Kink-Antikink Collisions in the �6 Model

Patrick Dorey,1,2 Kieran Mersh,1 Tomasz Romanczukiewicz,3 and Yasha Shnir1

1Department of Mathematical Sciences, Durham University, United Kingdom2LPTHE, CNRS-Universite Pierre et Marie Curie-Paris 6, 4 Place Jussieu, Paris, France

3Institute of Physics, Jagiellonian University, Krakow, Poland(Received 17 February 2011; published 25 August 2011)

We study kink-antikink collisions in the one-dimensional nonintegrable scalar �6 model. Although the

single-kink solutions for this model do not possess an internal vibrational mode, our simulations reveal a

resonant scattering structure, thereby providing a counterexample to the standard belief that the existence

of such a mode is a necessary condition for multibounce resonances in general kink-antikink collisions.

We investigate the two-bounce windows in detail, and present evidence that this structure is caused by the

existence of bound states in the spectrum of small oscillations about a combined kink-antikink

configuration.

DOI: 10.1103/PhysRevLett.107.091602 PACS numbers: 11.10.Lm, 11.27.+d

Introduction.—Solitary wave solutions in nonlinear fieldtheories have been studied for many decades. Perhaps thesimplest examples are the kink solutions which appear inmodels in one spatial dimension with a potential with twoor more degenerate minima. These models have applica-tions in condensed matter physics [1], field theory [2,3]and as simplified models of domain wall collisions incosmology [4–6]. The double well case is exemplified bythe nonintegrable�4 model. A particularly striking featureof this theory is the intricate structure of kink-antikinkscattering, first observed numerically in the 1970s [7] andmuch studied thereafter [8–13]. For initial velocities abovea critical value vc � 0:2598 the two incident waves alwaysescape to infinity after collision, with the emission of someradiation. Below vc, the incident waves generically be-come trapped, but there is also a complicated pattern ofnarrow ‘‘resonance windows’’ within which they are againable to escape to infinity.

The accepted explanation of the appearance of thesewindows, both in this model and in similar theories suchas the parametrically modified sine-Gordon model [10], isthat they are related to a reversible exchange of energybetween the translational and vibrational modes of theindividual kinks [9]. At the initial impact, some kineticenergy is transferred into internal ‘‘shape’’ modes of thekink and antikink. They then separate and propagate al-most independently, but for initial velocities less than vc

they no longer have enough translational energy to escapetheir mutual attraction, and so they return and collide asecond time. At this point some of the energy stored in theshape modes can be returned to the translational modes,tipping the energy balance back again and allowing thekink and antikink to escape to infinity, provided that thereis an appropriate resonance between the interval betweenthe two collisions, and the period of the internal modes.More generally, sufficient energy might be returned to thetranslational modes after three or more kink-antikink

collisions, leading to an intricate nested structure of reso-nance windows, a picture which has been confirmed byboth numerical and analytical studies [9–13].This mechanism requires that the kink and antikink

each support at least one internal vibrational mode,within which energy can be stored prior to being trans-ferred back to translational modes after a number ofbounces. This suggested, as stated for example in [12],that the existence of an internal kink mode is a necessarycondition for the appearance of resonance windows. Theparametrically modified sine-Gordon model lent furthersupport to this view: depending on the value of a pa-rameter, kinks and antikinks do or do not possess aninternal mode; correspondingly, resonance windows door do not appear [10]. Resonance windows have alsobeen observed in vector soliton collisions [14,15] and inthe scattering of kinks on impurities [16]. Again, themechanism always relies on the presence of a localizedinternal mode, either of a single kink or of an impurity,or both.In the present Letter we revisit this question in the

context of kink-antikink scattering in the �6 model [17].The potentials mentioned in [10] include a general poly-nomial form, but the �6 model has a number of featureswhose relevance to resonant scattering have not been ap-preciated to date, even in the recent work [18]. As for the�4 model, it is nonintegrable, but, as for the modified sine-Gordon model in the regime of no resonant scattering, asingle �6 kink does not have an internal oscillatory mode[17]. Nevertheless, certain kink-antikink collisions exhibitresonant scattering, thus providing a counterexample to theusual lore. We elucidate the mechanism for this process,finding that the underlying reason is a reversible transfer ofkinetic energy from the kinks to a collective bound statetrapped by the �KK pair.The model.—The one-dimensional �6 theory can be

defined by the rescaled Lagrangian density [17]

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L ¼ 1

2@��@��� 1

2�2ð�2 � 1Þ2: (1)

The model has three vacua �v 2 f�1; 0; 1g. Statickinks and antikinks, interpolating between neighboring

vacua, can be found from the one-kink solution �KðxÞ ��ð0;1ÞðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ tanhxÞ=2p

using the discrete symmetries

of the model under � ! �� and/or x ! �x, so� �KðxÞ � �ð1;0ÞðxÞ ¼ �Kð�xÞ, �ð0;�1ÞðxÞ ¼ ��KðxÞ and

�ð�1;0ÞðxÞ ¼ ��Kð�xÞ. These all have mass M ¼ 1=4.The model also possesses perturbative meson states,

fluctuations about the vacua � ¼ 0, �1. Fluctuationsaround the static solutions �sðxÞ which interpolate be-tween these vacua can be treated by setting �ðx; tÞ ¼�sðxÞ þ �ðxÞei!t. The linearized field equations are��xx þUðxÞ� ¼ !2� where the potential UðxÞ is

UðxÞ ¼ 15�4s � 12�2

s þ 1: (2)

For an isolated kink there are no localized solutions to thisequation beyond the usual translational zero mode, reflect-ing the absence of internal oscillatory modes. The states ofthe continuum spectrum can be written in terms of hyper-geometric functions [17].

Numerical results.—The initial conditions we took cor-respond to a widely separated kink-antikink pair propagat-ing towards a collision point. To find a numerical solutionof the partial differential equation (PDE) describing theevolution of the system, we used a pseudospectral methodon a grid containing 2048 nodes with periodic boundaryconditions. For the time stepping function we used asymplectic (or geometric) integrator of 8th order to ensurethat the energy is conserved. The time and the spatial stepswere �x ¼ 0:25 and �t ¼ 0:025. To check numericalstability we repeated selected calculations with �t ¼ 0:05and �t ¼ 0:0125.

Since the model (1) contains two distinct classes ofvacua, 0 and �1, kink-antikink collisions in the sectorsbuilt on the �v ¼ 0 and the �v ¼ �1 vacua have to beanalyzed separately. In the former case the initial configu-ration, which we denote as K �K or ð0; 1Þ þ ð1; 0Þ, can betaken as a superposition �ðxÞ ¼ �Kðxþ aÞ þ� �Kðx� aÞ�1 where a > 0 is the initial (half-)separation parameter,which we set equal to 12.5; in the latter case we take�ðxÞ ¼ � �Kðxþ aÞ þ�Kðx� aÞ and denote this as �KKor ð1; 0Þ þ ð0; 1Þ. In neither case are there internal vibra-tional modes bound to the individual kinks. By the standardpicture, we would therefore predict that neither shouldexhibit any resonance structure.

In the case of ð0; 1Þ þ ð1; 0Þ K �K collisions, this is indeedwhat we found [Figs. 1(a) and 1(b)]. For v < vcp � 0:289

the pair always become trapped, while for v > vcp

the collision yields a reflected pair of solitons togetherwith some radiation: ð0; 1Þ þ ð1; 0Þ ! ð0;�1Þ þ ð�1; 0Þ.However the ð1; 0Þ þ ð0; 1Þ �KK collision reveals a verydifferent picture, illustrated in Figs. 1(c)–1(e). An intricate

pattern of escape windows can be seen, up to a criticalvelocity vcr � 0:0457, after which the kinks always haveenough energy to separate. This is very similar to thebehavior of the�4 model, even though there are no internalmodes of the scattering kinks into which kinetic energy canbe transferred. Thus, we have to look for another explana-tion of our results.The mechanism.—The spectrum of linear perturbations

around a single�6 kink differs from that for the�4 kink inthat the potential UðxÞ is not symmetrical with respect toreflections x ! �x. As a result the potential for a kink-antikink pair depends on the order in which they appear onthe line: a well-separated K �K pair [Fig. 2(a)] has a raisedcentral plateau, while an equally separated �KK pair[Fig. 2(b)] has instead a wide central well. If the velocitiesof the kinks are relatively small, we expect—and willjustify later—that the adiabatic approximation can beused to find the spectrum of small fluctuations about thesetwo configurations, using Eq. (2) for the appropriate

FIG. 1 (color online). fK; �Kg collisions in the ð0; 1Þ þ ð1; 0Þ[(a), (b)] and ð1; 0Þ þ ð0; 1Þ sectors [(c), (d), (e)]. Plots (a) and (c)show the fitted final velocity of the kink as a function of initialvelocity, with the dotted line indicating the result for a purelyelastic collision, while (b) and (d) depict the field values mea-sured at the collision center. The final plot, (e), shows the timesto the first, second, and third kink-antikink collisions in theð1; 0Þ þ ð0; 1Þ sector.

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choices of �s. Figure 3(b) shows our numerical resultsfor the ð1; 0Þ þ ð0; 1Þ �KK pair as a function of separationparameter a. The two lowest states are quasizero modes,rapidly approaching zero from opposite sides as the sepa-ration grows. On top of these is a tower of meson stateswith separation-dependent energies. The analogous plotfor the ð0; 1Þ þ ð1; 0Þ K �K pair, by contrast, shows onlythe two quasizero modes.

So, a new feature of �KK collisions in the �6 model isthat although there are no internal modes of the individualkinks, energy can be stored after an initial impact in thetrapped meson states of the composite ð1; 0Þ þ ð0; 1Þ con-figuration. This opens the possibility that, for collisionswith vi < vcr satisfying a suitable resonance condition, thisenergy might be returned to the translational modes on asubsequent recollision, thereby allowing the kink and anti-kink to return to infinity.

To support this picture, we first assign a ‘‘bouncenumber’’ to each window, equal to the number of collisionsthat the kink and antikink undergo before their final escapeto infinity. Within each window the bounce number isconstant. The first two-bounce windows, discernible onFig. 1(e) as the intervals of velocity within which thetime to the third collision diverges, are centered at vi �0:0228, followed by a ‘‘false’’ window at vi � 0:0273,then a third at vi � 0:0303, and so on. The regions nearto these windows exhibit nested structures of higher-bounce windows, similar to those seen in kink-antikinkinteractions in the �4 model [11,12].Our analysis implies that each two-bounce window

should be associated with a distinct integer n, the numberof collective mode oscillations between its two collisions.In Fig. 4(a) we illustrate this for the three windows justdiscussed. It is easily checked that the number of oscilla-tions of �ð0; tÞ between the two collisions increases byone from one plot to the next. Furthermore, our numericalresults clearly indicate that these oscillations are thoseof the lowest collective mode of the �KK system. Inthe two-bounce windows we considered, we observedmaximal kink-antikink separations between bounces upto R � 2a � 12. Figure 3(a) shows the Fourier transformof the value of the field at the collision center between thetimes of the first two bounces for vi ¼ 0:045 48. The timebetween these bounces is T � 660, during most of whichthe separation between the kinks is close to 12. The Fouriertransform shows four peaks at frequencies 1.045, 1.28,1.61, 1.92, with the strength of the second peak aboutone order smaller than that of the first, and the higherones strongly suppressed. The positions of these peaksrelative to the energies of the first 4 parity-even modes[Fig. 3(a); note that the symmetry of the initial conditionrules out any coupling to parity-odd modes] are consistentwith a half-separation between the kink and antikink ofa � 6:13, in good agreement with the observed value. Thisalso lends support to our use of the adiabatic approxima-tion in the analysis of fluctuations about the kink-antikinkconfiguration.In Fig. 4(b) we show the dependence of the times

TðvÞ between the two collisions in the two-bounce win-dows on the number nðvÞ of oscillations in between thesecollisions. The results are a good fit to the linear relation!T ¼ 2�nþ �, where ! ¼ 1:0452, the same as the fre-quency of the lowest collective mode at 2a � 12 and thevalue found from the Fourier transform at vi ¼ 0:045 48,and � ¼ 6:0277. Requiring � to lie between 0 and 2�,as here, fixes the number n of collective mode oscillationsassigned to the three windows shown in Fig. 4(a) to be12, 13, and 14, respectively, while the initial velocityvi ¼ 0:045 48 used for the Fourier transform plot is veryclose to the n ¼ 109 two-bounce window.A linear fit with this value of! is exactly as predicted by

a resonance mechanism driven by the first mode, and

FIG. 2 (color online). The potentialUðxÞ (solid lines) for linearperturbations about the composite ð0; 1Þ þ ð1; 0Þ K �K (a) andð1; 0Þ þ ð0; 1Þ �KK (b) configurations (dashed lines).

FIG. 3. (a) Peaks of the Fourier transform of the field at theorigin in the two-bounce window for v ¼ 0:045 48, vs (b) even(solid lines) and odd (dashed lines) bound states for the ð1; 0Þ þð0; 1Þ configuration, as a function of the half-separation a.

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closely resembles similar relations in the �4 model [9],although the detailed mechanism here is different. In par-ticular, in resonant �4 scattering, energy is exponentiallylocalized on the kink and antikink, while in the �6 theorythis energy is stored in an extended meson state residing inthe potential well formed between the kink and antikink.This results in a small amount of radiation pressure on thekinks after their initial impact, which we observed forvi ¼ 0:05, just larger than vc, in a small (of order 10�6)acceleration of the bouncing kinks away from each other,when measured at times t � 300 after collision so that thekinks had become well separated. A corresponding simu-lation in �4 theory showed that any such effect is at least104 times weaker there.

Nevertheless, we can pursue the similarity between thetwo models by considering the asymptotic attractive forcesbetween kink and antikink in the absence of radiationpressure, which are FðaÞ � 2e�R for the �KK pair, andFðaÞ � 2e�2R for K �K pair, where R ¼ 2a is the kink-antikink separation. The arguments of [9] would thenpredict TðvÞ / ðv2

cr � v2i �� for impact velocities vi just

below vcr, with � ¼ 0:5. Our fits based on this functionalform favor a smaller value for �, in the range 0.39–0.45. Itwill be interesting to see whether a more refined treatment,taking radiation pressure into account, could account forthese discrepancies.

A novel feature of resonant �6 scattering is the‘‘missing’’ window at n ¼ 13. For resonant �4 scattering,two-bounce windows are also missing, for n < 3, but oncethey set in they are found for all n, at least up to initialvelocities very close to vc [9]. By contrast, in�

6 scatteringwe found the first two-bounce window at n ¼ 12, then agap at n ¼ 13, and then windows for all higher values of nthat we examined. A similar structure is reproduced whenlooking at the three-bounce windows next to a given two-bounce window, and we suspect that the pattern will con-tinue at all higher levels. It is possible that an explanationfor this behavior will be found in a careful treatment of thehigher modes of the bound-state spectrum.Conclusions.—Our investigation of �6 kink collisions

has shown that resonance phenomena havewider relevanceto kink scattering than had previously been thought. Inparticular, similar behavior should be seen in any modelin which kinks interpolate between degenerate but non-equivalent vacua. The new mechanism enabling reso-nances to occur is the formation of meson bound statesin the potential well created in the space between theconstituents of a suitably ordered kink-antikink pair, anddoes not require the existence of an internal mode localizedon a single kink. The resulting pattern of resonance win-dows is more complicated than that for �4 scattering, withgaps appearing at all levels. It remains a major challenge,deserving further study, to find a robust mechanism toexplain these gaps theoretically.We thank M. Peyrard for helpful discussions, and Hadi

Susanto and Roy Goodman for bringing references[14,15,18], respectively, to our attention. P. E. D. thanksLPTHE and Y. S. thanks the University of Oldenburgfor hospitality. The work was supported in part by theSTFC (P. E. D., K.M), the CNRS (P. E. D.), and theA. von Humboldt Foundation (Y. S.).

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