [16] 2 and spatially asymmetric currents [17]. From the theo- ] retical point of view, symmetry breaking conditions to S I. INTRODUCTION generatesolitonratchetswerediscussedinRef.[11]. The P mechanism underlying soliton ratchets was proposed in n. Transport phenomena induced by the interplay be- Ref. [9] for the case of a perturbed asymmetric double i tween non-equilibrium fluctuations, symmetry breaking sine-Gordon equation driven by a symmetric AC driver l n and nonlinearity, have recently attracted a great deal of andextendedinRef.[10]tothecaseofthedampedsine- [ interest. In particular, point particle ratchets described Gordon (SG) equation with asymmetric AC fields. In 1 byordinarydifferentialequationshavebeenlargelyinves- both cases, the phenomenon was ascribed to the exis- v tigated due to their relevance in several fields, including tence of an internaloscillationon the kink profile which, 4 molecular motors and Josephson junctions (see reviews in the presence of damping, couples to the translational 4 [1,2]). In simple terms, the point particle ratchets ap- mode of the kink and produces transport. The internal 0 pearastheunidirectionalmotionofadampedanddriven vibrationwasshowntobespatiallyasymmetric,thusgiv- 1 particle, which is achieved under the influence of only ing directionality to the motion, and phase lockedto the 0 stochastic and deterministic forces of zero average, in- externalforce. Thismechanism,alsoknownasthe inter- 6 0 dependently on initial conditions. The phenomenon was nal mode mechanism, has been confirmed for a number / ascribed to the breaking of the symmetries connecting of systems such as the asymmetric double sine-Gordon n orbits with opposite velocities in the phase space [3–5] equation with symmetric driver [9,18] and the SG sys- i l and to the phase locking of the particle dynamics to the tem with temporal asymmetric forces [10,19,20]. n : external driver [6,7]. In contrast to the continuous case, however, discrete v Ratchetphenomenaininfinite-dimensionalsystemsde- soliton ratchets have been scarcely investigated (some Xi scribedbynonlinearpartialdifferentialequationsofsoli- work on spatially asymmetric discrete soliton ratchets tontypehavealsobeeninvestigated. Thishasbeendone has been done in Refs. [21,22]). In this case, one can ex- r a for both spatially asymmetric potentials with temporar- pectthatthe presenceofthe Peierls-Nabarrobarriercan ily symmetric AC fields [8,9] and for symmetric poten- strongly influence the transport of discrete solitons. It tials with temporarily asymmetric AC fields [10–13]. In is therefore of interest to investigate the conditions un- both cases, the ratchet effect appears as a unidirectional der which the discrete soliton ratchets can exist. The motion of the soliton, which resembles the drift dynam- present paper is just devoted to this investigation. More ics observed for point particle ratchets; from here the precisely,westudytheratchetdynamicsinducedbytem- nameofsoliton ratchetsoriginates[9]. Thesolitonratch- porarily asymmetric forces of zero mean on topological ets induced by asymmetrical external fields have been solitons(kinksandantikinks)ofthediscretesine-Gordon implemented experimentallyinlong Josephsonjunctions (DSG)system. Thisequationmodelsanumber ofphysi- [14] by means of nonlinear harmonic mixings. This ap- calsystemssuchasarraysofJosephsonjunctions,crystal proachhasbeenshowntobe effectivealsoinotherphys- dislocationsorcharge-densitywaves(see[23,24]). Inpar- ical contexts [15] (for a detailed list of references see ticular,weinvestigatetheconditionsfortheoccurrenceof the review paper [2]). The existence of soliton ratch- solitonratchetsandstudy thedependence oftheaverage ets in long Josephson junctions was also experimentally solitonvelocityonthesystemparameters. Acomparison 1 withtheresultsderivedforcontinuoussolitonratchetsin Here u is the displacement of the nth particle from its n Ref. [10] is also provided. equilibrium position, ∆u u 2u +u is the n n+1 n n−1 ≡ − From our study it emerges that discrete soliton ratch- discrete Laplacian, κ is the coupling constant measuring ets are much more complicated than their continuum thediscretenessofthelattice,αisthedampingcoefficient counterpart. Inparticular,the meanvelocityof the kink and E(t) is an externaldriving field. In the following we in most cases appears to be a piece-wise function of the assume E(t) to be of the form parameters which resembles a devil’s staircase. We find E(t)=E cos(ωt)+E cos(mωt+θ) , (2) that kink transport becomes very effective on the cor- 1 2 responding orbits (limit cycles) which are phase locked where m is an integer even number. Notice that the su- to the external driver. Transport is possible also in the perpositionoftwoharmonicsmakesthe periodicforceto presence of more complicated dynamics such as chaotic be asymmetric in time for almost all values of θ, a fea- andintermittency orbits,especially whenthe systembe- ture which can be used to break the temporal symmetry comes very discrete (this is achieved when the coupling of the system (see below). In this context, it is of in- constantisverysmall). Inthesecases,however,the kink terest to investigate the condition under which a driving transport is not very efficient since the drift velocity is force of zero mean of type (2) can induce kink’s unidi- rather small. Except for the very discrete case, domi- rectional motion similar to the one observed in the con- nated by the pinning of the kink to the lattice, we find tinuum SG case [10]. In this regard, we remark that that the internal mode mechanism remains valid also in in the lower approximation, a discrete kink of the form the DSG ratchet in all cases in which the transport is u (t)=4arctan exp[n X(t)] ,canbeviewedasasin- n observed. gle particle [24] {and its−dynam}ics is described in terms We remark that the soliton ratchets induced by tem- of collective coordinates: the center of mass X(t) and porarilyasymmetricfieldsmaybeaneffectivewaytocon- the kink velocity X˙(t). In this approach, the effective trol the transport properties of a large variety of contin- point-particle equation of the motion becomes uous and discrete systems. From the experimental point of view, indeed, it is much easier to produce ratchets by X¨ +αX˙ +V′ (X)+E˜(t)=0 , (3) PN meansofasymmetricfieldsthanbybreakingtheinternal whereV (X)=V (X+1),V′ (X) sin2πX isthe spatial symmetry of the system [25], since in the former PN PN PN ∼ Peierls-Nabarro (PN) potential accounting for the dis- case no structural changes of the system are required. creteness of the lattice and E˜(t) E(t) is the effective The paper is organized as follows. In Section II, we ∼ driving field of the kink [we assume E˜(t) to be propor- presentthe modeland derivethe necessaryconditionfor tional to E(t)]. An important parameter of the problem the directed kink motion in terms of a simple symmetry is the frequency of kink oscillations in the bottom of the analysis,whichisbasedonapointparticledescriptionof PN potential (the PN frequency), which can be written the kink dynamics. In this section we also describe the as [30] desymmetrizationmechanismandconfirmthe results by means of numerical simulations. In the next section, we ω = 2πα κ3/2exp( π2√κ/2), α 30π . (4) studythedependenceofdiscretekinkratchetsonthesys- PN p 0 − 0 ≃ temparameters. Inparticular,weinvestigatethe depen- Within this approximation, the unidirectional motion of dence of the meanvelocityof the kink onthe amplitude, the kink corresponds to a limit cycle of Eq. (3), which is phase, and frequency of the AC driver as well as on the phase locked to the frequency of the external driver. On dampingandthecouplingconstant. Moreover,thevalid- this orbit, the average kink velocity is expressed as ityoftheinternalmodemechanisminthediscretecaseis k ω discussed. In Section IV, we consider the soliton ratchet v = X˙(t) = , (5) h i h i l · 2π in a finite lattice and discuss possible applications of the phenomenon to arrays of small Josephson junctions. Fi- with k and l being integer numbers. Notice that in this nally, in Section V, we summarize the main conclusions resonantregime, the kink travels k sites during the time of the paper. lT =2πl/ω,sothat,exceptforashiftinspace,itsprofile is completely reproduced after this time interval (in the pendulum analogy, this orbit corresponds to k full rota- II. THE MODEL tions of the pendulum during l periods of the external drive). In the following,wewill referto the phaselocked The AC-driven and damped discrete sine-Gordon dynamics also as to resonances. (DSG) equation is introduced in a dimensionless form as follows A. Symmetry properties and conditions for u¨n κ∆un + sinun+αu˙n+E(t)=0, n=1,2,...N. transport − (1) InanalogywiththecontinuousSGcase[10,11],onecan expect that the directed kink motion arises when all the 2 symmetries of Eq. (3), which relate kink solutions with we have that Eq. (8) is violated for any θ (if m is even), oppositevelocities,arebroken. Qualitativeconditionsfor so that the kink transport should become possible. the occurrence of this directed motion can be obtained from the analysis of the symmetry properties of Eq. (3). 4 x 10 Inthisapproach,themany-particleproblemisreducedto the one-particle ratchet studied before [3,4] [we neglect 3 oscillations of the discrete kink profile (to be discussed (a) later) which also contribute to the phenomenon]. Our analysis is based on the simple observationthat the sign of the soliton velocity X˙(t) can be changed by means of 2 the following symmetry operations: Dˆ :t t+T/2, X X, (6) t X → →− Dˆ :X X+X , t t+2t , (7) T → 0 →− 0 1 whereDˆ denotesashiftintime followedbyareflection X isspaceandDˆ isashiftinspacefollowedbyareflection T in time (here t is a constantand X is either an integer 0 0 orahalfinteger). NoticethatEq. (3)isalwaysinvariant under the symmetry DˆX provided the external driver 100 200 n 300 400 500 satisfies 4 x 10 E(t)= E(t+T/2). (8) 3 − Notice that V ( X)=V (X) is always satisfied be- PN PN (b) − cause the sine-Gordon potential is symmetric. In the zero damping limit (α 0), Eq. (3) becomes invariant also under the symmetr→y Dˆ with X =1, provided the 2 T 0 external driver satisfies the condition t E(t+t )=E( t+t ), (9) 0 0 − 1 with t being a constant, which depends on the shape of 0 E(t). In the overdamped limit (α ), Eq. (3) can be →∞ rewrittenasX˙ +V′ (X)+E˜(t)=0fromwhichonecan PN see that it becomes invariant under the symmetry Dˆ T with X0 =1/2, providedthe external driver satisfies the 100 200 300 400 500 condition n FIG.1. Contourplot ofthetemporalevolution ofthepar- E(t+t )= E( t+t ). (10) 0 − − 0 ticle velocities u˙n(t) for (a) E1 = 0.3, E2 = 0.15 and (b) It should be remarked here that V′ (X) = V′ (X + E1 = 0.45, E2 = 0. Other parameters are: κ = 1, α = 0.05, PN − PN ω = 0.1, θ = 2, D = 0.002, N = 500. Periodic boundary 1/2) is always satisfied since for the DSG equation we have that V′ (X) sin2πX). From the above symme- conditions havebeen applied. PN ∼ try properties it follows that one can break all symme- From these arguments it is also clear that an external tries relating orbits with opposite velocities by properly periodicdriverofzeromean,whichconsistsofthesuper- choosing the driving force E(t). Thus, for the general case α=0, we have only the symmetry Dˆ for Eq. (3). position of only the first two harmonics, is the simplest 6 X driver that can be used to induce the soliton ratchets in This symmetry can be broken by choosing any function the DSG system. E(t) which violates Eq. (8). In the zero damping limit (α 0), we have, besides Dˆ , also the symmetry Dˆ X T → withX =1. Inthiscase,onemustchooseadrivingfield 0 B. Numerical study of transport vs symmetries E(t) which violates both Eq. (8) and Eq. (9). Similarly, intheoverdampedlimit(α ),afunctionE(t)which →∞ To verify the validity of the previous analysis and to violates both Eq. (8) and Eq. (10) should be chosen. checkthedesymmetrizationofthe orbitsasafunctionof From these considerations it follows that a simple si- thedriverparametersE ,E ,andθ,werecoursetodirect nusoidaldriver cannot support the kink transportin the 1 2 numerical integration of Eqs. (1). In order to be sure lattice, since Eqs. (6)-(10) in this case are always satis- thatthe systemexploresthe whole phasespace andthat fied. For a biharmonic driver of the type (2), however, 3 the phenomenon does not depend on initial conditions, investigations (see below and Sec. IV) show that this is we perform the first step simulations in the presence of true in almost all cases, except for very narrow intervals white noise. inthe parameterspacewheretwoattractorscancoexist. 0.02 (2,1) 0.1 0.05 (1,1) (1,1) 0.01 v=+ω/(2π) > v 0 < v 0 −0.05 (k=−1,l=1) (−1,1) −0.1 (a) −0.01 v=−ω/(2π) (−2,1) −1.57 0 1.57 −0.02 0 0.01 0.02 E /E 0.02 (± 1,2) FIG.3. Velocities of two kink2s,1running into the opposite directionsatα=0.05,ω=0.1,E1=0.21,θ=0asafunction > v 0 of E2/E1. Dashed line denotes that the given solution is no < longer stable. −0.02 This fact makes possible to investigate Eqs. (1) in the (b) absence of noise without averaging over the initial con- −1.57 0 1.57 ditions. In Fig. 2 we plot the time average kink velocity 0.005 v as a function of the phase difference θ in the case h i of a biharmonic driver with m = 2. In contrast to the > continuous case for which v was shown to have a sinu- v 0 soidal dependence on θ [10h,1i2], in the discrete case, we < find a complicated piece-wise dependence v (θ), which h i resembles a sinusoidalfunction only slightly. This is due −0.005 (c) to the fact that in the most of cases the dynamics is phase lockedto the externaldriverand the kink velocity −π −π/2 0 π/2 π is given by Eq. (5). θ Noticethatforaweakdamping[seepanel(a)ofFig.2] several resonances (k = l = 1, k = 2,l = 1) are FIG. 2. Dependence of the average kink velocity on the ± ± clearly visible, while for a strong damping [panel (c)], phase difference θ for E1 = E2 = 0.2, α = 0.05 (a), the kink becomes pinnedto the lattice for largeintervals α = 0.2 (b) and α = 0.5 (c). Other parameters are: κ = 1, ofθ (depinningofthekinkwouldrequirestrongerfields). ω =0.35. Corresponding pairs of rotation numbers (k,l) are FromFigs.2(a-c)aqualitativeunderstandingofthesym- given nearby the most pronounced resonances. Dashed lines metries to break, in order to achieve unidirectional mo- mark thecoordinate axes. tion, can be obtained. In particular, one can see that White noise has been included by adding in the r.h.s. closeto the underdampedlimit (α=0.05),the meanve- ofEqs.(1)astochastictermξ (t)ofzeromean, ξ (t) = locity becomes zero in the intervals θ [ 2.26, 1.81], 0, and with the autocorrelationn function ξ (t)ξh n(t′)i= θ [0.80,1.06], and θ [1.11,1.33] [se∈e F−ig. 2(a−)]. For 2αDδ δ(t t′). The resultingLangevinheqnuatiomns hiave lar∈ger values of the dam∈ping constant [see Fig.2(b)], we mn been integr−ated numerically by using the fourth-order have v = 0 in the intervals θ [ 2.66, 1.88] and h i ∈ − − Runge-Kutta method, adopting either free ends or peri- θ [0.48,1.26], while for α = 0.5, the kink stays pinned ∈ odicboundaryconditions: u (t)=u (t) 2π(positive for θ [ 2.5, 0.95] and θ [0.64,2.19]. By increasing and negative signs refer to nk+inNks and anntik±inks, respec- α, the∈ex−trema−l values of v∈ shift to 0 or π with the h i ± tively). growth of α. Moreover, the length of the intervals in θ Figure 1 illustrates the dynamics of a kink of the where v = 0 increases, while the average kink velocity h i damped DSG equation under the influence of noise, decreasesasαincreases. Thisbehaviorisanobviouscon- driven by a biharmonic driver (2) with m = 2 [panel sequenceoftheslowingdowneffectofthedampingonthe (a)], and by a single harmonic driver [panel (b)]. We see kink motion. In the overdamped limit α , the in- → ∞ thatwhilethesingleharmonicdriverisunabletoproduce tervals become centered around the points θ = π/2 for ± directed motion, the biharmonic driver is quite effective whichE(t)=E1cosωt+E2cos(2ωt+π/2)= E(π/ω − − to produce the kink transport. Fromthis figure,one can t) andthus the symmetry (6)is satisfied. Instead,in the also observe that there is only one attractor correspond- underdamped case (α 0), the centers of these inter- → ing to the unidirectional motion of the kink. Further vals gradually shift to the positions θ = 0, π for which ± 4 E(t)=E( t) and the symmetry (7) is satisfied. the kink motion. − We recall that in the continuum case [10], the func- The emergence of the directed kink motion can also tional dependencies obtained in the underdamped and be seen as a consequence of the desymmetrizationof the overdamped limits were found to be v sinθ and basinsofattractionofthe twolimitcyclescorresponding h i ∼ v cosθ, respectively. Thus, the transition from one tokinksmovingwithoppositevelocities(forsingleparti- h i ∼ limittoanotherinthe discretecaseis similartothe con- cleratchets,thedesymmetrizationofthe orbitshasbeen tinuouscaseinthesensethatthedirectedsolitonmotion shown in Refs. [3,31]). disappears in correspondence with values of θ for which the respective symmetry is restored. Xc 229978 (a) 229956 t 400 500 600 700 800 900 195 174 0.1 (b) 190 (c) 172 Xc 185 170 Xc (k=1,l=1) 180 168 x 10−3 175 166 0 170 18000 18200 18400 18600 18800 15000 16000 17000 18000 (b) t t 0 −2.5 242 265 > (± 1,2) −5 240 (d) Xc260 (e) <v (k=−1,l=1) −7.5 Xc 238 225505 0.129 0.1295 0.13 236 245 0 18400 18500 18600 18700 18800 16500 17000 17500 18000 t t −0.1 (a) −0.01 FIG. 5. Coordinate of the kink center Xc as a func- (−2,1) tion of time for different values of the driving amplitude −0.02 E1=E2 =0.129(a);0.12981(b,c)and0.1309(d,e). Inpanels (c) and (e), Xc(t) has been plotted after each oscillation pe- (−3,1) −0.03 riod T (see text for details). Other parameters are: α=0.1, 0.13 0.14 0.15 ω=0.35, κ=1, θ=−0.5. −0.2 0.1 0.15 0.2 0.25 As soon as the symmetry is broken (by switching on E the field E2), one of the basins of attraction begins to 1 shrink and eventually disappears as E increases. As a FIG. 4. Dependence of the average kink velocity on the 2 result, only the limit cycles which correspond to motion amplitude of the driver E1 = E2 for θ = −0.5 (decreasing in one direction survive, as one can see from Fig. 3. In dependence) and θ = 2 (increasing dependence). Other pa- particular,fortheparametersgiveninthisfigure,wefind rameters are: α=0.1, κ=1, ω=0.35. Numbersin brackets that the attractor corresponding to the limit cycle with show the respective pair of rotation numbers (k,l). Insets rotation numbers k = 1,l = 1 disappears at E /E = show more details for case θ=−0.5. 2 1 0.0118, i.e., already for a rather weak asymmetry of the field (E E ). In the discrete case, however, the soliton velocity is 2 1 ≪ zero not only for those values of θ which restore the symmetries (6) and (7), but also for some finite inter- III. DEPENDENCE OF THE PHENOMENON ON val around these values. SYSTEM PARAMETERS Inthecaseofoddvaluesofm,thecondition(8)issat- isfied for any value of θ if E = 0. We have indeed that 2 E(t+T/2)=E cos(ωt+π)+6 E cos(mωt+mπ+θ)= In this section, we investigate the dependence of kink 1 2 E(t) and therefore the symmetry Dˆ is always valid. transport on the system parameters and the internal X − mode mechanism. This observation implies that there should be no kink transport in the system, a result which is indeed con- firmed by direct numericalsimulationsof the full system A. Dependence on the driving amplitude with m = 3, both in the presence and in the absence of noise. Also, in analogy with the continuous case, we find that for the same parameter values the discrete an- For continuous ratchets it was shown previously [10] tikink ratchets always move in the direction opposite to that the average kink velocity is proportional to E12E2, 5 so that, provided the respective symmetries are broken, The same behavior is seen in panel (c) of Fig. 5 for a the directed motion can occur for arbitrary small values larger time scale. Note that the intermittency behavior of the driving amplitudes. For the DSG equation, the occursnotonlyaroundthedepinningthreshold,butalso dependence of the average kink velocity on the driver for larger values of the driving amplitude. Also notice, amplitudes is depicted in Fig. 4 from which one can see fromtheinset(b)ofFig.4,thatthedepinning threshold thatthekinkvelocityisnotasmoothmonotonicfunction has an hysteretic behavior. In this regard, we remark of E , but a piecewise function with plateaux of differ- that the numerical investigation has been performed by 1,2 ent lengths,resembling a “devil’s staircase”[the same as increasing the driving amplitude in small steps, taking in Fig. 2(a) but monotonic]. The plateau values of the the final state for a given step as an initial condition kink velocities are given by Eq. (5) and correspond to of the next one. When the pinned state loses stability, dynamical regimes, which are limit cycles with rotation the system finds itself on a chaotic attractor, which cor- numbers(k,l)phase-lockedtothedriver. Noticethatthe responds to the directed motion. By increasing further largest resonant step ( v = 0.0557) is achieved at the E we obtain larger values of the kink velocity, while if 1 h i rotation numbers corresponding to the main resonance we move backwards, we observe that two attractors can (k = 1,l = 1). One can easily observe the smaller coexist - one of them corresponding to a pinned kink ± resonant steps with k = 2,l = 1 and k = 1,l = 2, oscillating around its center of mass, and the other to ± ± for which the kink velocities equal v = 0.1114 and a kink performing ratchet dynamics. If E is further 1 h i v =0.02785,respectively. Higher order resonances can decreasing, the moving state loses stability and the sys- h i also be identified, but they are not well visible in the tem jumps back to the pinned state. The width of the figure. hysteresis(interval betweentwo bifurcations) appearsto The main feature of the discrete case is the existence be rather small, i.e., 0.129065 < E < 0.12979. Simi- 1 ofathresholdinthedriveramplitude(depinning thresh- lar hysteretic phenomena also appear at larger values of old), above which the transport can occur and below the driver amplitudes. For example, one can find that which the kink is pinned to a lattice site, oscillating in the interval 0.1660 < E < 0.1678, two limit cycles 1 around its center of mass: with k = 1,l = 2 and k = 1,l = 1 coexist. A more de- tailed numericalinvestigationof the v (E ) dependence X = Nn=1n(un+1−un−1). (11) shows that the observed devil’s staihrciase1is incomplete c P 2(u u ) and there are gaps inside it, which depend on the value N 1 − of θ. As the v (E ) dependence becomes steeper, less 1 In Fig. 5, the kink position Xc is depicted as a function phase locked shtaites are found. In particular for the case of time for different values of the driving amplitude. We θ =2, we see that in the transition from the k=1,l=2 see that when the driving amplitude is below threshold to the k =1,l =1 state, all intermediate rational values [panel(a)],thekinkperformsperiodicoscillationsaround of k and l are missing. its center ofmass, while above the threshold[panels (b)- (e)],thestandingkinkbecomesunstableandthedirected motion starts. By further increasing the driving ampli- B. Dependence on the driving frequency tudeabovethethreshold,thekinkmotionbecomeseither phaselockedtotheexternaldriverorchaoticwithanin- Thedependenceofthemeankinkvelocityonthedriver termittent behavior [see panels (b) and (c)]. Panels (d) frequency was investigated in Ref. [10] for the continu- and (e) of Fig. 5 show the phase-locked dynamics with ous SG model. In this case, it was shown that the aver- rotation numbers k = 1,l = 5. Notice that in this case, age velocity depends on the driving frequency as v the kink travels over a fixed number of sites during each sin[θ θ (ω;α)]/ω3, wheretanθ (ω;α)=[α/(2ω)]h [i3∼+ 0 0 period T in agreementwith Eq. 5. By further increasing (α/ω)−2]. This fact implies that max v de·cays θ∈[0,2π[ E , the the kink dynamics starts to switch between pe- h i 1 withthegrowthofω. Forafixedvalueofθ,however,this riodic (or quasiperiodic) regime and intermittency. Sim- dependence isdefinedbythe mutualrelationsbetweenω ilar transitions in the case of a single harmonic driver and α, so that v can experience oscillations and sign have also been reported in Refs. [23,28]. In the intermit- h i reversals before tending to zero either in a decreasing or tency regime, the kink dynamics switches in an unpre- in increasing way. Similar behavior is expected also for dictable manner between two attracting limit cycles [see the DSG equation, although in this case, the problem is panels(b)and(c)ofFig.5]. Inthiscase,thekinkdwells complicatedbythepresenceofthePeierls-Nabarro(PN) sometime intervalonacertainsite,pinnedbythe corre- frequency ω in Eq. (3). In analogy with Fig. 2, one PN sponding minimum of the PN barrier,before jumping to can expect the dependence v (ω) to be also a piecewise the next site. From panel (b) one can also see that the h i function with a “devil’s staircase” character. dwelling time is not rationally related to the driving pe- In Fig. 6, we depict the kink velocity, normalized to riod T and changes randomly from jump to jump. Thus the driving frequency ( v T =2πv/ω) as a function of ω in the intermittent regime the kink does not travel the h i for two values of the damping constant (notice that for samenumberofsitesduringanintegernumberofperiods phase locked dynamics, v T coincides with the ratio of of E(t) and the global dynamics is chaotic. h i 6 rotation numbers k/l). One can see that the dynamics whileinset(c)showsdetailsofthe v dependenceinthe h i is characterized by a series of resonances, the most pro- neighborhood of the second resonance. The coupling of nounced one being at the main frequency (k =1,l =1). the kink motion with linear waves occurs in the interval We also observe that the resonances with k > l are less 0.447 < ω < 0.517. At the beginning of this interval, pronounced and the subharmonic resonances with k < l the kink displaysa chaotic tailas shownin Fig.6(a). As arepracticallynotvisibleinthefigure. Thecaseofsmall frequency increases, the chaotic tail becomes more and damping and larger driver amplitude (depicted in the more regular. This is shown in inset (b) of the figure, figure by circles) is characterized by an almost mono- from which one can see that at ω =0.507 the oscillating tonic decay, while in the opposite case (larger damping tail is almost monochromatic. At ω = 0.517 the oscil- and smaller driving amplitude), there are wide pinned lating tail turns into a localized oscillating mode which regions and some peak around the value of ω [notice decays at infinity. A further increase of the driving fre- PN from Eq. (4) that ω 0.17 for κ=1]. quency causes the decay of the width of the mode. PN ≃ 800 470 11110123ΟΟΟΟΟΟ un2ππ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ un2ππ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 600 8000 t 9000 10000 446605 Xc 1 (a) <v>T56789 ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ −π00++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++1++++++++++++++0++++++++++++++++0++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++2+++++++++++++++0++++++++++++++0+++++++++++++++++++++++++++++++++++++++++++n+300 4(000a0..45) 500 ΟΟ++0+Ο+++++++Ο++++++++++++++++++++++1++++++++0++++++++0++++++++++++++++++++++++++++++++++1++++++++5++++++++0++++++++++++++++++++++++++++++++++2++++++0n0 (c25)0(b3)00 Xc240000 8600 t 8800 9000 333667050 Xc 234 t 234 ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ <v>T 0000....00123.Ο×4Ο0Ο×ΟΟ×ΟΟ×ΟΟ×0Ο.×4×5××××Ο××Ο×0Ο×.Ο×5ΟΟ0Ο×Ο×Ο××Ο×Ο×Ο×0Ο×ΟΟ.Ο×5Ο5Ο×ΟΟ×ΟΟΟΟ0Ο.Ο×60ω 0 2500 5000 7500 10000 12500 15000 17500 1××××××××× ΟΟΟΟΟ×××ΟΟΟΟΟΟΟΟΟΟΟΟΟΟ 500 1 -01 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××Ο×ΟΟΟ××ΟΟΟ×Ο×Ο×ΟΟΟΟΟΟΟ×ΟΟΟΟ×ΟΟΟΟ×ΟΟΟ×Ο×××Ο××Ο×Ο×Ο×Ο×ΟΟ×ΟΟ×ΟΟ×ΟΟ×ΟΟΟ×Ο×Ο×Ο×Ο××Ο××Ο×Ο×Ο×ΟΟΟ×Ο×Ο××Ο×Ο×Ο×ΟΟ×ΟΟ×ΟΟ×ΟΟ×ΟΟΟΟΟΟ×ΟΟΟ×ΟΟ×ΟΟ×ΟΟΟ×Ο×Ο×Ο×Ο×Ο×ΟΟ×ΟΟΟ×Ο×Ο×Ο×Ο×Ο×ΟΟ×ΟΟΟ×Ο×Ο×Ο×××Ο×Ο×Ο×Ο×ΟΟΟ×Ο×Ο×ΟΟ×ΟΟ×Ο×ΟΟ×Ο×Ο×Ο×Ο×Ο× 2 (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 400 t ω X c 380 3 300 370 Xc FIG.6. Dependence of the normalized kink velocity hviT on the driving frequency for E1 =E2 =0.1, θ =2, α=0.05 360 t (◦) and E1 = E2 = 0.02, θ = 1.5, α = 0.15 (×). In both 200 16000 17000 18000 cases κ = 1. The insets (a) and (b) show the kink profile 0 5000 10000 15000 20000 25000 30000 for the first set of parameters when ω =0.45 and ω =0.507, respectively. Inset (c) shows more detailed behavior around ω=0.5. FIG. 7. Time evolution of the position of the kink cen- ter for: (a) κ = 0.5; E1 = E2 = 0.21 (curve 1), 0.19 In both the cases we find that, except for the reso- (curve 2), 0.17 (curve 3), 0.11 (curve 4) and (b) κ = 0.25; nances ω ω and 2ω ω , where ω is the frequency E1 = E2 = 0.14 (curve 1), 0.24 (curve 2), 0.22 (curve 3). L L L ≈ ≈ Other parameters are: θ = 1.5, α = 0.1, ω = 0.35. The of linear waves upper inset of panel (a) corresponds to details of the case ω2(q)=2κ(1 cosq)+1, (12) E1 = E2 = 0.19 and the lower inset corresponds to the case L − E1 = E2 = 0.11. The inset of panel (b) corresponds to the the kink transport becomes effective mainly at low fre- case E1 = E2 = 0.22. In all figures, the data have been quencies ω ωL and disappears for driving frequencies plotted with the intervalT =2π/ω. ≪ ω > ω . Also, from Fig. 6 one can see that the kinks L become pinned to the lattice for values of ω significantly A similar scenario occurs also at ω ω , but in this L smallerthanωL. Attheresonancesω ωLand2ω ωL, casethekinkvelocityismuchsmallera≈ndpracticallynot ≈ ≈ weobservethatthekinkdynamicsbecomecoupledtolin- visible in Fig. 6. Notice that in the frequency interval earwavesinthe system(plasmonsinthe caseofarrayof 0.895 < ω < 1.05, the kink becomes again coupled to Josephson junctions). Insets (a) and (b) of Fig. 6 show linear modes and displays a chaotic tail, which becomes the profiles of the kink solution in these resonant cases, 7 more regular as ω further increases. At the beginning in the sense that the forward and backward jumps be- of this window, the dynamics in the tails are strongly come of larger amplitude with a consequent decrease of chaotic and they are accompanied by the formation of the averagevelocity. large-amplitude localized excitations (breathers). In the left panel of Fig. 8 the diagram of possible dy- Byreducingthedrivingamplitude,thefrequencywin- namical regimes in the plane (κ,E ) is shown. We ob- 1 dows, in which the coupling with linear waves occurs, serve that by changing the coupling constant κ, one can decrease. We find that for E = E = 0.04, the first passfromregular(phaselocked)dynamicstochaos. The 1 2 coupling window occurs at 0.480 < ω < 0.495 and the regular dynamics are dominant for κ > 1 and when κ secondoneat0.955<ω <0.994. Inthiscase,nochaotic is decreases, the windows of chaotic motion appear. By dressingofthekinkisobserved[thecouplingoccurswith further decreasing κ, the number of chaotic windows in- veryfew (orsingle)linearmodesandthe kinklooksvery creases and the chaotic motion becomes dominant. At similar to that in Fig. 6(b)]. For the driving amplitude κ=0.25, almost no regular dynamics exist. Notice that E =E =0.01,the coupling with linear wavesdoes not the diagram in Fig. 8 does not show the complete pic- 1 2 take place at all, and around the resonant frequencies ture and the details of transitions to different dynamical ω and ω /2,the kink remainspinned to the lattice. As regimes because in order to limit the computation time, L L thedrivingamplitudeincreases,beyondacertainthresh- we have used in the numerical calculations the step of old chaotic oscillations can completely destroy the kink. 0.01 in the amplitude of the driving field E =E and a 1 2 From these results, we conclude that the coupling of the coarse step of 0.25 in κ. We find that for driving ampli- kink dynamics with linear waves depends very much on tudes E =E &0.3, the chaotic dynamics of the whole 1 2 the amplitude of the externalfield and this effect is non- lattice destroy kink solutions. linear in E . Near the resonances with linear modes, 1 the kink dynamics become more chaotic (diffusive), re- sembling the one ofa Brownianparticle (the kink makes manyrandomjumps backwardandforwardandthe uni- directionalmotioncanbeseenonlyatalargetimescale). slkrtwtccihotaroimeaneptenIltncnkilostehcttl.ncokoeatκenetnhgnICnsndoiit=spcst.wieκatFhoDsrin0l,efoieg.onewstcn5[.fophot3siil[m7etme2oiptht,n]snoaaeo,ywdκ,nnsnsssteeoeuwt→hnlnXpesdeecmtr(eceaooa∞idpinpn)finoiois]vasccnrtcacetgmohrhstnarttewteoatdrhhitibrgoeeeoeiaκsiannnvlcpttetcea.=teeeooselmkrnsuurItai0diennhppczu.si2keotelnsoiio5sdrungoudafanigb[reltntplypohleaccmyteevtoonthnhohonncepedledeolssruestueeti(ctnaynlavbpio,ncostne)lnteeintnt]tnetr.htimaoκgonseFcffu.crtafdtotuiorAnohhinemmsndeees-- E00001....0123 ⊕ΟΟΟΟΟΟΟΟΟΟΟΟ♦♦×××××××××××××××× ⊕⊕⊕⊕ΟΟΟΟΟΟΟ♦××××××××××××××××××× ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟΟ♦××××××× ⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟΟΟΟΟ♦♦×××××××××× ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟΟ×× ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟ ∇∇×××××××× ∇∇∇×∇∇∇∇∇∇∇∇×∇∇∇∇××∇∇∇×∇∇∇∇×××××∇∇∇∇∇∇×∇∇∇ΟΟΟΟΟΟΟΟΟΟΟΟΟΟ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇×××××××××××××××××××××××××××Ο◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊ΟΟΟ♦Ο×♦◊Ο◊◊♦Ο♦Ο◊♦◊Ο♦◊◊♦Ο×♦◊♦Ο◊Ο◊∇×Ο∇××♦◊◊∇◊∇∇∇∇◊◊Ο◊∇∇♦♦Ο♦Ο×◊×Ο♦Ο♦×∇∇∇∇∇×◊◊◊◊◊◊Ο♦♦Ο♦Ο×Ο♦×Ο♦◊♦×◊♦Ο♦Ο×Ο♦Ο♦×♦◊◊♦ΟΟΟ◊◊◊◊◊◊ΟΟ♦♦♦∇∇♦××××××××××××××××◊◊◊◊◊◊◊◊◊◊◊Ο♦◊◊Ο◊◊◊♦◊<<◊Ο♦◊◊vv♦--Ο000>>♦Ο...♦♦♦♦00111−Ο◊⊕Οπ×ΟΟ×⊕ΟΟ×ΟΟΟ×××Ο×ΟΟΟΟΟΟ×Ο×Ο×××××Ο××Ο×⊕Ο×ΟΟ×Ο×♦♦ΟΟΟ♦♦♦♦♦♦♦◊⊕Ο×⊕♦Ο×Ο⊕Ο×⊕⊕⊕⊕ΟΟ×⊕⊕Ο×⊕Ο0Ο×♦Ο×♦ΟΟ♦⊕Ο×♦⊕⊕Ο⊕⊕⊕×ΟΟ×Ο×ΟΟ×Ο××Ο×Ο×⊕Ο×Ο×⊕ΟΟ×Ο×Ο×θΟ×⊕Ο×⊕Ο×ΟΟ×⊕Ο×Ο×ΟΟ⊕Ο×((⊕Ο×ab⊕Ο⊕Ο×))ΟΟ⊕Ο×ΟΟΟπ 0000....0123E1 this figure, we conclude that the unidirectional motion 0 0.5 1.0 1.5 2.0 -0.05 0.00 0.05 0.10 0.15 exists also for small values of κ with dynamics which is κ <v> chaoticratherthanphaselocked(seeinsetsofthefigure). Chaotic motion, leading to transport is found to be either of the intermittent type [curve 4 of panel (a)] or diffusive type [curves 1-3 of panel (a)]. The intermittent regime is more often observed for κ & 1 and the behav- ior is the same as described in the previous subsection, i.e., the kink oscillates in the minima of the PN poten- tial until it jumps to the adjacent site [see lower inset of panel (a)]. The motion is characterized by the fact that the jumps of the kink occur randomly in time, but alwaysinthesamedirection. Forκ.1thediffusivemo- tion is the most typical scenario. In this case, the kink jumps randomly forward and backward [see upper inset of panel (a) and inset of panel (b)], but on average the motion remains unidirectional. A decrease of the cou- pling constant makes the dynamics even more diffusive 8 FIG. 8. Left panel. Diagram that demonstrates main understandingofthis behavior,wehaveplottedininsets regimes of the kink ratchet motion for different values of the (a) and (b) the mean velocity as a function of the phase couplingconstantκandthedrivingamplitudeE1 =E2(other difference θ. The inset (a) refers to the case κ=0.75 for parameters are: θ =1.5, α=0.1, and ω =0.35). In the dia- twodifferentvalues ofthe driveramplitude. We see that gram ◦ stands for the regime when kink is pinned (hvi=0), while for E = 0.19 the dependence v (θ) is piecewise, ⊕ stands for the periodic or quasiperiodic ratchet motion, × 1 h i butsomewhatsimilar toFig.2(i.e., it hasone minimum stands for the chaotic regime, and (cid:7) corresponds to the case andonemaximum),thedependencebecomesmoreirreg- when kinksolutions do not exist. ular as E increases, with small islands of phase locked Right panel. Dependence of the average kink velocity on the 1 regimes with opposite velocities. A similar phenomenon driving amplitude E1 = E2 for different values of the cou- is observed, when κ decreases to κ = 0.5 [inset (b)]. pling constant κ = 0.5(∆), κ= 0.75 (×), κ= 1 (⋄), κ= 1.5 ((cid:7)), and κ = 2 (◦). Other parameters are as in the left Thisbehaviorislinkedtothewellknownphenomenonof panel. The insets show dependence of the average velocity crises of attractors, i.e., to sudden appearance(or disap- on the phase difference θ. Inset (a) corresponds to κ=0.75, pearance) of an attractor as a system parameter varies E1=E2=0.19 (◦) and E1 =E2=0.25 (×),inset (b) corre- (see [33]). By further decreasing the coupling constant, spondstoκ=0.5,E1=E2 =0.22(⊕). Solidlinesaredrawn more and more attractor crises are found. as a guide to an eye. From the insets of Fig. 8, we also see that, in contrast withthecaseκ=1inFig.2,thedependence v = v (θ) h i h i IntherightpanelofFig.8,weshowthedependenceof losesanyresemblancewithasinusoidalfunctionforsmall the averagekink velocity on the driving amplitude E = values of κ. From Fig. 8, it is also evident that the de- 1 E for different values of the coupling constant κ. We pinning amplitude of the kink motion increases when κ 2 observe that already for κ=2, the curve is very smooth decreases. For very small κ, e.g., κ=0.0625 (not shown (except for the smallresonantsteps k=3,l=1 andk = in Fig. 8), no directed motion is found - the kink is ei- 3,l = 2) and the behavior becomes very similar to that therpinnedbythelatticeordestroyedbychaoswhenthe reported for the continuous limit ( v E2E ) [10]. By driving amplitude becomes large enough. This behavior h i ∼ 1 2 increasing the driving amplitudes E , the dependence is not surprising, since we know that high discreteness 1,2 becomes non-monotonic, a fact which was also observed normally prevents solitons from free propagation. inthecontinuousSGcaseandascribedtotheinteraction of the kink with internal oscillation modes [10]. On the other hand, decrease of the coupling constant makes the phase locking steps more pronounced. For κ = 1.5, the 2π dependence hvi(E1) almost coincides with the one for Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+(Οa+)Ο+Ο+Ο+Ο+ κ=2(forlargeramplitudes,however,thethreeresonant un Ο+ steps k = l = 1, k = 3, l = 2, and k = 2, l = 1 become π Ο+ much more visible). Ο+ Further decrease of the coupling constant makes the Ο+ dependence even less monotonic: for κ = 1 (shown by 018Ο+0Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο2+0Ο+0Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+ 220 240 ⋄ n in the figure), the non-monotonicity is quite weak, with asmallinterval0.145<E1 <0.154inwhichthekinkve- un locity drops back to zero. For E1 &0.2, the dependence 6.60 (b) is also non-monotonic because of the interaction of the 3.05 kink with vibrational modes localized on it. For κ=0.5 andκ=0.75,theaveragekinkvelocitybecomesstrongly -0.50 240 non-monotonic. 210 130 Notice from the right panel of Fig. 8 that several re- versals of the kink motion along the E axis occur. In n 180 6t5 1 0 this regard, we remark that the sign of v depends on h i the relations between ω, α, and ωPN. Decrease of κ can FIG. 9. Evolution of the kink profile un(t). Upper panel change ωPN significantly [see Eq. (4)], thus effecting the showsinitialkinkprofile(+)andkinkprofileafterintegration sign of the velocity. The reversals of v as a function of time 7T (◦), shifted two sites backwards. Parameters of the h i E1, however, are not fully explained by these arguments model are: E1=E2 =0.135, θ=1.5, α=0.1, ω=0.35, and andwebelievethat,inanalogywiththecontinuouscase, κ=1. thecouplingofthekinkwithsmallamplitudewavesplays also an important role. Weremarkthattheattractorcrisesleadingtoreversal By comparing the left and the right panels of Fig. 8, phenomena are out of the range of validity of the single one can also see that the non-monotonic jumps occur particle approximation since they involve many degrees in correspondence with the transitions from chaotic to of freedom. In particular, in the limit of weak coupling, regular (phase locked) regimes. In order to get a better the point particle approximation (3) as well as the sym- metry analysis discussed in Sec. II become not valid. 9 D. Internal mode mechanism and dependence on the is also observedfrom the three-dimensionalplot of un(t) damping parameter in panel (b). Similar results are obtained for different values of the driving amplitudes, as one can see from Inthecaseofcontinuoussolitonratchets,itwasshown Fig. 10. In this case, the kink is lockedon the resonance (see Refs. [9,10,19,20,18]) that a contribution to the with rotationnumbers (1,2) (notice that the kink repro- ratchet phenomenon comes also from the internal oscil- duces itself completely after the time 2T). In panel (b), lation of the kink via the internal mode mechanism. We the complete dynamics inthe time interval[0,4T]is also expectthiseffecttobetruealsointhepresentcase,both shown. The fact that the center of mass motion and for large values (i.e., close to the continuum limit) and the oscillations on the kink profile (internal mode) are intermediate andsmallvalues ofthe coupling constantκ perfectly synchronizedsuggeststhe existence ofcoupling (for very small values of κ, however, the kink becomes between the internal and translational modes similarly pinned to the lattice and the ratchetphenomenondisap- as for the continuous case. pears as discussed above). The existence of the internal mode mechanism in the Inthefollowingweinvestigatetheinternalmodemech- discrete case is also supported by the influence of the anism by fixing κ = 1 and performing direct numerical dampingconstantonthephenomenon. Inthecontinuous simulations of Eqs. (1). In particular, we show the exis- case, it was shown that the coupling between the trans- tence of a local oscillation on the kink profile which is, lational and internal modes decreases with the damping perfectly synchronized (phase locked) with the kink mo- and for the case of asymmetric potentials with symmet- tion. We havefoundthatwhenthe dynamicsofthe kink ric drivers, it was proved that in the limit α 0, the → center of mass is phase locked to the resonance (k,l), coupling completely disappears [18]. In the case of bi- the internal mode oscillation is also locked to the same harmonicasymmetric forces,the nonlinearityinduces an resonance. This is shown in Fig. 9 for the discrete kink effectivebias component,whichgivesrisetoapointpar- ratchet,whichisphaselockedtotheexternaldriverwith ticle contribution also in the absence of damping. rotation numbers k = 2, l = 7. In the panel (a) of thisfigure,theinitialandfinalkinkprofilesaredepicted, fromwhichonecanconcludethattheseprofilesperfectly coincide after l =7 periods of the externaldriver(notice that the final configurationhas been shifted by two sites backwards in order to demonstrate full coincidence). 0.4 0.3 2π Ο+Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊ E1 un Ο+◊ (a) 0.10 ⊕ π Ο+◊ 0.2 ∆E ⊕ ⊕ ⊕⊕ Ο+◊ 0.05 ⊕⊕ ⊕ ⊕ 0Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊◊ 0.1 ⊕⊕ ⊕⊕ un 180 n 200 220 0.000.0⊕ 0.1 0.2 0.3 ⊕⊕⊕⊕⊕0⊕⊕.4 α 6.8 0.0 0.0 0.1 0.2 0.3 0.4 3.1 α (b) -0.6 FIG. 11. Existence diagram of the main resonance 240 (k = l = 1) on the plane (α,E1 = E2) for ω = 0.35, θ = 2, 210 and κ = 1. The inset shows the dependence of the width of n resonance∆E ondampingcoefficientα. Whentheresonance 180 72 36. 0 t consists of several islands, ∆E is computed as a sum of the widths of individual islands. FIG. 10. Same as in Fig. 9. All parameters are the same except E1 =E2 =0.17. In panel (a), the initial kink config- The contribution of the internal mode to the ratchet uration is shown by (+), the configuration after time T (⋄) dynamics, however, should be sensitive to the damping, and after time 2T (◦), shifted by one site backwards. Panel ifthe coupling is controlledby the damping andthe uni- (b) shows three-dimensional dynamical picture for the kink directional motion should become less effective for small evolution in the time interval[0,4T]. valuesofthedampingconstant. Ontheotherhand,when the damping in the system becomes too large, the dy- The phase lockingofthe motionto the externaldriver namics are stronglyreduced(or stopped), so thata non- 10

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Discrete soliton ratchets driven by biharmonic fields PDF

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Preview Discrete soliton ratchets driven by biharmonic fields

Discrete soliton ratchets driven by biharmonic fields Yaroslav Zolotaryuk† and Mario Salerno‡ † Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine Kyiv 03143, Ukraine ‡ Dipartamento di Fisica “E. R. Caianiello” and Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia (CNISM), Universit´a di Salerno, I-84081 Baronissi, Salerno, Italy (February 4, 2008) Directed motion of topological solitons (kinksor antikinks) in thedampedand AC-drivendiscrete sine-Gordon system is investigated. We show that if the driving field breaks certain time-space symmetries, the soliton can perform unidirectional motion. The phenomenon resembles the well 6 known effects of ratchet transport and nonlinear harmonic mixing. Direction of the motion and its 0 velocitydependsontheshapeoftheACdrive. Necessaryconditionsfortheoccurrenceoftheeffect 0 are formulated. In comparison with thepreviously studied continuumcase, thediscrete case shows 2 a number of new features: non-zero depinning threshold for the driving amplitude, locking to the n rationalfractionsofthedrivingfrequency,anddiffusiveratchetmotioninthecaseofweakintersite a coupling. J 0 Pacs: 05.45.-a,07.90.+c, 89.20.-a demonstrated by using asymmetric magnetic fields [16] 2 and spatially asymmetric currents [17]. From the theo- ] retical point of view, symmetry breaking conditions to S I. INTRODUCTION generatesolitonratchetswerediscussedinRef.[11]. The P mechanism underlying soliton ratchets was proposed in n. Transport phenomena induced by the interplay be- Ref. [9] for the case of a perturbed asymmetric double i tween non-equilibrium fluctuations, symmetry breaking sine-Gordon equation driven by a symmetric AC driver l n and nonlinearity, have recently attracted a great deal of andextendedinRef.[10]tothecaseofthedampedsine- [ interest. In particular, point particle ratchets described Gordon (SG) equation with asymmetric AC fields. In 1 byordinarydifferentialequationshavebeenlargelyinves- both cases, the phenomenon was ascribed to the exis- v tigated due to their relevance in several fields, including tence of an internaloscillationon the kink profile which, 4 molecular motors and Josephson junctions (see reviews in the presence of damping, couples to the translational 4 [1,2]). In simple terms, the point particle ratchets ap- mode of the kink and produces transport. The internal 0 pearastheunidirectionalmotionofadampedanddriven vibrationwasshowntobespatiallyasymmetric,thusgiv- 1 particle, which is achieved under the influence of only ing directionality to the motion, and phase lockedto the 0 stochastic and deterministic forces of zero average, in- externalforce. Thismechanism,alsoknownasthe inter- 6 0 dependently on initial conditions. The phenomenon was nal mode mechanism, has been confirmed for a number / ascribed to the breaking of the symmetries connecting of systems such as the asymmetric double sine-Gordon n orbits with opposite velocities in the phase space [3–5] equation with symmetric driver [9,18] and the SG sys- i l and to the phase locking of the particle dynamics to the tem with temporal asymmetric forces [10,19,20]. n : external driver [6,7]. In contrast to the continuous case, however, discrete v Ratchetphenomenaininfinite-dimensionalsystemsde- soliton ratchets have been scarcely investigated (some Xi scribedbynonlinearpartialdifferentialequationsofsoli- work on spatially asymmetric discrete soliton ratchets tontypehavealsobeeninvestigated. Thishasbeendone has been done in Refs. [21,22]). In this case, one can ex- r a for both spatially asymmetric potentials with temporar- pectthatthe presenceofthe Peierls-Nabarrobarriercan ily symmetric AC fields [8,9] and for symmetric poten- strongly influence the transport of discrete solitons. It tials with temporarily asymmetric AC fields [10–13]. In is therefore of interest to investigate the conditions un- both cases, the ratchet effect appears as a unidirectional der which the discrete soliton ratchets can exist. The motion of the soliton, which resembles the drift dynam- present paper is just devoted to this investigation. More ics observed for point particle ratchets; from here the precisely,westudytheratchetdynamicsinducedbytem- nameofsoliton ratchetsoriginates[9]. Thesolitonratch- porarily asymmetric forces of zero mean on topological ets induced by asymmetrical external fields have been solitons(kinksandantikinks)ofthediscretesine-Gordon implemented experimentallyinlong Josephsonjunctions (DSG)system. Thisequationmodelsanumber ofphysi- [14] by means of nonlinear harmonic mixings. This ap- calsystemssuchasarraysofJosephsonjunctions,crystal proachhasbeenshowntobe effectivealsoinotherphys- dislocationsorcharge-densitywaves(see[23,24]). Inpar- ical contexts [15] (for a detailed list of references see ticular,weinvestigatetheconditionsfortheoccurrenceof the review paper [2]). The existence of soliton ratch- solitonratchetsandstudy thedependence oftheaverage ets in long Josephson junctions was also experimentally solitonvelocityonthesystemparameters. Acomparison 1 withtheresultsderivedforcontinuoussolitonratchetsin Here u is the displacement of the nth particle from its n Ref. [10] is also provided. equilibrium position, ∆u u 2u +u is the n n+1 n n−1 ≡ − From our study it emerges that discrete soliton ratch- discrete Laplacian, κ is the coupling constant measuring ets are much more complicated than their continuum thediscretenessofthelattice,αisthedampingcoefficient counterpart. Inparticular,the meanvelocityof the kink and E(t) is an externaldriving field. In the following we in most cases appears to be a piece-wise function of the assume E(t) to be of the form parameters which resembles a devil’s staircase. We find E(t)=E cos(ωt)+E cos(mωt+θ) , (2) that kink transport becomes very effective on the cor- 1 2 responding orbits (limit cycles) which are phase locked where m is an integer even number. Notice that the su- to the external driver. Transport is possible also in the perpositionoftwoharmonicsmakesthe periodicforceto presence of more complicated dynamics such as chaotic be asymmetric in time for almost all values of θ, a fea- andintermittency orbits,especially whenthe systembe- ture which can be used to break the temporal symmetry comes very discrete (this is achieved when the coupling of the system (see below). In this context, it is of in- constantisverysmall). Inthesecases,however,the kink terest to investigate the condition under which a driving transport is not very efficient since the drift velocity is force of zero mean of type (2) can induce kink’s unidi- rather small. Except for the very discrete case, domi- rectional motion similar to the one observed in the con- nated by the pinning of the kink to the lattice, we find tinuum SG case [10]. In this regard, we remark that that the internal mode mechanism remains valid also in in the lower approximation, a discrete kink of the form the DSG ratchet in all cases in which the transport is u (t)=4arctan exp[n X(t)] ,canbeviewedasasin- n observed. gle particle [24] {and its−dynam}ics is described in terms We remark that the soliton ratchets induced by tem- of collective coordinates: the center of mass X(t) and porarilyasymmetricfieldsmaybeaneffectivewaytocon- the kink velocity X˙(t). In this approach, the effective trol the transport properties of a large variety of contin- point-particle equation of the motion becomes uous and discrete systems. From the experimental point of view, indeed, it is much easier to produce ratchets by X¨ +αX˙ +V′ (X)+E˜(t)=0 , (3) PN meansofasymmetricfieldsthanbybreakingtheinternal whereV (X)=V (X+1),V′ (X) sin2πX isthe spatial symmetry of the system [25], since in the former PN PN PN ∼ Peierls-Nabarro (PN) potential accounting for the dis- case no structural changes of the system are required. creteness of the lattice and E˜(t) E(t) is the effective The paper is organized as follows. In Section II, we ∼ driving field of the kink [we assume E˜(t) to be propor- presentthe modeland derivethe necessaryconditionfor tional to E(t)]. An important parameter of the problem the directed kink motion in terms of a simple symmetry is the frequency of kink oscillations in the bottom of the analysis,whichisbasedonapointparticledescriptionof PN potential (the PN frequency), which can be written the kink dynamics. In this section we also describe the as [30] desymmetrizationmechanismandconfirmthe results by means of numerical simulations. In the next section, we ω = 2πα κ3/2exp( π2√κ/2), α 30π . (4) studythedependenceofdiscretekinkratchetsonthesys- PN p 0 − 0 ≃ temparameters. Inparticular,weinvestigatethe depen- Within this approximation, the unidirectional motion of dence of the meanvelocityof the kink onthe amplitude, the kink corresponds to a limit cycle of Eq. (3), which is phase, and frequency of the AC driver as well as on the phase locked to the frequency of the external driver. On dampingandthecouplingconstant. Moreover,thevalid- this orbit, the average kink velocity is expressed as ityoftheinternalmodemechanisminthediscretecaseis k ω discussed. In Section IV, we consider the soliton ratchet v = X˙(t) = , (5) h i h i l · 2π in a finite lattice and discuss possible applications of the phenomenon to arrays of small Josephson junctions. Fi- with k and l being integer numbers. Notice that in this nally, in Section V, we summarize the main conclusions resonantregime, the kink travels k sites during the time of the paper. lT =2πl/ω,sothat,exceptforashiftinspace,itsprofile is completely reproduced after this time interval (in the pendulum analogy, this orbit corresponds to k full rota- II. THE MODEL tions of the pendulum during l periods of the external drive). In the following,wewill referto the phaselocked The AC-driven and damped discrete sine-Gordon dynamics also as to resonances. (DSG) equation is introduced in a dimensionless form as follows A. Symmetry properties and conditions for u¨n κ∆un + sinun+αu˙n+E(t)=0, n=1,2,...N. transport − (1) InanalogywiththecontinuousSGcase[10,11],onecan expect that the directed kink motion arises when all the 2 symmetries of Eq. (3), which relate kink solutions with we have that Eq. (8) is violated for any θ (if m is even), oppositevelocities,arebroken. Qualitativeconditionsfor so that the kink transport should become possible. the occurrence of this directed motion can be obtained from the analysis of the symmetry properties of Eq. (3). 4 x 10 Inthisapproach,themany-particleproblemisreducedto the one-particle ratchet studied before [3,4] [we neglect 3 oscillations of the discrete kink profile (to be discussed (a) later) which also contribute to the phenomenon]. Our analysis is based on the simple observationthat the sign of the soliton velocity X˙(t) can be changed by means of 2 the following symmetry operations: Dˆ :t t+T/2, X X, (6) t X → →− Dˆ :X X+X , t t+2t , (7) T → 0 →− 0 1 whereDˆ denotesashiftintime followedbyareflection X isspaceandDˆ isashiftinspacefollowedbyareflection T in time (here t is a constantand X is either an integer 0 0 orahalfinteger). NoticethatEq. (3)isalwaysinvariant under the symmetry DˆX provided the external driver 100 200 n 300 400 500 satisfies 4 x 10 E(t)= E(t+T/2). (8) 3 − Notice that V ( X)=V (X) is always satisfied be- PN PN (b) − cause the sine-Gordon potential is symmetric. In the zero damping limit (α 0), Eq. (3) becomes invariant also under the symmetr→y Dˆ with X =1, provided the 2 T 0 external driver satisfies the condition t E(t+t )=E( t+t ), (9) 0 0 − 1 with t being a constant, which depends on the shape of 0 E(t). In the overdamped limit (α ), Eq. (3) can be →∞ rewrittenasX˙ +V′ (X)+E˜(t)=0fromwhichonecan PN see that it becomes invariant under the symmetry Dˆ T with X0 =1/2, providedthe external driver satisfies the 100 200 300 400 500 condition n FIG.1. Contourplot ofthetemporalevolution ofthepar- E(t+t )= E( t+t ). (10) 0 − − 0 ticle velocities u˙n(t) for (a) E1 = 0.3, E2 = 0.15 and (b) It should be remarked here that V′ (X) = V′ (X + E1 = 0.45, E2 = 0. Other parameters are: κ = 1, α = 0.05, PN − PN ω = 0.1, θ = 2, D = 0.002, N = 500. Periodic boundary 1/2) is always satisfied since for the DSG equation we have that V′ (X) sin2πX). From the above symme- conditions havebeen applied. PN ∼ try properties it follows that one can break all symme- From these arguments it is also clear that an external tries relating orbits with opposite velocities by properly periodicdriverofzeromean,whichconsistsofthesuper- choosing the driving force E(t). Thus, for the general case α=0, we have only the symmetry Dˆ for Eq. (3). position of only the first two harmonics, is the simplest 6 X driver that can be used to induce the soliton ratchets in This symmetry can be broken by choosing any function the DSG system. E(t) which violates Eq. (8). In the zero damping limit (α 0), we have, besides Dˆ , also the symmetry Dˆ X T → withX =1. Inthiscase,onemustchooseadrivingfield 0 B. Numerical study of transport vs symmetries E(t) which violates both Eq. (8) and Eq. (9). Similarly, intheoverdampedlimit(α ),afunctionE(t)which →∞ To verify the validity of the previous analysis and to violates both Eq. (8) and Eq. (10) should be chosen. checkthedesymmetrizationofthe orbitsasafunctionof From these considerations it follows that a simple si- thedriverparametersE ,E ,andθ,werecoursetodirect nusoidaldriver cannot support the kink transportin the 1 2 numerical integration of Eqs. (1). In order to be sure lattice, since Eqs. (6)-(10) in this case are always satis- thatthe systemexploresthe whole phasespace andthat fied. For a biharmonic driver of the type (2), however, 3 the phenomenon does not depend on initial conditions, investigations (see below and Sec. IV) show that this is we perform the first step simulations in the presence of true in almost all cases, except for very narrow intervals white noise. inthe parameterspacewheretwoattractorscancoexist. 0.02 (2,1) 0.1 0.05 (1,1) (1,1) 0.01 v=+ω/(2π) > v 0 < v 0 −0.05 (k=−1,l=1) (−1,1) −0.1 (a) −0.01 v=−ω/(2π) (−2,1) −1.57 0 1.57 −0.02 0 0.01 0.02 E /E 0.02 (± 1,2) FIG.3. Velocities of two kink2s,1running into the opposite directionsatα=0.05,ω=0.1,E1=0.21,θ=0asafunction > v 0 of E2/E1. Dashed line denotes that the given solution is no < longer stable. −0.02 This fact makes possible to investigate Eqs. (1) in the (b) absence of noise without averaging over the initial con- −1.57 0 1.57 ditions. In Fig. 2 we plot the time average kink velocity 0.005 v as a function of the phase difference θ in the case h i of a biharmonic driver with m = 2. In contrast to the > continuous case for which v was shown to have a sinu- v 0 soidal dependence on θ [10h,1i2], in the discrete case, we < find a complicated piece-wise dependence v (θ), which h i resembles a sinusoidalfunction only slightly. This is due −0.005 (c) to the fact that in the most of cases the dynamics is phase lockedto the externaldriverand the kink velocity −π −π/2 0 π/2 π is given by Eq. (5). θ Noticethatforaweakdamping[seepanel(a)ofFig.2] several resonances (k = l = 1, k = 2,l = 1) are FIG. 2. Dependence of the average kink velocity on the ± ± clearly visible, while for a strong damping [panel (c)], phase difference θ for E1 = E2 = 0.2, α = 0.05 (a), the kink becomes pinnedto the lattice for largeintervals α = 0.2 (b) and α = 0.5 (c). Other parameters are: κ = 1, ofθ (depinningofthekinkwouldrequirestrongerfields). ω =0.35. Corresponding pairs of rotation numbers (k,l) are FromFigs.2(a-c)aqualitativeunderstandingofthesym- given nearby the most pronounced resonances. Dashed lines metries to break, in order to achieve unidirectional mo- mark thecoordinate axes. tion, can be obtained. In particular, one can see that White noise has been included by adding in the r.h.s. closeto the underdampedlimit (α=0.05),the meanve- ofEqs.(1)astochastictermξ (t)ofzeromean, ξ (t) = locity becomes zero in the intervals θ [ 2.26, 1.81], 0, and with the autocorrelationn function ξ (t)ξh n(t′)i= θ [0.80,1.06], and θ [1.11,1.33] [se∈e F−ig. 2(a−)]. For 2αDδ δ(t t′). The resultingLangevinheqnuatiomns hiave lar∈ger values of the dam∈ping constant [see Fig.2(b)], we mn been integr−ated numerically by using the fourth-order have v = 0 in the intervals θ [ 2.66, 1.88] and h i ∈ − − Runge-Kutta method, adopting either free ends or peri- θ [0.48,1.26], while for α = 0.5, the kink stays pinned ∈ odicboundaryconditions: u (t)=u (t) 2π(positive for θ [ 2.5, 0.95] and θ [0.64,2.19]. By increasing and negative signs refer to nk+inNks and anntik±inks, respec- α, the∈ex−trema−l values of v∈ shift to 0 or π with the h i ± tively). growth of α. Moreover, the length of the intervals in θ Figure 1 illustrates the dynamics of a kink of the where v = 0 increases, while the average kink velocity h i damped DSG equation under the influence of noise, decreasesasαincreases. Thisbehaviorisanobviouscon- driven by a biharmonic driver (2) with m = 2 [panel sequenceoftheslowingdowneffectofthedampingonthe (a)], and by a single harmonic driver [panel (b)]. We see kink motion. In the overdamped limit α , the in- → ∞ thatwhilethesingleharmonicdriverisunabletoproduce tervals become centered around the points θ = π/2 for ± directed motion, the biharmonic driver is quite effective whichE(t)=E1cosωt+E2cos(2ωt+π/2)= E(π/ω − − to produce the kink transport. Fromthis figure,one can t) andthus the symmetry (6)is satisfied. Instead,in the also observe that there is only one attractor correspond- underdamped case (α 0), the centers of these inter- → ing to the unidirectional motion of the kink. Further vals gradually shift to the positions θ = 0, π for which ± 4 E(t)=E( t) and the symmetry (7) is satisfied. the kink motion. − We recall that in the continuum case [10], the func- The emergence of the directed kink motion can also tional dependencies obtained in the underdamped and be seen as a consequence of the desymmetrizationof the overdamped limits were found to be v sinθ and basinsofattractionofthe twolimitcyclescorresponding h i ∼ v cosθ, respectively. Thus, the transition from one tokinksmovingwithoppositevelocities(forsingleparti- h i ∼ limittoanotherinthe discretecaseis similartothe con- cleratchets,thedesymmetrizationofthe orbitshasbeen tinuouscaseinthesensethatthedirectedsolitonmotion shown in Refs. [3,31]). disappears in correspondence with values of θ for which the respective symmetry is restored. Xc 229978 (a) 229956 t 400 500 600 700 800 900 195 174 0.1 (b) 190 (c) 172 Xc 185 170 Xc (k=1,l=1) 180 168 x 10−3 175 166 0 170 18000 18200 18400 18600 18800 15000 16000 17000 18000 (b) t t 0 −2.5 242 265 > (± 1,2) −5 240 (d) Xc260 (e) <v (k=−1,l=1) −7.5 Xc 238 225505 0.129 0.1295 0.13 236 245 0 18400 18500 18600 18700 18800 16500 17000 17500 18000 t t −0.1 (a) −0.01 FIG. 5. Coordinate of the kink center Xc as a func- (−2,1) tion of time for different values of the driving amplitude −0.02 E1=E2 =0.129(a);0.12981(b,c)and0.1309(d,e). Inpanels (c) and (e), Xc(t) has been plotted after each oscillation pe- (−3,1) −0.03 riod T (see text for details). Other parameters are: α=0.1, 0.13 0.14 0.15 ω=0.35, κ=1, θ=−0.5. −0.2 0.1 0.15 0.2 0.25 As soon as the symmetry is broken (by switching on E the field E2), one of the basins of attraction begins to 1 shrink and eventually disappears as E increases. As a FIG. 4. Dependence of the average kink velocity on the 2 result, only the limit cycles which correspond to motion amplitude of the driver E1 = E2 for θ = −0.5 (decreasing in one direction survive, as one can see from Fig. 3. In dependence) and θ = 2 (increasing dependence). Other pa- particular,fortheparametersgiveninthisfigure,wefind rameters are: α=0.1, κ=1, ω=0.35. Numbersin brackets that the attractor corresponding to the limit cycle with show the respective pair of rotation numbers (k,l). Insets rotation numbers k = 1,l = 1 disappears at E /E = show more details for case θ=−0.5. 2 1 0.0118, i.e., already for a rather weak asymmetry of the field (E E ). In the discrete case, however, the soliton velocity is 2 1 ≪ zero not only for those values of θ which restore the symmetries (6) and (7), but also for some finite inter- III. DEPENDENCE OF THE PHENOMENON ON val around these values. SYSTEM PARAMETERS Inthecaseofoddvaluesofm,thecondition(8)issat- isfied for any value of θ if E = 0. We have indeed that 2 E(t+T/2)=E cos(ωt+π)+6 E cos(mωt+mπ+θ)= In this section, we investigate the dependence of kink 1 2 E(t) and therefore the symmetry Dˆ is always valid. transport on the system parameters and the internal X − mode mechanism. This observation implies that there should be no kink transport in the system, a result which is indeed con- firmed by direct numericalsimulationsof the full system A. Dependence on the driving amplitude with m = 3, both in the presence and in the absence of noise. Also, in analogy with the continuous case, we find that for the same parameter values the discrete an- For continuous ratchets it was shown previously [10] tikink ratchets always move in the direction opposite to that the average kink velocity is proportional to E12E2, 5 so that, provided the respective symmetries are broken, The same behavior is seen in panel (c) of Fig. 5 for a the directed motion can occur for arbitrary small values larger time scale. Note that the intermittency behavior of the driving amplitudes. For the DSG equation, the occursnotonlyaroundthedepinningthreshold,butalso dependence of the average kink velocity on the driver for larger values of the driving amplitude. Also notice, amplitudes is depicted in Fig. 4 from which one can see fromtheinset(b)ofFig.4,thatthedepinning threshold thatthekinkvelocityisnotasmoothmonotonicfunction has an hysteretic behavior. In this regard, we remark of E , but a piecewise function with plateaux of differ- that the numerical investigation has been performed by 1,2 ent lengths,resembling a “devil’s staircase”[the same as increasing the driving amplitude in small steps, taking in Fig. 2(a) but monotonic]. The plateau values of the the final state for a given step as an initial condition kink velocities are given by Eq. (5) and correspond to of the next one. When the pinned state loses stability, dynamical regimes, which are limit cycles with rotation the system finds itself on a chaotic attractor, which cor- numbers(k,l)phase-lockedtothedriver. Noticethatthe responds to the directed motion. By increasing further largest resonant step ( v = 0.0557) is achieved at the E we obtain larger values of the kink velocity, while if 1 h i rotation numbers corresponding to the main resonance we move backwards, we observe that two attractors can (k = 1,l = 1). One can easily observe the smaller coexist - one of them corresponding to a pinned kink ± resonant steps with k = 2,l = 1 and k = 1,l = 2, oscillating around its center of mass, and the other to ± ± for which the kink velocities equal v = 0.1114 and a kink performing ratchet dynamics. If E is further 1 h i v =0.02785,respectively. Higher order resonances can decreasing, the moving state loses stability and the sys- h i also be identified, but they are not well visible in the tem jumps back to the pinned state. The width of the figure. hysteresis(interval betweentwo bifurcations) appearsto The main feature of the discrete case is the existence be rather small, i.e., 0.129065 < E < 0.12979. Simi- 1 ofathresholdinthedriveramplitude(depinning thresh- lar hysteretic phenomena also appear at larger values of old), above which the transport can occur and below the driver amplitudes. For example, one can find that which the kink is pinned to a lattice site, oscillating in the interval 0.1660 < E < 0.1678, two limit cycles 1 around its center of mass: with k = 1,l = 2 and k = 1,l = 1 coexist. A more de- tailed numericalinvestigationof the v (E ) dependence X = Nn=1n(un+1−un−1). (11) shows that the observed devil’s staihrciase1is incomplete c P 2(u u ) and there are gaps inside it, which depend on the value N 1 − of θ. As the v (E ) dependence becomes steeper, less 1 In Fig. 5, the kink position Xc is depicted as a function phase locked shtaites are found. In particular for the case of time for different values of the driving amplitude. We θ =2, we see that in the transition from the k=1,l=2 see that when the driving amplitude is below threshold to the k =1,l =1 state, all intermediate rational values [panel(a)],thekinkperformsperiodicoscillationsaround of k and l are missing. its center ofmass, while above the threshold[panels (b)- (e)],thestandingkinkbecomesunstableandthedirected motion starts. By further increasing the driving ampli- B. Dependence on the driving frequency tudeabovethethreshold,thekinkmotionbecomeseither phaselockedtotheexternaldriverorchaoticwithanin- Thedependenceofthemeankinkvelocityonthedriver termittent behavior [see panels (b) and (c)]. Panels (d) frequency was investigated in Ref. [10] for the continu- and (e) of Fig. 5 show the phase-locked dynamics with ous SG model. In this case, it was shown that the aver- rotation numbers k = 1,l = 5. Notice that in this case, age velocity depends on the driving frequency as v the kink travels over a fixed number of sites during each sin[θ θ (ω;α)]/ω3, wheretanθ (ω;α)=[α/(2ω)]h [i3∼+ 0 0 period T in agreementwith Eq. 5. By further increasing (α/ω)−2]. This fact implies that max v de·cays θ∈[0,2π[ E , the the kink dynamics starts to switch between pe- h i 1 withthegrowthofω. Forafixedvalueofθ,however,this riodic (or quasiperiodic) regime and intermittency. Sim- dependence isdefinedbythe mutualrelationsbetweenω ilar transitions in the case of a single harmonic driver and α, so that v can experience oscillations and sign have also been reported in Refs. [23,28]. In the intermit- h i reversals before tending to zero either in a decreasing or tency regime, the kink dynamics switches in an unpre- in increasing way. Similar behavior is expected also for dictable manner between two attracting limit cycles [see the DSG equation, although in this case, the problem is panels(b)and(c)ofFig.5]. Inthiscase,thekinkdwells complicatedbythepresenceofthePeierls-Nabarro(PN) sometime intervalonacertainsite,pinnedbythe corre- frequency ω in Eq. (3). In analogy with Fig. 2, one PN sponding minimum of the PN barrier,before jumping to can expect the dependence v (ω) to be also a piecewise the next site. From panel (b) one can also see that the h i function with a “devil’s staircase” character. dwelling time is not rationally related to the driving pe- In Fig. 6, we depict the kink velocity, normalized to riod T and changes randomly from jump to jump. Thus the driving frequency ( v T =2πv/ω) as a function of ω in the intermittent regime the kink does not travel the h i for two values of the damping constant (notice that for samenumberofsitesduringanintegernumberofperiods phase locked dynamics, v T coincides with the ratio of of E(t) and the global dynamics is chaotic. h i 6 rotation numbers k/l). One can see that the dynamics whileinset(c)showsdetailsofthe v dependenceinthe h i is characterized by a series of resonances, the most pro- neighborhood of the second resonance. The coupling of nounced one being at the main frequency (k =1,l =1). the kink motion with linear waves occurs in the interval We also observe that the resonances with k > l are less 0.447 < ω < 0.517. At the beginning of this interval, pronounced and the subharmonic resonances with k < l the kink displaysa chaotic tailas shownin Fig.6(a). As arepracticallynotvisibleinthefigure. Thecaseofsmall frequency increases, the chaotic tail becomes more and damping and larger driver amplitude (depicted in the more regular. This is shown in inset (b) of the figure, figure by circles) is characterized by an almost mono- from which one can see that at ω =0.507 the oscillating tonic decay, while in the opposite case (larger damping tail is almost monochromatic. At ω = 0.517 the oscil- and smaller driving amplitude), there are wide pinned lating tail turns into a localized oscillating mode which regions and some peak around the value of ω [notice decays at infinity. A further increase of the driving fre- PN from Eq. (4) that ω 0.17 for κ=1]. quency causes the decay of the width of the mode. PN ≃ 800 470 11110123ΟΟΟΟΟΟ un2ππ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ un2ππ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 600 8000 t 9000 10000 446605 Xc 1 (a) <v>T56789 ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ −π00++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++1++++++++++++++0++++++++++++++++0++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++2+++++++++++++++0++++++++++++++0+++++++++++++++++++++++++++++++++++++++++++n+300 4(000a0..45) 500 ΟΟ++0+Ο+++++++Ο++++++++++++++++++++++1++++++++0++++++++0++++++++++++++++++++++++++++++++++1++++++++5++++++++0++++++++++++++++++++++++++++++++++2++++++0n0 (c25)0(b3)00 Xc240000 8600 t 8800 9000 333667050 Xc 234 t 234 ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ <v>T 0000....00123.Ο×4Ο0Ο×ΟΟ×ΟΟ×ΟΟ×0Ο.×4×5××××Ο××Ο×0Ο×.Ο×5ΟΟ0Ο×Ο×Ο××Ο×Ο×Ο×0Ο×ΟΟ.Ο×5Ο5Ο×ΟΟ×ΟΟΟΟ0Ο.Ο×60ω 0 2500 5000 7500 10000 12500 15000 17500 1××××××××× ΟΟΟΟΟ×××ΟΟΟΟΟΟΟΟΟΟΟΟΟΟ 500 1 -01 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××Ο×ΟΟΟ××ΟΟΟ×Ο×Ο×ΟΟΟΟΟΟΟ×ΟΟΟΟ×ΟΟΟΟ×ΟΟΟ×Ο×××Ο××Ο×Ο×Ο×Ο×ΟΟ×ΟΟ×ΟΟ×ΟΟ×ΟΟΟ×Ο×Ο×Ο×Ο××Ο××Ο×Ο×Ο×ΟΟΟ×Ο×Ο××Ο×Ο×Ο×ΟΟ×ΟΟ×ΟΟ×ΟΟ×ΟΟΟΟΟΟ×ΟΟΟ×ΟΟ×ΟΟ×ΟΟΟ×Ο×Ο×Ο×Ο×Ο×ΟΟ×ΟΟΟ×Ο×Ο×Ο×Ο×Ο×ΟΟ×ΟΟΟ×Ο×Ο×Ο×××Ο×Ο×Ο×Ο×ΟΟΟ×Ο×Ο×ΟΟ×ΟΟ×Ο×ΟΟ×Ο×Ο×Ο×Ο×Ο× 2 (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 400 t ω X c 380 3 300 370 Xc FIG.6. Dependence of the normalized kink velocity hviT on the driving frequency for E1 =E2 =0.1, θ =2, α=0.05 360 t (◦) and E1 = E2 = 0.02, θ = 1.5, α = 0.15 (×). In both 200 16000 17000 18000 cases κ = 1. The insets (a) and (b) show the kink profile 0 5000 10000 15000 20000 25000 30000 for the first set of parameters when ω =0.45 and ω =0.507, respectively. Inset (c) shows more detailed behavior around ω=0.5. FIG. 7. Time evolution of the position of the kink cen- ter for: (a) κ = 0.5; E1 = E2 = 0.21 (curve 1), 0.19 In both the cases we find that, except for the reso- (curve 2), 0.17 (curve 3), 0.11 (curve 4) and (b) κ = 0.25; nances ω ω and 2ω ω , where ω is the frequency E1 = E2 = 0.14 (curve 1), 0.24 (curve 2), 0.22 (curve 3). L L L ≈ ≈ Other parameters are: θ = 1.5, α = 0.1, ω = 0.35. The of linear waves upper inset of panel (a) corresponds to details of the case ω2(q)=2κ(1 cosq)+1, (12) E1 = E2 = 0.19 and the lower inset corresponds to the case L − E1 = E2 = 0.11. The inset of panel (b) corresponds to the the kink transport becomes effective mainly at low fre- case E1 = E2 = 0.22. In all figures, the data have been quencies ω ωL and disappears for driving frequencies plotted with the intervalT =2π/ω. ≪ ω > ω . Also, from Fig. 6 one can see that the kinks L become pinned to the lattice for values of ω significantly A similar scenario occurs also at ω ω , but in this L smallerthanωL. Attheresonancesω ωLand2ω ωL, casethekinkvelocityismuchsmallera≈ndpracticallynot ≈ ≈ weobservethatthekinkdynamicsbecomecoupledtolin- visible in Fig. 6. Notice that in the frequency interval earwavesinthe system(plasmonsinthe caseofarrayof 0.895 < ω < 1.05, the kink becomes again coupled to Josephson junctions). Insets (a) and (b) of Fig. 6 show linear modes and displays a chaotic tail, which becomes the profiles of the kink solution in these resonant cases, 7 more regular as ω further increases. At the beginning in the sense that the forward and backward jumps be- of this window, the dynamics in the tails are strongly come of larger amplitude with a consequent decrease of chaotic and they are accompanied by the formation of the averagevelocity. large-amplitude localized excitations (breathers). In the left panel of Fig. 8 the diagram of possible dy- Byreducingthedrivingamplitude,thefrequencywin- namical regimes in the plane (κ,E ) is shown. We ob- 1 dows, in which the coupling with linear waves occurs, serve that by changing the coupling constant κ, one can decrease. We find that for E = E = 0.04, the first passfromregular(phaselocked)dynamicstochaos. The 1 2 coupling window occurs at 0.480 < ω < 0.495 and the regular dynamics are dominant for κ > 1 and when κ secondoneat0.955<ω <0.994. Inthiscase,nochaotic is decreases, the windows of chaotic motion appear. By dressingofthekinkisobserved[thecouplingoccurswith further decreasing κ, the number of chaotic windows in- veryfew (orsingle)linearmodesandthe kinklooksvery creases and the chaotic motion becomes dominant. At similar to that in Fig. 6(b)]. For the driving amplitude κ=0.25, almost no regular dynamics exist. Notice that E =E =0.01,the coupling with linear wavesdoes not the diagram in Fig. 8 does not show the complete pic- 1 2 take place at all, and around the resonant frequencies ture and the details of transitions to different dynamical ω and ω /2,the kink remainspinned to the lattice. As regimes because in order to limit the computation time, L L thedrivingamplitudeincreases,beyondacertainthresh- we have used in the numerical calculations the step of old chaotic oscillations can completely destroy the kink. 0.01 in the amplitude of the driving field E =E and a 1 2 From these results, we conclude that the coupling of the coarse step of 0.25 in κ. We find that for driving ampli- kink dynamics with linear waves depends very much on tudes E =E &0.3, the chaotic dynamics of the whole 1 2 the amplitude of the externalfield and this effect is non- lattice destroy kink solutions. linear in E . Near the resonances with linear modes, 1 the kink dynamics become more chaotic (diffusive), re- sembling the one ofa Brownianparticle (the kink makes manyrandomjumps backwardandforwardandthe uni- directionalmotioncanbeseenonlyatalargetimescale). slkrtwtccihotaroimeaneptenIltncnkilostehcttl.ncokoeatκenetnhgnICnsndoiit=spcst.wieκatFhoDsrin0l,efoieg.onewstcn5[.fophot3siil[m7etme2oiptht,n]snoaaeo,ywdκ,nnsnsssteeoeuwt→hnlnXpesdeecmtr(eceaooa∞idpinpn)finoiois]vasccnrtcacetgmohrhstnarttewteoatdrhhitibrgoeeeoeiaκsiannnvlcpttetcea.=teeeooselmkrnsuurItai0diennhppczu.si2keotelnsoiio5sdrungoudafanigb[reltntplypohleaccmyteevtoonthnhohonncepedledeolssruestueeti(ctnaynlavbpio,ncostne)lnteeintnt]tnetr.htimaoκgonseFcffu.crtafdtotuiorAnohhinemmsndeees-- E00001....0123 ⊕ΟΟΟΟΟΟΟΟΟΟΟΟ♦♦×××××××××××××××× ⊕⊕⊕⊕ΟΟΟΟΟΟΟ♦××××××××××××××××××× ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟΟ♦××××××× ⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟΟΟΟΟ♦♦×××××××××× ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟΟ×× ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟΟ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟΟΟ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ΟΟΟΟΟΟ ∇∇×××××××× ∇∇∇×∇∇∇∇∇∇∇∇×∇∇∇∇××∇∇∇×∇∇∇∇×××××∇∇∇∇∇∇×∇∇∇ΟΟΟΟΟΟΟΟΟΟΟΟΟΟ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇∇×××××××××××××××××××××××××××Ο◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊◊ΟΟΟ♦Ο×♦◊Ο◊◊♦Ο♦Ο◊♦◊Ο♦◊◊♦Ο×♦◊♦Ο◊Ο◊∇×Ο∇××♦◊◊∇◊∇∇∇∇◊◊Ο◊∇∇♦♦Ο♦Ο×◊×Ο♦Ο♦×∇∇∇∇∇×◊◊◊◊◊◊Ο♦♦Ο♦Ο×Ο♦×Ο♦◊♦×◊♦Ο♦Ο×Ο♦Ο♦×♦◊◊♦ΟΟΟ◊◊◊◊◊◊ΟΟ♦♦♦∇∇♦××××××××××××××××◊◊◊◊◊◊◊◊◊◊◊Ο♦◊◊Ο◊◊◊♦◊<<◊Ο♦◊◊vv♦--Ο000>>♦Ο...♦♦♦♦00111−Ο◊⊕Οπ×ΟΟ×⊕ΟΟ×ΟΟΟ×××Ο×ΟΟΟΟΟΟ×Ο×Ο×××××Ο××Ο×⊕Ο×ΟΟ×Ο×♦♦ΟΟΟ♦♦♦♦♦♦♦◊⊕Ο×⊕♦Ο×Ο⊕Ο×⊕⊕⊕⊕ΟΟ×⊕⊕Ο×⊕Ο0Ο×♦Ο×♦ΟΟ♦⊕Ο×♦⊕⊕Ο⊕⊕⊕×ΟΟ×Ο×ΟΟ×Ο××Ο×Ο×⊕Ο×Ο×⊕ΟΟ×Ο×Ο×θΟ×⊕Ο×⊕Ο×ΟΟ×⊕Ο×Ο×ΟΟ⊕Ο×((⊕Ο×ab⊕Ο⊕Ο×))ΟΟ⊕Ο×ΟΟΟπ 0000....0123E1 this figure, we conclude that the unidirectional motion 0 0.5 1.0 1.5 2.0 -0.05 0.00 0.05 0.10 0.15 exists also for small values of κ with dynamics which is κ <v> chaoticratherthanphaselocked(seeinsetsofthefigure). Chaotic motion, leading to transport is found to be either of the intermittent type [curve 4 of panel (a)] or diffusive type [curves 1-3 of panel (a)]. The intermittent regime is more often observed for κ & 1 and the behav- ior is the same as described in the previous subsection, i.e., the kink oscillates in the minima of the PN poten- tial until it jumps to the adjacent site [see lower inset of panel (a)]. The motion is characterized by the fact that the jumps of the kink occur randomly in time, but alwaysinthesamedirection. Forκ.1thediffusivemo- tion is the most typical scenario. In this case, the kink jumps randomly forward and backward [see upper inset of panel (a) and inset of panel (b)], but on average the motion remains unidirectional. A decrease of the cou- pling constant makes the dynamics even more diffusive 8 FIG. 8. Left panel. Diagram that demonstrates main understandingofthis behavior,wehaveplottedininsets regimes of the kink ratchet motion for different values of the (a) and (b) the mean velocity as a function of the phase couplingconstantκandthedrivingamplitudeE1 =E2(other difference θ. The inset (a) refers to the case κ=0.75 for parameters are: θ =1.5, α=0.1, and ω =0.35). In the dia- twodifferentvalues ofthe driveramplitude. We see that gram ◦ stands for the regime when kink is pinned (hvi=0), while for E = 0.19 the dependence v (θ) is piecewise, ⊕ stands for the periodic or quasiperiodic ratchet motion, × 1 h i butsomewhatsimilar toFig.2(i.e., it hasone minimum stands for the chaotic regime, and (cid:7) corresponds to the case andonemaximum),thedependencebecomesmoreirreg- when kinksolutions do not exist. ular as E increases, with small islands of phase locked Right panel. Dependence of the average kink velocity on the 1 regimes with opposite velocities. A similar phenomenon driving amplitude E1 = E2 for different values of the cou- is observed, when κ decreases to κ = 0.5 [inset (b)]. pling constant κ = 0.5(∆), κ= 0.75 (×), κ= 1 (⋄), κ= 1.5 ((cid:7)), and κ = 2 (◦). Other parameters are as in the left Thisbehaviorislinkedtothewellknownphenomenonof panel. The insets show dependence of the average velocity crises of attractors, i.e., to sudden appearance(or disap- on the phase difference θ. Inset (a) corresponds to κ=0.75, pearance) of an attractor as a system parameter varies E1=E2=0.19 (◦) and E1 =E2=0.25 (×),inset (b) corre- (see [33]). By further decreasing the coupling constant, spondstoκ=0.5,E1=E2 =0.22(⊕). Solidlinesaredrawn more and more attractor crises are found. as a guide to an eye. From the insets of Fig. 8, we also see that, in contrast withthecaseκ=1inFig.2,thedependence v = v (θ) h i h i IntherightpanelofFig.8,weshowthedependenceof losesanyresemblancewithasinusoidalfunctionforsmall the averagekink velocity on the driving amplitude E = values of κ. From Fig. 8, it is also evident that the de- 1 E for different values of the coupling constant κ. We pinning amplitude of the kink motion increases when κ 2 observe that already for κ=2, the curve is very smooth decreases. For very small κ, e.g., κ=0.0625 (not shown (except for the smallresonantsteps k=3,l=1 andk = in Fig. 8), no directed motion is found - the kink is ei- 3,l = 2) and the behavior becomes very similar to that therpinnedbythelatticeordestroyedbychaoswhenthe reported for the continuous limit ( v E2E ) [10]. By driving amplitude becomes large enough. This behavior h i ∼ 1 2 increasing the driving amplitudes E , the dependence is not surprising, since we know that high discreteness 1,2 becomes non-monotonic, a fact which was also observed normally prevents solitons from free propagation. inthecontinuousSGcaseandascribedtotheinteraction of the kink with internal oscillation modes [10]. On the other hand, decrease of the coupling constant makes the phase locking steps more pronounced. For κ = 1.5, the 2π dependence hvi(E1) almost coincides with the one for Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+(Οa+)Ο+Ο+Ο+Ο+ κ=2(forlargeramplitudes,however,thethreeresonant un Ο+ steps k = l = 1, k = 3, l = 2, and k = 2, l = 1 become π Ο+ much more visible). Ο+ Further decrease of the coupling constant makes the Ο+ dependence even less monotonic: for κ = 1 (shown by 018Ο+0Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο2+0Ο+0Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+Ο+ 220 240 ⋄ n in the figure), the non-monotonicity is quite weak, with asmallinterval0.145<E1 <0.154inwhichthekinkve- un locity drops back to zero. For E1 &0.2, the dependence 6.60 (b) is also non-monotonic because of the interaction of the 3.05 kink with vibrational modes localized on it. For κ=0.5 andκ=0.75,theaveragekinkvelocitybecomesstrongly -0.50 240 non-monotonic. 210 130 Notice from the right panel of Fig. 8 that several re- versals of the kink motion along the E axis occur. In n 180 6t5 1 0 this regard, we remark that the sign of v depends on h i the relations between ω, α, and ωPN. Decrease of κ can FIG. 9. Evolution of the kink profile un(t). Upper panel change ωPN significantly [see Eq. (4)], thus effecting the showsinitialkinkprofile(+)andkinkprofileafterintegration sign of the velocity. The reversals of v as a function of time 7T (◦), shifted two sites backwards. Parameters of the h i E1, however, are not fully explained by these arguments model are: E1=E2 =0.135, θ=1.5, α=0.1, ω=0.35, and andwebelievethat,inanalogywiththecontinuouscase, κ=1. thecouplingofthekinkwithsmallamplitudewavesplays also an important role. Weremarkthattheattractorcrisesleadingtoreversal By comparing the left and the right panels of Fig. 8, phenomena are out of the range of validity of the single one can also see that the non-monotonic jumps occur particle approximation since they involve many degrees in correspondence with the transitions from chaotic to of freedom. In particular, in the limit of weak coupling, regular (phase locked) regimes. In order to get a better the point particle approximation (3) as well as the sym- metry analysis discussed in Sec. II become not valid. 9 D. Internal mode mechanism and dependence on the is also observedfrom the three-dimensionalplot of un(t) damping parameter in panel (b). Similar results are obtained for different values of the driving amplitudes, as one can see from Inthecaseofcontinuoussolitonratchets,itwasshown Fig. 10. In this case, the kink is lockedon the resonance (see Refs. [9,10,19,20,18]) that a contribution to the with rotationnumbers (1,2) (notice that the kink repro- ratchet phenomenon comes also from the internal oscil- duces itself completely after the time 2T). In panel (b), lation of the kink via the internal mode mechanism. We the complete dynamics inthe time interval[0,4T]is also expectthiseffecttobetruealsointhepresentcase,both shown. The fact that the center of mass motion and for large values (i.e., close to the continuum limit) and the oscillations on the kink profile (internal mode) are intermediate andsmallvalues ofthe coupling constantκ perfectly synchronizedsuggeststhe existence ofcoupling (for very small values of κ, however, the kink becomes between the internal and translational modes similarly pinned to the lattice and the ratchetphenomenondisap- as for the continuous case. pears as discussed above). The existence of the internal mode mechanism in the Inthefollowingweinvestigatetheinternalmodemech- discrete case is also supported by the influence of the anism by fixing κ = 1 and performing direct numerical dampingconstantonthephenomenon. Inthecontinuous simulations of Eqs. (1). In particular, we show the exis- case, it was shown that the coupling between the trans- tence of a local oscillation on the kink profile which is, lational and internal modes decreases with the damping perfectly synchronized (phase locked) with the kink mo- and for the case of asymmetric potentials with symmet- tion. We havefoundthatwhenthe dynamicsofthe kink ric drivers, it was proved that in the limit α 0, the → center of mass is phase locked to the resonance (k,l), coupling completely disappears [18]. In the case of bi- the internal mode oscillation is also locked to the same harmonicasymmetric forces,the nonlinearityinduces an resonance. This is shown in Fig. 9 for the discrete kink effectivebias component,whichgivesrisetoapointpar- ratchet,whichisphaselockedtotheexternaldriverwith ticle contribution also in the absence of damping. rotation numbers k = 2, l = 7. In the panel (a) of thisfigure,theinitialandfinalkinkprofilesaredepicted, fromwhichonecanconcludethattheseprofilesperfectly coincide after l =7 periods of the externaldriver(notice that the final configurationhas been shifted by two sites backwards in order to demonstrate full coincidence). 0.4 0.3 2π Ο+Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊ E1 un Ο+◊ (a) 0.10 ⊕ π Ο+◊ 0.2 ∆E ⊕ ⊕ ⊕⊕ Ο+◊ 0.05 ⊕⊕ ⊕ ⊕ 0Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊Ο+◊◊ 0.1 ⊕⊕ ⊕⊕ un 180 n 200 220 0.000.0⊕ 0.1 0.2 0.3 ⊕⊕⊕⊕⊕0⊕⊕.4 α 6.8 0.0 0.0 0.1 0.2 0.3 0.4 3.1 α (b) -0.6 FIG. 11. Existence diagram of the main resonance 240 (k = l = 1) on the plane (α,E1 = E2) for ω = 0.35, θ = 2, 210 and κ = 1. The inset shows the dependence of the width of n resonance∆E ondampingcoefficientα. Whentheresonance 180 72 36. 0 t consists of several islands, ∆E is computed as a sum of the widths of individual islands. FIG. 10. Same as in Fig. 9. All parameters are the same except E1 =E2 =0.17. In panel (a), the initial kink config- The contribution of the internal mode to the ratchet uration is shown by (+), the configuration after time T (⋄) dynamics, however, should be sensitive to the damping, and after time 2T (◦), shifted by one site backwards. Panel ifthe coupling is controlledby the damping andthe uni- (b) shows three-dimensional dynamical picture for the kink directional motion should become less effective for small evolution in the time interval[0,4T]. valuesofthedampingconstant. Ontheotherhand,when the damping in the system becomes too large, the dy- The phase lockingofthe motionto the externaldriver namics are stronglyreduced(or stopped), so thata non- 10

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