Asymptotics for kink propagation in the discrete Sine-Gordon equation

Transcription

1 Physica D elsevier.com/locate/physd Asymptotics for kink propagation in the discrete Sine-Gordon equation L.A. Cisneros a,, A.A. Minzoni b a Graduate Program in Mathematical Sciences, Facultad de Ciencias, Universidad Nacional Autónoma de México, Apdo. 0-76, 0000 México, D. F., Mexico b FENOMEC, Department of Mathematics and Mechanics, IIMAS, Universidad Nacional Autónoma de México, Apdo. 0-76, 0000 México, D. F., Mexico Received 9 November 006; received in revised form 8 July 007; accepted 7 August 007 Available online August 007 Communicated by Y. Nishiura Abstract The evolution of a propagating kink in a Sine-Gordon lattice is studied asymptotically using an averaged Lagrangian formulation appropriately coupled to the effect of the radiation. We find that unlike the continuum case the interaction ith the Goldstone mode is important to explain the acceleration of the kink as it hops along the lattice. We develop a discrete WKB type solution to study the interaction of the kink and the radiation. In particular using this solution e sho ho to calculate the effect of the Peyrard and Kruskal resonant radiation in the energy loss of the kink. We obtain a set of modulation equation hich explains qualitatively the evolution of the kink ith remarkable quantitative agreement. c 007 Elsevier B.V. All rights reserved. Keyords: Kink; Peierls Nabarro potential; Radiation damping; Internal modes; Modulation averaged Lagrangian. Introduction Since 938 hen Frenkel and Kontorova [] introduced their model, hich describes a chain of harmonically coupled particles subjected to an external periodic substrate potential, several numerical and asymptotic results have been obtained. The pioneering ork on numerical simulation as carried out by Currie et al. [] shoing that an initial condition, hich is the configuration of the exact one-kink solution of the continuous Sine-Gordon equation, changes its shape a little by shrinking and radiates phonons resulting in spontaneous damping of the kink motion. Hoever, Currie et al. did not follo the evolution sufficiently long enough to observe in detail the final state of the initial travelling kink. Ishimori and Munakata [3] obtained the first results on asymptotic approximation, they studied the discreteness effects on the dynamics of a Sine-Gordon kink in the Frenkel and Kontorova lattice using the MacLaughlin and Scott perturbation formalism, that is, they considered the continuous Corresponding address: Department of Mathematics and Statistics, 46 Humanities Bldg, MSC03 50, University of Ne Mexico, Albuquerque, NM , USA. Tel.: ; fax: address: cisneros@math.unm.edu L.A. Cisneros. approximation of the Frenkel and Kontorova model Sine- Gordon lattice equation to all orders and they rote the model equation as a perturbed continuous Sine-Gordon equation, the corrections are the perturbation. They shoed that from the zeroth order a kink moves in a periodic potential field, also knon as Peierls Nabarro PN potential, hich causes obbling and pinning of the kink. Hoever no attempt as made to compare the asymptotic ith the numerical solution. Peyrard and Kruskal [4] studied the kink s dynamics in the discrete Sine-Gordon equation in the very discrete regimen. They studied in detail the complicated numerical evolution of a travelling kink shoing that all initial conditions have the same qualitative behavior. They discovered, solving the linearized Sine-Gordon equation ith a forcing term given by the travelling kink, that the kink ill emit radiation hen its velocity resonates ith the phase velocity of the lattice aves phonons. This radiation as found to be large at large kink velocities and small at small velocities. Using this, they explored the position of the knees and obtained an estimated mean kink position. Hoever no equations ere proposed to couple the radiation and the kink motion. In the continuous version and for initial conditions hich are not the exact kink solution, Smyth and Worthy [8] obtained the radiation loss by means of the momentum and energy conservation /$ - see front matter c 007 Elsevier B.V. All rights reserved. doi:0.06/j.physd

2 L.A. Cisneros, A.A. Minzoni / Physica D equations of the continuum Sine-Gordon equation. They calculated the momentum loss of the kink to the radiated aves. The radiated aves ere calculated in terms of the kink parameters obtaining a closed system of equations. Boesch et al. [5,6] introduced a collective-variable method using a projection-operator approach to study the emission of phonon radiation from a discrete Sine-Gordon kink, ith this approach they explained the radiative effects and the kink evolution. They obtained good quantitative comparisons beteen their collective-variable approach and the full numerical solution proving that the collective-variable method is a good idea to describe the kink. Hoever in their collective-variable approach they had to solve numerically all the differential equations to account for the radiation. Kevrekidis and Weinstein [7] used the normal forms to study the large time behavior of kink solution in the discrete Sine-Gordon equation for initial conditions hich are small perturbation of a stable kink. They studied in detail the kink s internal modes for the static case and they coupled them nonlinearly to the radiation modes in the continuous spectrum. In this paper e consider the complementary situation namely e develop a quantitative asymptotic theory for the kink propagation calculating the effect of the radiation shed by the kink. We use the discrete average Lagrangian as the counterpart of the one considered in [8]. In this approach the Poisson summation formula provides the Peierls Nabarro potential responsible for the hopping of the kink along the lattice. We find that unlike the continuum case only static kink solutions are possible. We also sho that approximately propagating solutions only exist for velocities smaller than a certain threshold hich explains hy fast kinks do not propagate long distances. The coupling beteen the kink and the radiation is studied using the existence of the resonant radiation discovered in [4] produced by the kink motion. This radiation is calculated using a discrete type of WKB solution. A careful study of the numerical solution in the neighborhood of the kink reveals that the dynamical counterpart of the Goldstone static mode is responsible for a small scale asymmetry of the kink. This is shon to produce a larger Peierls Nabarro potential hich in turn produces larger accelerations. When all these effects are included a remarkable agreement beteen the numerical and the asymptotic solution is found. The paper is organized as follos: Section formulates the problem and describes the relevant behavior of previously knon numerical solution hich is relevant to this ork. In Section 3 e present the averaged Lagrangian for the asymptotic kink evolution and e explain qualitatively the possible behaviors, the relevant calculations of this section are presented in Appendix A. In Section 4 e describe the effect of the radiation on the kink and derive the radiation damped modulation equations. The details of the discrete signalling problem are documented in Appendix B. Section 5 is devoted to the study of the effect of the Goldstone mode on the propagating kink. Finally, Section 6 is devoted to detailed comparison beteen the modulation solution, hich takes into account the Goldstone mode, and the full numerical solution. In Section 7 e present our conclusions. The details of several laborious calculations are in Appendix C.. Statement of the problem and numerical solution The equation of the motion for the Frenkel and Kontorova model, hich describes a chain of harmonically coupled particles subjected to an external periodic substrate potential, is the folloing discrete Sine-Gordon equation y n = y n y n y n d sin y n, < n < hich is obtained from the Lagrangian, L = y n y n y n d cos y n, here the constant d is the discreteness parameter, d large d is the continuous limit and d < is the discrete one, y n is the relative displacement of particle n. For d large Eq. can be approximate by the continuous Sine-Gordon equation. This is seen multiplying Eq. by d and making the change in variables τ = t/d, x n = n/d to obtain, u ττ = u xx sin u. This equation can be solved exactly [9] using the inverse scattering method obtaining a one parameter family of kink solutions given by, ux, τ = 4 arctan exp x vτ, 3 v here the velocity 0 v is arbitrary. The discrete equations of motion are not exactly integrable, hoever, these equations can be solved [4] numerically. We no reconsider briefly the numerical solution to reproduce the relevant results needed for our ork. We solve numerically a finite number, N, of equations using a Runge Kutta method ith initial conditions obtained from 3, that is y n t 0 = 4 arctan exp n 0, 4 d v0 y n t 0 = sech n 0, d v0 d v0 N n N 5 here 0 = v 0 t 0 at initial time t = t 0 and the boundary conditions are taken to be y N = π, y N = 0. To avoid boundary reflexion problems, e solve for a large number of particles typically particles so that the trailing aves of the front ave do not reach the artificial boundary. We estimate the numerical evolution of the position t and velocity v t = t using an interpolation for the

3 5 L.A. Cisneros, A.A. Minzoni / Physica D Fig.. Full numerical solution of system for N particles ith v 0 = 0.8, 0 = 0, d = 0.95, N = and t = a Initial condition 4, b numerical solution at t = , c backard radiation and d forard radiation. midpoint of the kink, as follos t m π y m t y m t y m t, 6 v t t ne t old, 7 t here m is the integer that satisfies y m t > π and y m t < π at t = t ne and t = t ne t old. We reproduce the full numerical solution of Peyrard and Kruskal [4] for the initial conditions v 0 = 0.8, 0 = 0, d = 0.95 and v 0 = 0.4, 0 = 0, d =. The numerical evolution of the kink for the initial velocity v 0 = 0.8 is shon in Fig.. As described in [4], the kink decelerates very rapidly and after moving a relatively short distance in the lattice to be contrasted ith the travelling ave in the continuum [8] it is trapped, see Figs. and 3. Also, there are trailing aves emitted as radiation predominantly at the back of the travelling kink and there are critical velocities at hich there is an abrupt change in the type of trailing aves emitted by the travelling kink, these critical velocities are identified ith the corresponding velocities to the knees of the curve in Fig.. We folloed the kink evolution until it becomes trapped also observing knees appearing at larger times. The kink evolution for the initial velocity v 0 = 0.4 has the same qualitative behavior as the initial velocity v 0 = 0.8 see Figs. 4 6, but there is a strong quantitative difference beteen the to cases. For larger initial velocity v 0 the kink is rapidly trapped hile for smaller initial velocity v 0 the kink travels almost like a free kink and eventually it is trapped. Several authors have studied the asymptotic of propagating kinks [3,0]. In these orks the damping mechanisms have been discussed in a qualitative ay and no quantitative results ere obtained. In [4] the main mechanism of radiation damping is found but the detailed coupling of the radiation to the kink as not addressed. On the other hand, the asymptotic evolution of the trapped kink has been studied by Kevrekidis and Weinstein [7] and shon to be dominated by the nonlinear interaction of the internal kink modes ith the radiation. We are interested in the complementary regime namely the asymptotic study of the kink dynamics before the trapping occurs. In particular, ho the internal modes of the kink couple to the radiation loss during the hopping of the kink along the lattice. 3. Asymptotic equations for the kink evolution Folloing Smyth and Worthy [8], ho studied the kink evolution for the continuous Sine-Gordon equation, e take the modulated kink solution in the form: y n t = K n t = 4 arctan exp n t t. 8 As in the continuum case, e assume that the idth t of this trial function ill adjust and the velocity v t = t ill be non-uniform, from the initial condition 4 e have v t 0 = v 0, 0 = t 0 = v 0 t 0, and t 0 = d v 0 at the initial time t = t 0. A previous study [0] assumed that is

4 L.A. Cisneros, A.A. Minzoni / Physica D Fig.. a Numerical evolution of velocity 7 for v 0 = 0.8, 0 = 0, d = 0.95, N = and t = 0.05 at t = , b first knee, c second knee and d third knee and trapped velocity. Fig. 3. a Numerical evolution of position 6 for v 0 = 0.8, 0 = 0, d = 0.95, N = and t = 0.05 at t = , b zoom of a, c numerical position of trapped kink and d zoom of c.

5 54 L.A. Cisneros, A.A. Minzoni / Physica D Fig. 4. Full numerical solution of system for N particles ith v 0 = 0.4, 0 = 0, d =, N = and t = a Initial condition 4, b numerical solution at t = , c backard radiation and d forard radiation. Fig. 5. a Numerical evolution of velocity 7 for v 0 = 0.4, 0 = 0, d =, N = and t = 0.05 at t = , b first knee, c second knee and d third knee and trapped velocity.

6 L.A. Cisneros, A.A. Minzoni / Physica D Fig. 6. a Numerical evolution of position 6 for v 0 = 0.4, 0 = 0, d =, N = and t = 0.05 at t = , b zoom of a, c numerical position of trapped kink and d zoom of c. fixed and relatively large for the kink scattering by impurities in a discrete chain and no detailed quantitative comparison ere given beteen numerics and asymptotics. The trial function 8 is then substituted in the Lagrangian to obtain the averaged Lagrangian L = d 8 n sech n n sech arctan sinh sech n here e have used the exact relations, y n y n = 4 arctan sinh n sech, y n = n sech n, n cos y n = sech., 9 The term proportional to has been neglected because it does not contribute to the dynamics. To evaluate the Lagrangian 9, e observe that taking relatively large continuum approximation, e have y n y n sech n. 0 Using 0 in 9 and the Poisson summation formula see Appendix A e obtain to leading order L = d 4 π 3, hich as expected recovers the Lagrangian obtained by Smyth and Worthy [8] for the continuous Sine-Gordon equation. To find the effects due to the discreteness of the lattice e use the Poisson summation formula in the full expression 9. The details are given in Appendix A. Only the leading order periodic correction hich is the Peierls Nabarro PN potential is retained. We obtain L K = d here h = 8 4π sinh π cos π π h, 3 arctan sinh sech x dx

7 56 L.A. Cisneros, A.A. Minzoni / Physica D can be interpolated as: h 8 0, for > Notice that for, h 4/ and Eq. recovers the continuum limit. The periodic PN terms are exponentially small hen. On the other hand they have an important contribution hen hich shos the strong lattice effects on the motion of the kink. Taking variations in the averaged Lagrangian ith respect to and, e obtain the ordinary differential equations for the kink parameters δ : = δ : = d π 3 sinh π sin π, 3 6 π d 6 π 3 π h. 4 For fixed and satisfying the continuum dispersion relation, = d, the first equation is the same one discussed by Braun and Kivshar [0] once the appropriate scaling and the approximation 0 are used. Eq. 3 has saddles in the nodes particles of the chain and centers in the middle of consecutive nodes. The unbounded orbits represent the jumping of the kink along the lattice. The equation for displays the fact that has no fixed point unless the kink itself is stationary and there are to possibilities, at a node or beteen to nodes, moreover notice that as expected the selection of the steady state is determined by exponentially small terms [7, ]. This explains the numerical observation that all kink initial conditions become trapped. On the other hand if the PN potential is neglected e find that the equation for has fixed points for, R = ith R = d h. 6 In Fig. 7 e sho a graph of 6. We observe that the discrete dispersion relation 5 is satisfied only if for d = hich is in contrast ith the continuous dispersion relation, S = /d =, hich can be satisfied for all 0. This explains hy lo velocity kinks propagate for longer distances since they are approximated fixed points and are ithin an exponentially small correction steady propagating solutions of the equations. On the other hand rapid kinks are not approximated steady propagating solutions. They must radiate a great quantity of energy to adjust to possible approximate kink solutions. To complete the modulation theory e must include the effect of radiation folloing [4,8]. Fig. 7. Graphs of R related to the discrete dispersion relation 5 and S = /d related to the continuous dispersion relation, both for d =. Fig. 8. Divided difference of yn num t K n t beteen the full numerical solution and the modulated kink 8 using 6 and 7 to estimate the modulated parameters for a t = 0, b t = 45 and c t = The effect of radiation To calculate the effect of radiation e calculate the energy loss proposed by Peyrard and Kruskal using the ideas of [8] to relate it to the motion of the kink. To this end the solution is assumed to be y n t K n t for l n l here l is a suitable cut off node and yn R t yc n t y n P K t outside the region occupied by the kink. This decomposition is suggested by the results in Fig. 8. In Fig. 8 the divided difference g n t = yn num t K n t yn num t K n t has to components. The first one is a relatively flat shelf hich is produced by long ave radiation as in the continuum case to the readjustment of. This is taken into account by yn c t. The Peyrard and Kruskal trailing ave radiation yn P K t hich is controlled by the leading peak in Fig. 8 is the one discussed in [4]. This radiation is absent in the continuum limit because it is caused by the resonance of the ave numbers ω/ produced by the moving source ith the lattice period for definite values of ω. With these assumptions the energy loss takes the form:

8 L.A. Cisneros, A.A. Minzoni / Physica D d dt l n= l = y n y n y n d cos y n y n y n d cos y n y n l y n n= l y n y n y n l l yn y y n n d sin y n. 7 No the left hand side is evaluated at the coherent kink and 7 becomes d dt d 4π sinh π cos π π 3 h = [ y R n ] l yn R y n R, 8 l here e have assumed a symmetric radiation shedding due to the small velocity. Since the radiation is small e calculate y R n using the linearized equation. y R n yn R y n R y n R d y n R = 0, 9 ith an appropriate signalling boundary condition, hich is discussed in Appendix B, on yn R at the leading and trailing edges of the kink. From Eq. 38 of the Appendix B, e obtain: [ y c n yc n / n ] l l = π 3c g t and from Eq. 46, e have [ ] yn P K yn P K / n = k l 8π R k exp π k v ρ 0 k v 3 cos k v l kt ψ k. The average of Eq. is the energy loss studied in [4]. The cross products beteen yn c and y n P K do not contribute since e average on the fast time scale to obtain: d dt d 4π sinh π cos π π 3 exp h = π 3c g t k ρπ k v 64π R k k v 3. Since the average of R k is approximately v, e obtain: [ 8 ] π 3 d sinh π sin π π 3 6 π d 6 π 3 π h = π 3c g t k 64π exp ρπ k v k v. Since = 0 during the oscillation to obtain a non-singular equation, e equate the energy loss of to the equation of motion of to obtain 6 π d 6 π 3 π h = c g t and = k d π 3 sinh π sin π 8π exp ρπ k v 3 k v 3. 4 Notice that in Eq. 4 the resonant damping is not of the form µ ith µ constant. It depends on v = to capture the fact that for large velocities the damping is relatively large due to the smallness of k /. Hoever as decreases, as is discussed in [4] and is described in Appendix B, k / gros and the equation cos d k v = k losses the small roots k. This produces the knee in the velocity evolution since the damping decreases abruptly hen the small roots are lost. Notice that this form of damping goes to zero as 0, hich is to be expected since the kink does not resonate ith the lattice as it becomes steady. Before e compare the solution of 3 and 4 ith the corresponding full numerical solution, e analyze qualitatively the nature of the modulational solutions. In Eq. 4 the damping decreases, thus the idth given by 3 adjusts to a ne larger value the kink becomes broader as decreases. This in turn loers the PN potential barrier. This allos the kink to propagate for relatively long distances since it settles onto an approximately propagating solution of 3 and 4. In Figs. 9 e sho a comparison beteen the full numerical solution and the modulational solution ith kink parameters evolving according to 3 and 4. In the Figs. 9 and 0 e sho the comparisons for ρ = 0.95 in the radiation damping. We note from Figs. 9 and 0 that for constant ρ = 0.95 e have a very good comparison ith the full numerical solution up to the time of the first knee. We observe that there is a relatively large acceleration. This process, as e discuss in Appendix B, cannot be obtained analytically since there are no explicit analytic solutions for the signalling problem

9 58 L.A. Cisneros, A.A. Minzoni / Physica D Fig. 9. Numerical comparison beteen the numerical position 6 and the solution of the system 3 and 4 for ρ = 0.95 and v 0 = 0.3. Fig.. Numerical comparison beteen the numerical velocity 7 and the solution of the system 3 and 4 for ρ t = 0.5 tanh t/60 and v 0 = 0.3. Fig. 0. Numerical comparison beteen the numerical velocity 7 and the solution of the system 3 and 4 for ρ = 0.95 and v 0 = 0.3. Fig.. Numerical comparison beteen the numerical position 6 and the solution of the system 3 and 4 for ρ t = 0.5 tanh t/60 and v 0 = 0.3. ith variable velocity v. We thus take the change in ρ to be of the form ρ t = 0.5 tanht/60 hich has the Fig. 3. Numerical comparison of Fig. in a smaller indo. correct time scale t 60 for the first knee and account for the mismatch beteen t and t. The results obtained are shon in Figs. and and are indeed very good. It is to be remarked that the same function ρ t is used for a variety of initial conditions obtaining very good agreement ith mean for initial velocities beteen 0.3 and 0.5. A closer comparison shos that the oscillation of the modulation equations is smaller than the numerical one see Fig. 3. This is because the PN potential barrier is very sensitive to the idth of the kink due to its exponential dependence. This discrepancy suggests that unlike the continuous problem here the comparison is very good there is another mechanism hich accounts for the larger oscillations. We no describe ho the dynamical counterpart of the internal modes, studied in [7] for the static case, accounts for the larger oscillations. 5. The dynamical effect of the internal modes The static discrete kink has internal modes as established by Kevrekidis and Weinstein [7]. There is an edge mode [7] ith eigenvalue close to the loer edge of the continuum band

10 L.A. Cisneros, A.A. Minzoni / Physica D The Lagrangian L K is the previously studied kink Lagrangian. The Lagrangian L A corresponds to the internal mode. Since the amplitude A is assumed to be small see Fig. 8, e have from Appendix C Fig. 4. Graph of 7 shoing the asymmetry of the peak for a t = 0, b t = 45 and c t = 00. gap, ±i d, 4. There is also a Goldstone mode hich d is the counterpart of the zero eigenvalue mode given in the continuum limit by the phase derivative of the kink solution due to the translation invariance. In the discrete case the translation invariance is lost and the zero eigenvalue of the continuum problem becomes an exponentially small eigenvalue for the discrete case. Moreover, this eigenvalue can be positive or negative depending on the kink s position. The Goldstone mode can be taken to leading order [7] to be A sechn /. To study the dynamical effect of this mode e observe from Fig. 4 that the peak of the divided difference δ n t = yn num t ynum n t 5 has an asymmetry hich evolves in time. On the other hand, n t = K n t K n t 6 has no asymmetry. Thus the difference, g n t = n t δ n t 7 hich is plotted in Fig. 8 is the asymmetry in the peak. From Fig. 8 e observe that the larger g n is located at the kink location and it is a dipole. This has the same form of the divided difference of the Goldstone mode. Moreover, this mode is seen to interact strongly ith the radiation. Thus for the modulation theory e take a trial function hich includes a modulation of the Goldstone mode in the form: y n t = 4 arctan exp n t t n t A t sech t. 8 The effect of the edge mode is accounted for by the time dependence of t. Using 8 e obtain the averaged Lagrangian in the form, L = L K L A L int. 9 L A = A A 36 A A A π 3 π 6 sinh π cos π sinh 4I cos π cos π. d π sinh π I, A4 4 4d cos π π π 3 sinh π I 3,. 30 The functions I, I, and I 3, are given in the Appendix C. As expected the quadratic terms of L A involve the PN contribution hich in the static case is responsible for the change in stability of the kink depending on its position. The effect of the dynamics is included in the terms and. The interaction terms in L int are of to types. One gives the added mass to the kink due to the internal mode and the second type of interaction produces a modification of the PN potential, V,, felt by the kink due to the presence of the internal mode. This interaction ill be shon to be responsible for the enhancement of the oscillation in velocity described in the previous section. This last Lagrangian is calculated in Appendix C in the form, L int = 4A sin π V, 4A A3 sin π V, sin π S,. 3 d 3d Thus the dynamical effect of the internal mode is obtained by solving the modulation equations given by the Lagrangian 9. There ill be ne equations for A and, the equation for ill be also modified. The equation for the position ill no have a larger PN potential due to the interaction ith the internal mode hich ill be shon to explain the larger amplitude of the numerical velocity oscillation hich as not captured using only the kink trial function. In detail the equations take the form: 4π 3 exp π d sin π

11 60 L.A. Cisneros, A.A. Minzoni / Physica D π A V, cos π π AV, cos π d 6 π d = l 3 3 π h A π d A π d = l 33 [ ] π A A sinh d. [ A3 d 9.3 ] 0. = l A 34 8 d π 6 π 8A A π 36 π cosh sinh A d π = l. 35 The loss for the equation is basically as before because the contribution of the Goldstone mode is very small see Fig. 8. We thus have: l = k l = c g t, A l A = c g t, l = c g t. 8π exp ρπ k v k v 3 These modulation equations ill no be used to explain quantitatively the kink dynamics using simple interaction description beteen the modulated kink and the internal mode generated by its motion through the lattice. 6. Solution to the modulation equations and comparison ith the full numerical results Before e compare the solution of modulational equations ith the full numerical solution e describe quantitatively the behavior of these modulation solutions. The equation for has a center at the right root of the discrete dispersion relation 5 and the idth of the kink is attracted toard this steady state due to the damping. On the other hand, the equation for has a center for = 0.85 and a saddle for = 0.35 hen = 0.9. This center is to be expected since it corresponds to the minimum of the Rayleigh quotient as a function of for the trial function approximating the Goldstone mode. The equation for A has a critical point for A = 0. The linear part at this point has to terms. The first one gives the value of the static Rayleigh quotient hich alternates sign depending on hether the kink is at a minimum or a maximum for the PN potential. The second term is the dynamic modification of this behavior due to the terms and. We thus see that due to the rapid oscillation of the kink parameters this term ill have a small average. It ill turn out that the average is small and negative. Thus, the A = 0 fixed point is a center ith a long time scale. The A 3 term prevents the run aay of the amplitude of the internal mode during the unstable part of the eigenvalue cycle covered by the hopping of the kink. Finally, the equation for the has a ne term in the PN potential proportional to A caused by the internal mode. This term is important since the integral in V, involves a double pole hich gives a relatively large contribution. Due to the long period of A this contribution is important for a long time. As before the change in the damping produces the knee taking into account the internal mode. This analysis explains qualitatively in simple terms the behavior of the travelling kink in the lattice. We no compare the modulational solution ith the numerical results. We use for the comparison ρ t = 0.5 tanh t/60. We no see that the amplitude agreement is very good due to the influence of the internal mode. As shon in Figs The same ρ t as used for other initial conditions and Figs. 8 and 9 display the velocity comparisons for v 0 = 0.4 and v 0 = 0.5, respectively. We see that the agreement is remarkable due to the complexity of the radiation process and the approximation used. At later stages after the first knee the radiation damping of the approximate equations is too strong and the modulation theory for the radiation cannot capture the details of the abrupt change in the radiation. Beyond the first knee the PN barrier becomes small and since the damping is also small the kink travels as a continuum kink for a small velocity. 7. Discussion and conclusions We have developed a modulation solution including the radiation for the motion of a Sine-Gordon kink in a discrete lattice. We have shon that unlike the continuum case not even approximate kink solutions are possible for a velocity > c = This explains hy fast initial kinks radiate a great deal of energy to adjust to approximate travelling kinks ith smaller velocity. For smaller velocities the dynamics of the kink can be accurately studied including the evolution of the to internal modes. The effect of the edge mode is captured as in the continuum case by the modulation of the kink idth and the flat shelf radiation. The effect of the

12 L.A. Cisneros, A.A. Minzoni / Physica D Fig. 5. Numerical comparison beteen the numerical position 6 and the solution of the system 3 35 for ρ t = 0.5 tanh t/60 and v 0 = 0.3. Fig. 7. Numerical comparison beteen the numerical velocity 7 and the solution of the system 3 35 for ρ t = 0.5 tanh t/60 and v 0 = 0.3 in a smaller indo. Fig. 6. Numerical comparison beteen the numerical velocity 7 and the solution of the system 3 35 for ρ t = 0.5 tanh t/60 and v 0 = 0.3. Goldstone mode is taken into account explicitly. The effect of the discrete Goldstone mode is of crucial importance since the kink travels jumping beteen a stable and an unstable state, at the minimum and maximum of the PN potential. As the internal mode gros or oscillates, the shape of the kink develops a narro asymmetry. This in turn is responsible for the modification of the PN potential increasing its value due to the small scale contribution. This enhances the velocity oscillation and the resonant radiation. The radiation mechanism discovered by Peyrard and Kruskal [4] as incorporated in detail into the modulational solution to give the damping term hich as not calculated in the previous studies. This as calculated by solving approximately an appropriately defined signalling problem for the linear discrete equation using a discrete analogue of the superposition of WKB type solutions. With these simple mechanisms e explained in very good quantitative detail the dynamics of the kink before setting into an equilibrium. We also observe that the equations for fixed are analogous to the ones studied by Kevrekidis and Weinstein [7]. Hoever, the higher order interactions that Fig. 8. Numerical comparison beteen the numerical velocity 7 and the solution of the system 3 35 for ρ t = 0.5 tanh t/60 and v 0 = 0.4. Fig. 9. Numerical comparison beteen the numerical velocity 7 and the solution of the system 3 35 for ρ t = 0.5 tanh t/60 and v 0 = 0.5. e neglected beteen the kink, the internal modes and the radiation in the dynamics become important in this regime. The

13 6 L.A. Cisneros, A.A. Minzoni / Physica D equations for the transition beteen the moving kink studied in this ork and the static kink studied in [7] need to be developed to capture the transition beteen the different behaviors of the radiation. This delicate question needs to be studied in further ork. In particular, ho to incorporate the small knees of Figs. and 5 ill require a more precise analysis of the radiation. We conclude by remarking ho the modulation solution explains in simple terms the mechanism hich couple the kink ith the internal modes and the radiation. This approach gives a simple and accurate asymptotic method hich can be used in other situations involving kink propagation in lattices. Acknoledgements The authors ould like to express their gratitude to Noel F. Smyth for advice concerning the full numerical solution, to Panayotis Panayotaros for useful discussions and to Cristobal Vargas for getting them interested in lattices. Also, they ould like to thank the anonymous revieer for the comments. L. A. Cisneros gratefully acknoledges support from the grant CONACYT-No.Reg.7057 from the Consejo Nacional de Ciencia y Tecnologia Mexico and the grant SNI-Exp763 from the Sistema Nacional de Investigadores. Appendix A For an analytic function f x the Poisson summation formula takes the form: f n = A 0 A n cos nπ here A n = B n = n= B n sin nπ, n= f z cos nπz dz, n = 0,,,... f z sin nπz dz, n = 0,,,.... The coefficients, A n and B n hich are exponentially small in the idth of f, control the size of the PN potential barrier. To leading order this formula gives, n sech = 4π sinh π cos π n n sech = π π 4 cos π 6 sinh 3 π 3 cosh π π sinh π arctan sinh n sech = cos π arctan sinh sech x cos π x dx. The first term in the last sum is obtained by means of an interpolation of the corresponding integral, hich cannot be calculated in closed form. The rest of the terms are at most Oe π. Appendix B. Radiation damping B.. The long ave radiation emitted by the shelf Long aves generated by the shelf are assumed to satisfy the linear continuous Sine-Gordon equation ith the boundary condition u x = g t at x = t, as in [8]. The boundary condition g t is determined, equating the excess of energy contained in the coherent structure ith the energy contained in the radiation shed. For the kink, the excess of kinetic energy in obtained from the linearization of L K at the fixed point 0 gives, π = cg tl u t u x 3 0 u dx, 36 c g t l here c g is the group velocity of the linear aves obtained from the linearized equation y n y n y n y n y d n = 0. That is, sin k c g = /d 4 sin k/ 37 ith k as the ave number. The ave number k is estimated by considering the frequency Ω of the linear oscillator for at the fixed point. Thus the aves are produced at a ave number k hich satisfies k = Ω, k = sin k/. This gives c g /0. Since u x = g at the boundary, then the integral 36 is estimated by the trapezoidal rule to obtain see [8], g = π 3c g 0 t. As in [8], this gives [ y c n yc n / n ] l l = g t = π 3c g 0 t. 38 In the case of the internal mode the excess energy includes the kinetic energy of the internal in mode L A hich is, A A π. 36 Thus in the general case, e have [ y c n yc n / n ] l l

14 = π A 0 A π. 39 c g t B.. The Peyrard and Kruskal radiation The Peyrard and Kruskal radiation is caused by the resonance of the kink motion ith the lattice. To determine its effect e need to solve, yn P K yn P K y n P K yn P K d y n P K = 0, n t 40 ith appropriate matching conditions to the kink for n t. From Fig. 8 e observe that this type of radiation is produced by the oscillation of the spatial difference of the kink. Thus a suitable difference of yn P K must be matched to a given function f n t, hich is determined by the shape of the difference of the kink and of the internal modes hich are also excited, as n approaches t. Clearly this problem has no exact solution. Moreover, e need to derive a suitable matching condition in order to calculate the radiation. To this end e begin by studying the Laplace transform of the boundary condition in the form: 0 f n t e st dt. The integral cannot be evaluated in closed form. We thus take t vt here v =, since e expected the deviation vt to be small. In this case the integral can be evaluated by assuming v to be constant, since it is sloly varying, to obtain: v e ns/v B s, n = v e su/v f u du = e ns/v B s, n n n L.A. Cisneros, A.A. Minzoni / Physica D e su/v f u du. 4 The boundary condition 4 for 40 has a WKB form since for small vb is sloly varying and the phase in the exponential has large variations in n. We then take a WKB type of solution A n, s exp ϕ n, s for the Laplace transform of Eq. 40, s d ỹn P K ỹ P K n ỹ P K n = 0. To leading order the phase ϕ n, s satisfies the discrete dispersion relation, s exp ϕ n, s ϕ n, s d exp ϕ n, s ϕ n, s = 0. Assuming ϕ n, s ϕ n, s ϕ n, s, e obtain ϕ n, s = ln s, here s is the root of s d = 0. This gives the appropriate ave number of the discrete lattice at s = i. The function A n, s ill be determined in the matching. A suitable matching condition is obtained choosing the point n = N, and the points n = N and n = N. Fig. 8 suggests matching the spatial difference of the radiation to the spatial difference of the kink, thus e take the derivative to be u N u N / and it is to be matched to e ns/v B s, n. This gives u N u N = Be Ns/v A s exp ϕ N, s ϕ N, s exp ϕ N, s ϕ N, s = Be Ns/v. For matching, e take ϕ N = Ns/v and ϕ N = N s/v in the kink region. On the other hand ϕ N ϕ N is the radiation region and it satisfies the dispersion relation. Which gives exp ϕ N ϕ N = s. We thus determine A n, s in the form: A n, s = B e s/v s. 4 The approximate solution takes the form: y n t = n B s πi e s/v s est ds. 43 C The contour C lies in Re s > 0. The singularities of the integrand are to branch points extending from ±i 4 to d infinity. The poles are at the roots of the dispersion relation s = e s/v hich recovers the resonant aves described in [4] as the dominant contribution to the solution the function v is assumed sloly varying. The poles are on the imaginary axis and give real residues in the form: y n t = k e ink/v e it B i k e ik/v /v c.c, 44 / here the sum extends on the roots k of e s/v s = 0 as in [4]. Notice that the aves move to the left, also as v goes to zero the small roots k are lost as described in [4]. To complete the solution e calculate B i k. We have B i k = v N e i ku/v f u du. Since N is relatively large compared to the idth of the even f u. Then e can take, B i k = v e i ku/v f u du. 45 The function f u is provided by the derivative of the kink shape see Eq. 8 and the integral is extended to < u <. In this case f u = sech u. This integral is exponentially small since it is controlled by the poles of f u in Im < 0. Finally, e must take into account the fact that x t is not x vt. In fact e have from Fig. 9 that x vt < x t. On the other hand the functional dependence x vt is needed to construct the approximate solution for the radiation. Because of this e approximate x t by x vt ρ here ρ is a sloly varying function. To determine ρ e note from

15 64 L.A. Cisneros, A.A. Minzoni / Physica D Fig. 9 that ρ is basically constant and not too different from except at the point of larger acceleration hich is at the knee. Moreover, this point is ell-estimated using a constant value of ρ. We thus propose a decreasing ρ to adjust the mismatch ith a decrease around t = 60 hich is the time of the first knee estimated from the ρ = solution to the modulation equations. The actual values in ρ t = 0.5 tanh t/60 ere estimated by fitting, as in other problems, the result to a specific initial condition of v = 0.4. Then similar results ere obtained ith the same ρ t for a range of initial velocities beteen 0.3 and 0.5. It is to be noted that this approach ties the proposed modification of the idth in [4] to the details of the radiation mechanism. This shos that for discrete problems since there are larger accelerations at the knee, the effect of the radiation is more sensitive to the exact position of the kink than in the continuous case, here ρ = is sufficient since there are no knees [8]. We finally obtain estimating B i k by the residue at the first pole in the form, B i k = 4π v exp π k v ρ. 46 Using this expression into 44 e obtain, yn P K t = 8π v exp π k v ρ R k k k cos v n kt ψ k, 47 here i k v R k = i k e i k/v, 48 ψ k = arg / i k / i k e i k/v. 49 Appendix C From the trial function 8 it is obtained y n y n = 4 arctan sinh n sech A sinh sech n tanh n, sinh sech n y n = n sech n A n sech A n n sech tanh n, n A sech n cos y n sech n n sech tanh A n sech n sech A3 n 3 sech3 A4 n sech 4 4 sech 4 n n sech sech n sech n tanh, n here the last expression has been expanded in Taylor series as a function of A, near A = 0. By substituting the last expressions into the Lagrangian and neglecting the cross product of derivatives for example:,... because they are smaller terms, it is found that the averaged Lagrangian is, L = L K L A L int, here L K = L A = 8 A n sech A A n d sech n n sech arctan sinh sech n sech n tanh A n n n n sech tanh sinh tanh n A n d sech A4 4d and L int sech n sinh sech n n sech n sech 4 = A sech 4 n n sech n sech sech n sech,, n n tanh

16 A d A n sech 8 arctan sinh sinh sech n/ A3 3d sinh sech n/ sech 3 n L.A. Cisneros, A.A. Minzoni / Physica D n n sech tanh n / sech tanh n/ sech n tanh n. The series appearing in these Lagrangians are evaluated ith the Poisson summation formula, and to leading order e obtain n sech = sin π sech n tanh n sech x sech x tanh x sin π x dx = sin π V, n n sech 3 sech = sin π tanh n sech 3 x sech x tanh x sin π x dx = sin π S, n n sech tanh = 3 π 6 sinh π cos π n n n sech tanh = π cos π 8 x sech x tanh x cos π x dx sinh sech n tanh n sinh sech n = sinh 4 cos π = sinh 4I cos π n sech 4 sinh sech x tanh x cos πx dx sinh sech x = 4 3 8π π 3 sinh π cos π n n sech sech =.75 cos π. sech x x sech cos π x dx =.75. I, cos π n n sech 4 sech = cos π sech 4 x x sech cos π x dx = I 3, cos π. The first term in the last to sums is obtained by means of an interpolation of the corresponding integral. References [] Y. Frenkel, T. Kontorova, Zh. Eksp. Teor. Phys , 340; J. Phys [] J.F. Currie, S.E. Trullinger, A.R. Bishop, J.A. Krumhansl, Phys. Rev. B [3] Y. Ishimori, T. Munakata, J. Phys. Soc. Japan [4] M. Peyrard, M. Kruskal, Physica 4D [5] R. Boesch, C.R. Willis, M. El-Batanouny, Phys. Rev. B [6] R. Boesch, C.R. Willis, Phys. Rev. B [7] P.G. Kevrekidis, M.I. Weinstein, Physica D [8] N.F. Smyth, A.L. Worthy, Phys. Rev. E [9] S.P. Novikov, S.V. Manakov, L.P. Pitaevsky, V.E. Zakarov, Theory of Solitons. The Inverse Scattering Transform, Consultants Bureau, Ne York, 984. [0] O.M. Braun, Yu.S. Kivshar, Phys. Rev. B [] P.G. Kevrekidks, C.K.R.T. Jones, T. Kapitula, Phys. Lett. A