Reply to “Comment on ‘Existence of internal modes of sine-Gordon kinks’ ”

Reply to “Comment on ‘Existence of internal modes of sine-Gordon kinks’ ” Niurka R. Quintero* Departamento de Física Aplicada I, E.U.P., Universidad de Sevilla, c/ Virgen de África 7, 41011 Sevilla, Spain Angel Sánchez† Grupo Interdisciplinar de Sistemas Complejos (GISC) and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain and Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain Franz G. Mertens‡ Physikalisches Institut, Universität Bayreuth, D-85440 Bayreuth, Germany 共Received 16 November 2005; published 9 June 2006兲 In this reply to the comment by C. R. Willis, we show, by quoting his own statements, that the simulations reported in his original work with Boesch 关Phys. Rev. B 42, 2290 共1990兲兴 were done for kinks with nonzero initial velocity, in contrast to what Willis claims in his paper. We further show that his alleged proof, which assumes among other approximations that kinks are initially at rest, is not rigorous but an approximation. Moreover, there are other serious misconceptions which we discuss in our paper. As a consequence, our result that quasimodes do not exist in the sG equation 关Phys. Rev. E 62, R60 共2000兲兴 remains true. DOI: 10.1103/PhysRevE.73.068602 PACS number共s兲: 05.45.Yv, 03.50.⫺z I. INTRODUCTION In his paper 关1兴, Willis claims that in 关2兴he used a “rigorous” projection operator collective variable formalism for nonlinear Klein-Gordon equations to “prove” the continuum sine-Gordon 共sG兲equation has a long-lived quasimode whose frequency ␻ s=1.004 is in the continuum just above the lower phonon band edge with a lifetime 共1/ ␶ s兲 =0.003⌫0.In关3兴we performed two numerical investigations which show that neither intrinsic internal modes nor quasimodes exist in contrast to previous results. Willis further claims that he “proves” that our first numerical investigation could not possibly observe the quasimode in principle and our second numerical investigation actually demonstrates the existence of the sG quasimode. He states that his analytic calculations and verifying simulations were performed for a stationary sine-Gordon soliton at the origin, but that we explained our analytic calculations and confirming simulations in terms of the Doppler shift of the phonons when stationary sine-Gordon solitons have a zero Doppler shift. In the following we argue that his paper is irrelevant because the results Willis refers to in 关2兴were obtained for moving kinks 共which we prove by quoting statements from 关2兴兲, and because his alleged “proof” is not a mathematical proof but an approximate calculation as any collective coordinate theory 关4兴. II. ON THE NONSTATIONARITY OF KINKS After carefully reading their paper 关2兴several times, we are afraid that Willis’s paper suffers from misconceptions. Quoting literally from 关2兴, p. 2296, first paragraph beginning after Eq. 共3.2兲, it reads: “We impose initial conditions in our simulations in two different ways. The first way is by specifying the field 关¼兴thus giving the equilibrium kink a nonzero initial velocity.” Subsequently, three out of four figures of the paper refer to this type of simulation. For instance, the caption of Fig. 1 reads: “The initial condition for t=0 is a kink with velocity determined by specifying the kink shape at ¼.” Therefore, in spite of Willis’s claims that their simulations were for stationary kinks, that is not the case, and he cannot dismiss our explanation of his observations based on this. Kinks moved in his simulations and correspondingly their phonon spectra were shifted, as we explained in 关3兴. III. MISCONCEPTIONS IN THE PRECEDING PAPER In his paper 关1兴Willis claims: 共i兲“What they measured was the phonon absorption spectrum of the linearized sine-Gordon 共sG兲equation ¼.” This is simply not correct. We simulated the full sG equation: in page R61 of 关3兴, in the beginning of Sec. III, this is perfectly clear: “We have computed the numerical solution of the perturbed sG equation¼.” We never linearized anything and worked with the full equation. 共ii兲“Consequently, their two equations of motion for P共t兲 and l共t兲, which they state are the ‘basis of their theoretical analysis’ of Ref. 关3兴have no relevance to our analytic derivation and confirming simulations.” This is not correct. Actually, those equations are the way we find that, were there any internal modes or quasimodes in the kink, they would be excited parametrically by the ac driving. In spite of his claim that he “proved in Sec. III that an ac driver cannot create a sG quasimode,” we have shown, by means of those equations, that parametric excitation at half the frequency of the *Electronic address: [email protected] †URL: http://gisc.uc3m.es/ ˜ anxo ‡Electronic address: [email protected] PHYSICAL REVIEW E 73, 068602 共2006兲 1539-3755/2006/73共6兲/068602共3兲©2006 The American Physical Society068602-1 mode must be observed. We refer the interested reader to 关5–7兴to learn more about this effect. On this issue, it is particularly important to realize the inconsistencies in Willis’s reasoning: Just after Eq. 共16兲of 关1兴Willis claims: “Thus an ac force field cannot possibly excite a phonon mode. Consequently, their first numerical investigation in Ref. 关3兴could not possibly detect the presence of the sG quasimode and thus it has no relevance to the existence or nonexistence of the sG quasimode.” If this were true 共which it is not兲, Willis would be forced to conclude that a biharmonic force f共t兲=−关 ⑀ 1cos共 ␻ t兲+ ⑀ 2cos共2 ␻ t+ ␪ 兲兴 is also unable to excite the phonons. But this is not the case: On the contrary, Willis has recently studied 关8兴, by using the same approximate method that he developed in 1990, the action of this biharmonic force and dissipation on the ratchet motion of the kink. In this paper, he claims: “We use a rigorous collective variable for nonlinear Klein-Gordon equations to prove that the rectification of the current is due to the excitation of an internal mode ⌫共t兲, which describes the oscillation of the slope of the kink, and due to a dressing of the bare kink by the ac driver.” In Willis’s own words, dressing of the kinks means phonons excited in the system, in contradiction with his remark on our work. We also note that Willis’s discussion of our theoretical treatment contains misconception about “exact analytical calculations” in his earlier paper, which unfortunately are not exact 共cf. Sec. V below兲. 共iii兲“The simulations and analysis by the authors in Ref. 关3兴were done for an appreciably discrete sG equation.” This is not correct. Our discretization was of the same order as that of 关2兴: 0.05 in our case, 0.02 in theirs 关cf. 关2兴, p. 2296, below Eq. 共3.1兲兴. If ours is irrelevant, so is the one in 关2兴, and vice versa. 共iv兲On the other hand, Willis does not trust our numerical simulations and explicitly refers to the “flawed design of their 共ours in 关3兴兲 simulation,” claims that all we observe in Fig. 2 of 关3兴is “a complicated interference pattern,” arising from “reflection of phonons at the boundary at time 200.” We were fully aware of the possibility of phonon reflection at the boundaries, this being the reason why we used free boundary conditions, to minimize this effect. Furthermore, we did not trust a visual inspection of our Fig. 2 as he does, and carried out a Fourier analysis of the time evolution of the kink width, finding only phonons 共as Willis points out, present in the system due to emission and reflection兲. We did not find any Fourier peak arising from a hypothetical “quasimode,” in spite of the fact that the author of the paper concedes that “the first 200 s 关actually, time units兴of their simulation of the width l共t兲关actually, the simulation was for the full sine- Gordon equation, not of the width兴gives a very good representation of the sG quasimode.” How can it be then that it does not show up in the Fourier spectra, which we report in the paragraph below Fig. 2? IV. ON FURTHER EVIDENCE ON THE NONEXISTENCE OF QUASIMODES We would like to refer the reader to additional research that has significantly advanced the subject. Indeed, one of us 共N.R.Q.兲has published a very relevant investigation on this subject 关9兴, showing that the spectrum mentioned in the paper, see Eq. 共15兲, and which we used in our own paper 关Eq. 共15兲of 关3兴兴 is not correct, and in turn the parentheses should read 共n−1兲 ␲ /Lfor small n/L. This paper is very pertinent to this discussion because the author of the paper seems to be very worried that we are not aware of the “phonon dressing” of the kink. This work of Quintero and Kevrekidis is the best way to understand what are the consequences of the phonons on the kink behavior. Furthermore, there was only one other paper in the literature that observed something similar to a “quasimode” 关10兴, in fact using simulations of kinks driven by constant forces. This only independent confirmation of the “quasimode” was also proven wrong by us in 关11兴. In this case, it is not a matter of simulations and how they measured the “quasimode”: In the above reference we showed that the analytical calculations in this other paper were not correct. We encourage the reader to take this further report on the absence of quasimodes into account. V. ON THE APPROXIMATE CHARACTER OF COLLECTIVE COORDINATE THEORIES Finally, Willis does not seem to be informed about the nature and rigor of collective coordinate calculations. He states in the abstract that “prove the continuum sine-Gordon equation has a long lived quasimode¼.” Unfortunately and contrary to his belief, collective coordinate calculations cannot prove anything in the mathematical sense, in so far as they are approximate. Even if he tries to keep his calculation 共which, by the way, he reproduces literally from 关2兴, therefore adding no new evidence at all in favor of his claims兲 exact, he cannot, and he has to make a number of approximations including two linearizations and assuming Xand its derivative to be exactly zero 共see p. 2293 of 关2兴, 2nd paragraph on the right兲. Willis goes as far as claiming: “their Eq. 共3兲for l共t兲is incorrect because it contains none of the many terms proportional to ␹ 共t兲that appear in the exact equation of motion¼.” Our equation is completely correct, consistent with our derivation, and confirmed by numerical simulations. Our result is different from his simply because neither of them is exact. They begin with different ansätze and subsequently lead to different approximate equations. Collective coordinate and related calculations are interesting, often times very accurate and useful, but by no means exact and, mathematically speaking, constitute no proof. Indeed, it has been shown that such a procedure may lead to wrong predictions: For a detailed discussion, see 关4兴. In this respect, we want to stress that when Willis uses the paper 关12兴in favor of his argument, it is again an approximate result and a conjecture. It is true that Kälbermann calculates some solutions analytically, but he has to resort to numerical simulations to make his point. In addition, the relationship of Kälbermann’s “wobble solution” with Willis’s “quasimode” is dubious at best: Kälbermann finds a continuum of such possible quasimodes, which is not surprising because his wobble solution is a combination of a kink and a breather, and therefore is a one-parameter family of solutions; a longer discussion of this work is out of the scope of COMMENTS PHYSICAL REVIEW E 73, 068602 共2006兲 068602-2 this paper, but we want to stress that he never finds a specific quasimode as Willis claims, and Kälbermann’s only example has a frequency below the phonon band, very different from Willis’s quasimode. Furthermore, this wobble is, in Kälbermann’s own words, only “apparently stable” and all he can be positive about is that “the kink appears to be decaying to a wobble.” Therefore, there is no such thing as a mathematical proof of the existence of “quasimodes” in this paper either. VI. CONCLUSION Based on all the arguments summarized above, we are afraid that the preceding paper 关1兴contains misconceptions about our work. Willis’ work did not deal with stationary kinks, the simulations in Ref. 关2兴are as discrete as ours, collective coordinate theories are not rigorous mathematical proofs, there are further developments on this question, and Willis misreads our paper 关3兴. Therefore, Willis’s paper does not change our conclusions, and our result that quasimodes do not exist in the sG equation 关3兴stands in its own right. ACKNOWLEDGMENTS This work has been supported by the Ministerio de Ciencia y Tecnología of Spain through Grant Nos. FIS2005-973 共N.R.Q.兲, BFM2003-07749-C05-01, FIS2004-01001, and NAN2004-09087-C03-03 共A.S.兲and by “Acciones Integradas Hispano-Alemanas” HA2004-0034—D/04/39957. 关1兴C. R. Willis, preceding paper. 关2兴R. Boesch and C. R. Willis, Phys. Rev. B 42, 2290 共1990兲. 关3兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. E 62, R60 共2000兲. 关4兴A. Sánchez and A. R. Bishop, SIAM Rev. 40, 579 共1998兲. 关5兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. Lett. 84, 871 共2000兲. 关6兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. E 62, 5695 共2000兲. 关7兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Phys. Rev. E 64, 046601 共2001兲. 关8兴C. R. Willis and M. Farzaneh, Phys. Rev. E 69, 056612 共2004兲. 关9兴N. R. Quintero and P. G. Kevrekidis, Phys. Rev. E 64, 056608 共2001兲. 关10兴E. Majernikova, Y. B. Gaididei, and O. M. Braun, Phys. Rev. E 52, 1241 共1995兲. 关11兴N. R. Quintero, A. Sánchez, and F. G. Mertens, Eur. Phys. J. B 19, 107 共2001兲. 关12兴G. Kälbermann, J. Phys. A 37, 11603 共2004兲. 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