Radiationless energy exchange in three-soliton collisions

Radiationless energy exc hange in three-soliton collisions Sergey V. Dmitriev 1 , ∗ Pana yotis G. Kevr ekidis 2 , and Y uri S. Kivs har 3 1 Institute for Met als Sup erplas ticity Pr ob lems RAS, Khalturina 39, 450001 Ufa, R ussia 2 Dep artmen t of Mathematics and Statistics, University of Massachusetts, Amher st, MA 01003 USA 3 Nonline ar Physics Center, R ese ar ch Scho ol of Physic al Scienc e and Engine ering, Aus tr alian National University, C anb err a, 0200 ACT, Aust r alia W e revisit the problem of th e three-soliton collisions in th e we akly p ertu rb ed sine-Gordon eq uation and develop an effective three-particle mo del allow ing to explain many interes ting features observ ed in numerical simulations of the soliton collisions. In particular, we explain why collisi ons b etw een tw o kinks and one an tikink are observ ed to b e practicall y elastic or strongly inelastic dep end ing on relati ve initial positions of the kinks. The fact that the three- soliton coll isions b ecome more elastic with an increase in th e collision velocity also b ecomes clear in the framew ork of the three- particle mo del. The three-particle mo del do es not invo lve in ternal mod es of the kinks, but it giv es a qu alitativ e description to all the effects observed in the th ree-soliton collisions, includ ing the fractal scattering and the existence of short- live d t h ree-soliton b ound states. The radiationless energy ex change b etw een the colliding solitons in weakly p erturb ed integ rable systems takes place in t he vicinity of the separatrix multi-soliton solutions of the correspon d ing integrable equations, where ev en small p erturbations can result in a considerable c hange in t h e collisio n outcome. This conclusion is illustrated through the use of the reduced three-particle mo del. P ACS num bers: I. INTRO DUCTION The study of so liton collisions in nonintegrable sys- tems [1, 2 ] is interesting b ecause such sy s tems typically describ e more realis tic situations than the integrable sy s- tems wher e the interactions b etw een s o litons are known to b e purely elastic [3 ]. In nonintegrable systems, the co l- lision outcome can b e highly nontrivial and, depending on the degree of nonin tegra bilit y , the collision scenar io can have qualitatively different features. F or the classica l φ 4 equation, ∂ 2 u ∂ t 2 − ∂ 2 u ∂ x 2 + u − u 3 = 0 , (1) which is rather far from an integrable sy stem, kink c o l- lisions a re always acco mpanied by a certa in amount of radiation in the form of small-amplitude w av e pack ets, as well as by the excitation of the kink’s internal mo des [4]. The latter are resp o nsible for sev eral effects in the φ 4 kink-antikink collisio ns. In pa r ticular, the resonant energy exchange betw een the trans la tional motio n of the kinks and their internal mo des explains the fractal kink - antikink scattering [5]. This is a topic that was initiated by the numerical studies in Ref. [6] (see also Ref. [7] and references therein), and it is still under ac tive inv estiga- tion [8]. F or long time, the ex citation of the soliton in ternal mo des and the ra diation lo sses were thought to be tw o ma jor manifestations o f inelasticit y of the soliton colli- sions in nonintegrable mo dels. How ev er, a qua litatively different manifestation was recently iden tified, namely , ∗ Electronic address: dmitriev.sergey .v@gmail. com the radiationle s s ener gy exchange (REE ) b etw een collid- ing solitons [9 , 10, 11, 12, 13, 14, 15, 16, 1 7] (abbreviated as ” REE” in Ref. [10], a designation that w e will use hereafter). The energy transferred to soliton internal mo des in soliton collisions, for small ǫ , is typically propor tio nal to ǫ 2 , and the s a me is true for the radiation losses (here ǫ is the co efficient in front of a p er turbation term, added to an integrable equation). T erms pr o p ortional to ǫ 2 ap- pea r as the low est-or der co rrection ter ms in the collective v aria ble appr oaches used to describ e the s oliton’s inter- nal mo des [4 ]; the kink dyna mics in the discrete φ 4 equa- tion [18]; the kink and br e ather dynamics in the discrete sine-Gordon equatio n (SGE) [19, 20]; and the radiatio n from the discr ete SGE kink [21] and fr o m the solito n in the disc r ete nonlinear Schr¨ odinger equa tion (NLSE ) [1]. On the other hand, the degree of inelasticity due to the REE effect, when the latter is prese n t (see details b e- low) grows prop ortio na lly to ǫ [1 0, 11]. This means that for weakly p erturb ed integrable systems the REE effect bec omes a do mina nt manifestation of the inelasticity of collision, while the s o liton’s internal mo des and radiation bec ome increas ingly imp orta nt with str onger deviations from integrable c a se. The REE effect can also b e resp ons ible for the fractal soliton sc a ttering which was demonstra ted for the first time in [14] for the weakly p erturb ed SGE and later for the w eakly p erturb ed NLSE [15, 16]. In contrast to those studies, in Refs. [22, 23] the fractal sca ttering of vec- tor s olitons in the coupled NLSE was attributed to the resonance ener gy exchange b et ween the soliton’s tr ansla- tional and in ternal mo des, i.e., through the mec hanism similar to that o per ating for the φ 4 kinks [5, 6, 7, 8]. F racta l soliton sca ttering in the weakly p erturb ed NLSE was explained qualitatively in the fra me o f a v ery 2 simple mo del [15] and for the generalized NLSE in the context o f a more elab o rate collective v ariable a pproach [24], based on the metho d of K arpman a nd Solov’ev [25]. Remark ably , the soliton’s in ter nal mo de s were not in- volv ed into consideratio n in [15, 24] indicating that the underlying dominan t mechanism for the fractal scatter- ing was the RE E effect (rather than the int erna l mo de excitation). F or weakly p erturb ed in tegra ble sy s tems, parameters of the colliding solitons where the REE effect is observed can be found from the analysis of the corr esp onding inte- gr able equation. This w as done for the w eakly p erturb ed SGE in [12] and for the w eakly p erturb ed NLSE in [16] using the fact that the REE effect is obser ved in the vicin- it y of separ atrix multi-soliton solutions of the integrable equation. In the case of mo derate deviation from int egr ability , it b ecomes increa singly impo rtant to chec k if the degr ee of nonin tegra bility a nd the sign of perturbatio n allows for the exis tence of noticea ble soliton internal modes be- fore one can judge to which ex ten t the REE effect and the soliton in ternal modes contribute to the inela s ticit y of co llision (see, e.g., Sec. I I D in [17]). The effect of the RE E effect in the c a se of a mo derate degree of no n- int egr a bility ha s studied far less ex tensively than in the case o f weak per turbation, though v a luable results hav e bee n recently obta ined for the discrete NLSE [26] and for the generaliz ed NLSE with v a rious t yp es of the nonlinear term [2 4], where a gener al sys tem of ordinary differen tial equations was derived for the v elo cities, amplitudes, p o- sitions and phases of the solitary wav e s . The latter was shown to q ualitatively and quant itatively match the pre- dictions o f the full mo del. In this pa pe r, we study the REE effect in thr ee-soliton collisions of a weakly per turb ed sine-Gordo n equation. In the frame o f the three-par ticle mo del, we de mo nstrate that the REE effect is directly related to a separatrix solution, and it o ffers a very transpar ent explanation of the origin of fracta l solito n scattering. W e also classify in a general way the p o ten tial for emerg ence of such phe- nomena in three-kink collis ions of the weakly p er turb ed sine-Gordon mo del. The paper is org anized a s follows. In Sec. II, the results of numerical study of the degr ee of inelasticity of three-soliton co llisions in the F renk el-Ko ntoro v a model Eq. (2) ar e presented. First, the collisions b etw een three kinks/antikinks ar e analyzed in Sec. II A and then the kink-breather co llis ions are in vestigated in Sec. I I B. The three-particle model is introduced and analyzed in Sec. II I . The discussio n of the re s ults a nd our co nclusions a re presented in Sec. IV. II. THREE-SOLITON C OLLISIONS IN WEAKL Y D ISCRETE SGE T o s tudy the effects of non-integrability on the soliton collisions it is des irable to hav e a model with tunable deviation from a n integrable case [c.f. with Eq. (1) which do es not hav e s uch a par ameter]. The F renkel-Kontorov a (FK) mo de l, d 2 u n d t 2 − 1 h 2 ( u n − 1 − 2 u n + u n +1 ) + sin u n = 0 , (2) which is a discr etization of the integrable SGE, u tt − u xx + sin u = 0 , (3) is a co n venien t choice for such a study [12, 13, 14]. The (singular) per turbation parameter in Eq. (2) is ǫ = h 2 (with h b eing the lattice spacing ); the lowest order cor- rection to SGE due to the discretization can b e qua n ti- fied, up on a T aylor expansion of the second difference, as ( ǫ/ 12) u xxxx . The exa c t three-so liton solutions to SGE a r e well known [12, 27]. The solutions ar e the com binations o f single-solito n solutions, namely kinks ( K ) or a ntikinks ( K ), having the top ologic a l c harg es q = 1 and q = − 1, resp ectively , and tw o -soliton solutions, namely breathers ( B ), which are actually the kink-antikink oscillatory bo und states. Energy E and momentum P of one SGE kink ar e de- fined b y its velocity V a s follows E K = 8 δ, P K = 8 V δ, where δ − 1 = p 1 − V 2 . (4) Energy and momentum of a br eather are defined b y its frequency ω and velocity V : E B = 16 η ξ , P B = 16 η ξ V , where ξ − 1 = p 1 − V 2 , η = p 1 − ω 2 . (5) Below w e describ e the num erica l r e sults for the three- soliton collisions in the weakly discrete ( h 2 = 0 . 04 ) SGE Eq. (2). The exact three-solito n solutio ns to SGE were employ e d for setting the initial conditio ns. The equa tions of mo tion Eq. (2) were integrated with the us e of the St¨ ormer metho d of order six. W e register the pa r ameters of quasi- particles a fter their collision and compa r e them with those b efor e the collis ion. The lar ger the change in the parameters, the more inelastic the collis ion is. A. Three-kink colli sions W e num b er the kinks in a w ay that at t = 0 (be- fore the co llis ions) their initial p ositions are related as ( x 0 ) 1 < ( x 0 ) 2 < ( x 0 ) 3 and momenta as P K 1 > P K 2 > P K 3 . Here we consider only symmetric collisio ns with P K 1 > 0, P K 2 = 0, and P K 3 = − P K 1 . Cons ideration of non-symmetric collisions does not bring a ny new im- po rtant physical effects. F or the symmetric collisions it is c o nv enient to set ( x 0 ) 1 = − ( x 0 ) 3 so that the three- soliton collisio ns a re exp ected when ( x 0 ) 2 is clos e to the origin, otherwise the tw o successive tw o-s oliton collisions will take place. Thus, a mong the kink’s initial po sitions ( x 0 ) i the only essential parameter is ( x 0 ) 2 . 3 Finally , our three-kink sys tem is defined b y the top o- logical c harg es o f the kink s . There are eigh t pos sible v aria nts in assigning the charges to the three kinks, which, due to s ymmetry , can b e divided into three g roups of to po logically different collisions : K K K = K K K , K K K = K K K , and K K K = K K K = K K K = K K K . W e will r efer to each gr oup by referring to their first mem- ber . The co llision outcome is presented b y the momenta of kinks after collisio n, ˜ P K j , as the functions of ( x 0 ) 2 for a given P K 1 , whic h defines the initial momenta of the kinks, P K j , as described above. In some case s a kink- antikink pair can merge into a br e a ther. In tho se c ases we assumed that the kinks co nstituting the breather share its momentu m equally , in or der to plot their momenta. The res ults for the K K K collisio ns are shown in Fig . 1 (a) for P K 1 = 0 . 8. Similar results for the K K K c o llisions are s hown in Fig. 1 (b) also for P K 1 = 0 . 8. The results for the K K K co llis ions are shown in Fig . 2 (a) for P K 1 = 2 . 5 (larger collision velo cit y) and in Fig. 2 (b) for P K 1 = 0 . 8 (smaller collision velocity). In the panels (a’) and (b’) of Fig. 1 and Fig. 2 the examples of collisio ns ar e presented on the ( x, t ) pla ne by showing the r egions of ener gy density gr e a ter than a certain v alue, so that the cores of the solitons ar e clearly seen. These examples are g iven for the particular v al- ues of the initial co ordina te o f the middle kink, ( x 0 ) 2 , indicated by the arrows in the corresp o nding panels at left. First we no te that K K K and K K K collisions are a l- wa y s practically elastic reg ardless of the sp ecifics of ( x 0 ) 2 (see Fig. 1) and only K K K collisions are inelas tic for ( x 0 ) 2 close to the orig in (see Fig. 2). W e conclude that if a kink has p ositive or negative c harg e with equal prob- ability , then the REE in three-kink collisions can be ex- pec ted in t wo cases from eight. Of pa rticular imp ortance is the fact that in the s trongly inelastic K K K collisions shown in Fig. 2 the energy given to the kink’s internal mode s and to the radia tion is negligible in c omparison to the ener gy exchange b etw een the quas i-particles [12]. This is the main feature of REE effect in soliton c o llisions. The K K K collisions can be strong ly inela stic b ecause in this case the c o res o f all three k inks ca n mer ge. Tw o- kink collisions are pr actically elastic for the co nsidered case of w eak p ertur bation, h 2 = 0 . 04, as it c a n b e seen in Fig. 1 (b), (b’). T o explain wh y the tw o -kink colli- sions are elastic we note that Eq. (2) conserves e ne r gy and, for s ma ll p erturbatio n parameter h 2 , the momen- tum is a lso conserved with a high a ccuracy while the higher-or der conser v ation laws of SGE ar e de s troy ed by the weak discreteness. The conserv ation of energy and momentum s ets t wo co nstraints on the t wo parameters of the t wo-kink solution. A three - kink solution ha s o ne free parameter and REE be comes possible if all three kinks participate in a co llis ion. F or the K K K collisions we no te that the collision with a la rger velo city [see Fig . 2 (a), (a’)] r esults o nly in quan- - 2 0 - 1 0 0 1 0 2 0 0 5 0 1 0 0 1 5 0 2 0 0 - 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8 - 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8 K K K ( a ') t x K K K ~ ( a ) P K j ( x 0 ) 2 - 2 0 - 1 0 0 1 0 2 0 0 5 0 1 0 0 1 5 0 2 0 0 _ K _ K K K K K ( b ') t x - 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8 - 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8 _ K K K ~ ( b ) P K j ( x 0 ) 2 FIG. 1: Numerical results for (a), (a’) K K K and (b), (b’) K K K coll isions in FK model. The left panels sho w the mo- menta of the kink s after collision ˜ P K j as the functions of the initial p osition of middle k ink, ( x 0 ) 2 . In b oth cases m omenta of the kinks b efore the collision w ere P K 1 = − P K 3 = 0 . 8 and P K 2 = 0 and th ey are nearly same after the collision meaning that the collisions are practically elastic for an y ( x 0 ) 2 . The righ t panels sho w th e examples of collisions on the ( x, t ) plane for ( x 0 ) 2 = 0 by plotting th e regions with the energy density greater than certain val ue, so that the cores of the solitons are clearly seen. titative change of kink parameters , while collision with a smaller velocity [see Fig. 2 (b), (b’)] may r e sult in fusion of a kink-antikink pair in a breather. The result of K K K collisions is extr e mely s ensitive to v ariatio ns in ( x 0 ) 2 in the vicinity o f ( x 0 ) 2 = 0, esp ecially for small co llis ion velocities. A simple explana tio n of the fact that the collisio ns b e- t ween tw o kinks a nd an an tikink are a lways pr actically elastic for K K K and can be strongly inela s tic in the c a se of K K K will b e o ffer ed in Sec. I I I. In the ca se of weak pe r turbation w e never obser ved fractal pa tterns in the thre e -kink collisions (reca ll that in the φ 4 mo del such patterns can b e observed even in t wo-kink collisio ns but, as it was a lready mentioned, this mo del is far fro m an integrable one), while it can b e ob- served in the kink-brea ther collisions , as discussed b e low, and in the brea ther-breather co llisions [14]. B. Kink-breather collisions Without loss o f g enerality , w e assume P K + P B = 0 . Then w e hav e tw o par ameters, the momentum P B and frequency ω of the brea ther. The o utcome of the K B 4 - 2 0 - 1 0 0 1 0 2 0 0 2 0 4 0 6 0 8 0 - 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8 - 3 - 2 - 1 0 1 2 3 _ _ K K K K K K ( a ') t x _ K K K ~ ( a ) P K j ( x 0 ) 2 - 2 0 - 1 0 0 1 0 2 0 0 5 0 1 0 0 1 5 0 2 0 0 _ K B K K K ( b ') t x - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2 - 2 - 1 0 1 2 _ _ K K K B B K K K ~ ( b ) P K j ( x 0 ) 2 FIG. 2: Numerical results for K K K collisions with (a), (a’) P K 1 = 2 . 5 (larger collis ion vel o city) and (b ,b’) P K 1 = 0 . 8 (smaller collisi on velocity) in the FK model. The left pan els sho w the momenta of the kinks after collisio n ˜ P K j as t he fun c- tions of the initial p osition of middle kink, ( x 0 ) 2 . Collisions are strongly inelastic for ( x 0 ) 2 close to the origin. The right panels sho w th e examples of collisions on the ( x, t ) plane for (a’) ( x 0 ) 2 = 0 . 05 and (b’) ( x 0 ) 2 = 0 . 01 by plotting the re- gions with the energy densit y greater than a certain val ue, so that the cores of th e sol itons are clearly seen. Collisi on with a larger vel o city in (a), (a’) results only in quantitativ e change of kink parameters while collision with a smaller ve- locity in (b), (b ’) ma y result in fusion of a kink -antikink pair in a breather. collisions is studied a s a function of the initial sepa ra- tion b etw een the kink and the breather co nt rolle d by the initial kink p o sition ( x 0 ) K . In Fig. 3 we plot the momenta of kinks a fter collisio n, ˜ P K j (including the kinks constituting the breather, as- suming a s ear lie r that they share the brea ther’s momen- tum equally), as a function of ( x 0 ) K for (a) P B = 2 . 5 (larger collisio n velocity) and (b) P B = 1 . 6 (smaller co l- lision veloc it y). One can see that str ong REE is po ssible in the K B co llis ions. Note that in Fig. 3 only a small part of o ne p erio d of the output functions is shown for the r e gion with s trong REE. In (a) there is a ra nge of ( x 0 ) K where the breather o bta ins enough energy to split int o a k ink -antikink pa ir [ex a mple is shown in (a’)]. In (b), in addition to this p os sibility , there app ears a r e gion where the breather is reflected from the kink [exa mple is shown in (b’)]. F or larg e collis io n v elo cities [somewha t lar g er than in Fig. 3 (a), (a’)] the kink passes throug h the breather with no qualitative change in the collis io n outco me; there is o nly some energy and momen tum exc hange b etw een - 4 0 - 2 0 0 2 0 4 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 9 4 0 4 1 - 3 - 2 - 1 0 1 2 _ K K K B K ( a ') t x _ B B K K K ~ ( a ) P K j ( x 0 ) K - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 K B K B ( b ') t x 2 5 2 6 2 7 2 8 2 9 3 0 3 1 - 2 - 1 0 1 2 K K B K _ K B B ~ ( b ) P K j ( x 0 ) K FIG. 3: Numerical results for the kink-b reather collisio ns in FK mo del for (a), (a’) P B = 2 . 5 (larger collisi on vel o city) and (b,b’) P B = 1 . 6 (smaller collision velocity) and ω = 0 . 05 in b oth cases. The left panels show th e momenta of the kinks after th e collision ˜ P K j as a function of the initial p osition of the kink, ( x 0 ) K . The righ t panels sho w examples of collisio ns on t h e ( x, t ) plane for (a’) ( x 0 ) K = 40 . 25 and (b ’) ( x 0 ) K = 28 . 1. them. How ever, for sufficiently small collision velocity the collision outcome as the function of the initial sep- aration b etw een the kink a nd breather is a fractal. An example is presented in Fig. 4 for ω = 0 . 3, P B = 0 (kink and breather ha ve zer o initial velo cities), wher e the so liton’s momenta after collisio n ˜ P K j are shown as the functions of ( x 0 ) K . The structure presented in Fig. 4 can be describ ed as a chain of self-similar patterns. At each scale smoo th regions are separ ated by the apparently chaotic regions of tw o symmetry t yp es, one shown in (a) and (d) and another one in (b) and (c). (b) and (d) present blowups of the regions indicated in panel (a ); (c) is a blowup of the region indica ted in (b). Two examples of the kink- breather dynamics are given in Fig. 5 for (a) ( x 0 ) K = 0 . 2 4 and (b) ( x 0 ) K = 0 . 2 36 [indicated in Fig. 4 (b) b y the arrows A and B, resp ec- tively]. The three-particle solution has a certain lifetime L (in this example L ≈ 55) and then it splits in to a kink and a breather. Similar dynamics has b een re p o rted, e.g ., for the bre a ther-brea ther system in the weakly discr ete FK mo del [14], in the weakly p er tur bed NLSE [15], and recently for the g eneralized NLSE [24]. Thus, this type o f dynamics is rather g eneral. F or the tw o-solito n collisions in the weakly perturb ed NLSE w e hav e estimated nu- merically the proba bility p to o bserve the three-par ticle 5 0 . 2 4 3 0 . 2 4 4 0 . 2 4 5 0 . 2 4 6 0 . 2 4 7 0 . 2 4 8 - 0 . 6 - 0 . 3 0 . 0 0 . 3 0 . 6 0 . 2 1 0 . 2 2 0 . 2 3 0 . 2 4 0 . 2 5 - 0 . 6 - 0 . 3 0 . 0 0 . 3 0 . 6 0 . 2 8 5 0 . 2 8 6 0 . 2 8 7 0 . 2 8 8 0 . 2 8 9 0 . 2 9 0 - 0 . 6 - 0 . 3 0 . 0 0 . 3 0 . 6 j ~ ( c) P K j B A i n ( c ) ~ ( b ) P K - 0 . 6 - 0 . 3 0 . 0 0 . 3 0 . 6 - 0 . 6 - 0 . 3 0 . 0 0 . 3 0 . 6 j i n ( d) i n ( b) ~ ( a ) P K j ~ ( d ) P K ( x 0 ) K FIG. 4: F ractal kink-breather scattering observed for ω = 0 . 3, P B = 0 (kink and b reather h a ve zero initial velocities). The kinks’ momen ta after the collision ˜ P K j are sh own as the functions of th e initial p osition of the kink , ( x 0 ) K , at different scales. At eac h scale smo oth regi ons are separated by the apparently chao tic regions of tw o symmetry types, one show n in (a) and (d) and another one in (b ) an d (c). (b) and (d) present blo wups of the regions ind icated in p anel (a); (c) is a blo wup of th e region indicated in (b). - 1 0 - 5 0 5 1 0 0 5 0 1 0 0 K B ( a ) t x - 1 0 - 5 0 5 1 0 0 5 0 1 0 0 K B ( b ) t x FIG. 5: Examples of th e kink- breather dynamics for (a) ( x 0 ) K = 0 . 24 and (b) ( x 0 ) K = 0 . 236 [indicated in Fig. 4 (b) by th e arro ws A and B, resp ectively]. system with the lifetime L and found that p ∼ L − 3 [15]. Her e we carry out a similar estimation for the kink- breather solution in the FK mo del and the result is shown in Fig. 6. The numerical da ta can b e fitted as p ∼ L − 3 . 5 . There is evidence that for sufficient ly small frequency of the breather the kink-br e ather system in the FK mo del with a sma ll h 2 never splits [13]. All the imp orta nt features of the K B fra ctal scatter ing 1 0 0 0 1 0 0 0 0 1 E - 4 1 E - 3 0 . 0 1 0 . 1 1 p L FIG. 6: Probabilit y p to observ e the kink -breather system with th e lifetime L (in Fig. 5 w e hav e L ≈ 55). Numerical data is sho wn by dots and only the cases with L > 1000 w ere taken into account. Dashed line is th e guide for an eye and it has slope -3.5. including the existence o f the t wo qualitatively differen t sto chastic reg ions in the frac ta l structure will b e clarified in Sec. II I with the help of the three- particle mo del. II I. THREE-P AR T ICLE MODEL A. Description of the mo del A ttempting to explain the effects observed in the three- soliton collisions in weakly p erturb ed SGE rep orted in Sec. I I, w e consider the solitary w av es as effectiv e pa r- ticles, and study the dynamics of three such pa rticles in one-dimensional space. T he par ticles hav e mass m = 8, which is the rest mass of SGE kink, and they carry top o - logical c harg es q j = ± 1. Particles with q j = 1 ( q j = − 1) will b e called kink s (res pec tively , antikinks) by analo gy with the SGE solitons. W e assume that particles i and j , having co ordinates x i and x j , in terac t via the po tent ial U ij ( r ij ) = 16 + q i q j 16 cosh( r ij ) , r ij = x j − x i , (6) which q ualitatively a pproximates the interaction o f t wo SGE kinks. The potential of E q. (6) is attractive for q i 6 = q j and repulsive fo r q i = q j . The binding energ y of the kink-antikink pair is equa l to 16, which is the ener gy of tw o standing SGE kinks. Note that for the kink and antikink at an y finite distance r ij the p otential ener gy U ij ( r ij ) is less than 1 6. If the kinetic ener gy of relative motion of the particles is les s than 16 − U ij ( r ij ), then the particles cannot escap e the mutual attraction and they form a n o scillatory bo und state, i.e., a brea ther . 6 The Hamiltonian of the thr ee-particle system is H = m 2 3 X j =1 v 2 j + U 12 ( r 12 ) + U 13 ( r 13 ) + U 23 ( r 23 ) , (7) where v j = dx j /dt , and ther e is one mor e in tegral o f motion, na mely the co nserv a tion o f momentum. Without loss of generality , we as sume that the tota l moment um in the system is equa l to zero , i.e., m ( v 1 + v 2 + v 3 ) = 0. Int ro ducing new v a r iables x 2 − x 1 → √ 3 α + β , x 3 − x 1 → 2 β , t → √ 2 mt , (8) the Hamiltonia n of Eq. (7) can b e presented in the form H = 1 2  ˙ α 2 + ˙ β 2  + U 12 ( √ 3 α + β ) + U 13 (2 β ) + U 23 ( √ 3 α − β ) , (9) which is the Hamiltonian of a unit-mass pa rticle moving in the t wo-dimensional scatter ing p otential. Now we solve numerically thre e equations of motion which can b e derived from the Hamiltonia n Eq. (7) and, inv er ting Eq. (8), prese nt the three -particle dyna mics by the tra jectory o f the par ticle in the ( α, β )-plane. B. Separatrix three-soliton solutions to SGE Several separatr ix three-s o liton solutions to the ex actly int egr a ble SGE Eq. (3) hav e b een r ep orted in [1 2]. Here we repro duce t wo solutio ns impo rtant for our study . The separatr ix K K K solution is u K K K ( x, t ) = 4 ar ctan(exp x ) + 4 arctan R S , R = δ (sinh F − co s h G sinh x ) , S = δ (co sh G + sinh F s inh x ) − cosh F co sh x, F = − δ x, G = δ V t, δ − 1 = p 1 − V 2 . (10) In this highly symmetr ic solution the anti-kink is at rest and it is lo c a ted at the point of collision of t wo kink s moving with the velocities V and − V . The kink-breather separatr ix solution is u K B ( x, t ) = 4 ar ctan(exp x ) + 4 arctan X Y , X = η (sinh D − co s C sinh x ) , Y = η ( cos C + sinh D sinh x ) − cosh D cosh x, C = − ω t, D = η x, η = p 1 − ω 2 , (11) and it ha s only o ne par ameter ω becaus e it is a particular form of the K B solution wher e the kink and the br eather hav e zero velocities and zero distance b etw een them. In Fig. 7 we plot (a) the SGE solution Eq. (10) for V = 0 . 2, (b) the thr e e-particle dynamics in the ( x, t ) space for q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = − ( v 0 ) 3 , ( v 0 ) 2 = 0, ( x 0 ) 1 = - 2 0 0 2 0 - 1 0 0 - 5 0 0 5 0 1 0 0 t x ( a ) - 2 0 0 2 0 0 5 0 1 0 0 1 5 0 2 0 0 t x ( b) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 4 5 6 4 3 2 4 0 4 5 6 0 5 0 ( c ) FIG. 7: (Colo r online) (a) The SGE solution of Eq. (10) for V = 0 . 2; (b) the three-particle dynamics with q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = − ( v 0 ) 3 , ( v 0 ) 2 = 0, ( x 0 ) 1 = − ( x 0 ) 3 = − 25, and ( x 0 ) 2 = 0 in th e ( x , t ) space; (c) the red line shows the cor- respond ing tra jectory of the particle in the scattering p oten- tial in the ( α, β )-plane (isopotential lines are sho wn in blac k). The p article in (c) moves along th e p otential ridge and th is motion is unstable. The picture in (c) gives a v isual image of the separatrix K K K solution Eq. (10). - 8 - 4 0 4 8 0 2 0 4 0 6 0 8 0 1 0 0 t x ( a ) - 6 - 3 0 3 6 0 2 0 4 0 6 0 8 0 1 0 0 x ( b) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 6 0 4 0 4 0 3 2 4 5 4 5 6 4 5 0 ( c ) FIG. 8: ( Color online) (a) The SGE solution Eq. (11) for ω = 0 . 2; (b ) the th ree-particle dynamics with q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = ( v 0 ) 3 = 0, ( x 0 ) 1 = − ( x 0 ) 3 = − 4, and ( x 0 ) 2 = 0 in the ( x, t ) space; (c) the red line shows the corre- sp on d ing tra jectory of the particle in the scattering p otential in the ( α, β )-plane (isop otential lines are show n in blac k). The particle in ( c) oscillates along the p otential ridge and this mo- tion is unstable. The p icture in (c) giv es a v isual image of the separatrix K B solution Eq . (11). − ( x 0 ) 3 = − 25, and ( x 0 ) 2 = 0, and in (c) the red line shows the corres po nding dynamics in the ( α, β )-plane. In (c) the isolines of the scattering po tential are also shown (black lines ). The scattering potential in this case is a sup e rp osition of a ridge along β = 0 and t wo tro ughs along the lines β = ± √ 3 α . Note that the intersection of 7 the ridg e and the tw o tr oughs forms in the vicinity of the origin the ridge along the line α = 0; the tra jector y of the particle shown by the r e d line g o es exactly on the top of this ridg e. Obviously , this type of motio n is unstable and, as we will see in the following, small v a r iation in the initial conditions ma y r esult in qualitatively different dynamics of the particle. The picture pr esented in Fig . 7 (c) gives a v is ual image of the s e paratrix K K K so lution Eq. (10). In Fig. 8 w e plot (a) the SGE solution Eq. (11) for ω = 0 . 2, (b) the three-particle dynamics in the ( x, t ) space for q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = ( v 0 ) 3 = 0, ( x 0 ) 1 = − ( x 0 ) 3 = − 4, a nd ( x 0 ) 2 = 0, and in (c) the red line shows the co rresp onding tra jectory in the ( α, β )- plane. The particle in (c) oscilla tes along the p o tent ial ridge a nd, similar ly to the previo us example, this mo tio n is unstable. The picture presented in Fig. 8 (c) g ives a visual image of the separ atrix K B solution of Eq. (11). When the r e d line passes the or igin of the ( α, β )-plane, from Eq . (8) one ha s x 1 = x 2 = x 3 , i.e., all three parti- cles meet a t one p oint. In the SGE this cor resp onds to simult aneo us collisio n of all three kinks. Lo oking at Fig. 8(c) one ca n exp ect the p ossibility of oscillation of the particle along the ridge o f the scattering po tent ial with β = 0. This is indeed possible for the three-particle system but, from β = 0 one finds from Eq. 8 that x 1 ( t ) ≡ x 3 ( t ), which cannot b e rea lized in the FK mo del be c a use the k ink s have finite width. C. Three-kink col lisions In Fig. 9 we compare the K K K and K K K sym- metric collisio ns in the three-particle mo del for ( x 0 ) 1 = − ( x 0 ) 3 = − 25, ( x 0 ) 2 = 0 and ( v 0 ) 1 = − ( v 0 ) 3 = 0 . 6, ( v 0 ) 2 = 0. The top panels show the three- particle dy- namics in the ( x, t ) space. F o r each c ase, the bo ttom panels cor resp ondingly show the equip otential lines of the scattering potential of Eq. (9) (black) and the tra jectory of the particle (red line) in the ( α, β )-plane. In (a’) the scattering po ten tial for q 1 = q 2 = q 3 = 1 is a sup erp osi- tion of thre e troughs while in (b’) for q 1 = q 2 = − q 3 = 1 it is a s upe r p osition of a ridge and t wo troughs. The po- ten tial in Fig. 9 (b’) can be obtained fr o m that sho wn in Fig. 7 (c) and Fig. 8 (c) through a r o tation by − π / 3. In Fig. 9 (a), the like pa rticles rep el each o ther a nd, in (a’), the particle hits the potential bar rier a nd go es back. In (b), one can see that par ticles collide in tw o successive tw o -soliton collisions. In this case, the par ticle in (b’) pas s es the tw o p otential troughs one after ano ther and then mo ves a wa y from the origin in the direction symmetrically equiv alent to the direction it ca me from. Since the red line in (a’) and (b’) never go es through the origin, the three particles never meet at o ne p oint. In Fig. 1 0 we give t wo exa mples of near-separatr ix K K K symmetric collisions in the thr ee-particle mo del for ( x 0 ) 1 = − ( x 0 ) 3 = − 25 and ( v 0 ) 1 = − ( v 0 ) 3 = 0 . 6, ( v 0 ) 2 = 0. Recall that the separatr ix solution shown in - 1 0 0 1 0 2 0 4 0 6 0 - 1 0 0 1 0 2 0 4 0 6 0 t x ( a ) K K K t x ( b) _ K K K - 5 0 5 - 5 0 5 9 6 8 0 7 0 6 0 5 0 ( a ' ) - 5 0 5 - 5 0 5 3 2 4 0 4 5 5 0 6 0 ( b' ) FIG. 9: ( Color online) Comparison of (a,a’) K K K and (b,b’) K K K symmetric collisions. The t op panels show the three- particle dynamics in th e ( x , t ) sp ace. The tra jectories of kink s are sh own by th ic ker lines th an those of anti kink s. The b ot- tom pan els correspondingly show the eq u ip otentia l lines of the scattering p otential Eq. (9) (black) and th e tra jectory of particle (red line) in the ( α, β )-plane. The parameters are ( x 0 ) 1 = − ( x 0 ) 3 = − 25, ( x 0 ) 2 = 0 and ( v 0 ) 1 = − ( v 0 ) 3 = 0 . 6, ( v 0 ) 2 = 0. The c harges of the p articles are (a,a’) q 1 = q 2 = q 3 = 1 and (b ,b’) q 1 = q 2 = − q 3 = 1. - 1 0 0 1 0 2 0 4 0 6 0 - 1 0 0 1 0 2 0 4 0 6 0 t x ( a ) t x ( b) - 5 0 5 - 5 0 5 ( a ' ) - 5 0 5 - 5 0 5 ( b' ) FIG. 10: (Color online) Sensitivity of the result of near sep- aratrix collision to a small deviation from ( x 0 ) 2 = 0 demon- strated by setting (a,a’) ( x 0 ) 2 = 1 . 2 and (b,b’) ( x 0 ) 2 = 0 . 2. In ( a,a’) only a quantita tive change in the system can b e seen up on collision [compare with the actual t hree-kink collision in FK mo del sh o wn in Fig. 2(a’)]. In (b,b’), kink and antikink merge in a breather [compare with Fig. 2(b ’)]. Other pa- rameters: ( x 0 ) 1 = − ( x 0 ) 3 = − 25 and ( v 0 ) 1 = − ( v 0 ) 3 = 0 . 6, ( v 0 ) 2 = 0. 8 Fig. 7 cor resp onds to ( x 0 ) 2 = 0 but Fig. 10 co rresp onds to (a,a’) ( x 0 ) 2 = 1 . 2 a nd (b,b’) ( x 0 ) 2 = 0 . 2. In Fig. 10 (a,a’), the devia tion from the separatr ix is rather large and only quantitativ e changes in the par ticle parameters can be seen. This should b e compared with the actual three-kink collision in FK mo del shown in Fig. 2(a’). In (b,b’), kink and an tikink merge in a breather [compare with Fig. 2(b’)]. T aking in to a ccount the time reversibil- it y in the Hamiltonia n sy stems, this picture can b e also regar ded as a n illus tr ation of the bre a kup of a breather colliding with a kink. The three - particle mo del ex pla ins why the REE effect is more pronounced for the solitons colliding with a small relative velocity . F or the particle moving in the ( α, β )- plane along the s eparatrix [red line in Fig. 7 (c)], any p e r - turbation results in expone ntial in time deviatio n from the po tential r idge. High-speed collision results in fas ter passing of the s c a ttering p otential a nd the tra jecto r y of the particle ca nnot be considera bly changed. The situa- tion is opp osite for the slow particle, which co rresp onds to the co llis ion o f solitons with a sma ll re la tive velo city . D. Kink-breather collisions Here we select the parameter s of the three particle s so as to sim ula te the collisions betw een a kink and a breather. In pa rticular, w e se t the charges of particles as q 1 = − q 2 = q 3 = 1, their initial velocities as ( v 0 ) 1 = ( v 0 ) 2 = 0 . 3, ( v 0 ) 3 = − 0 . 6; the init ial p os itio ns of the particles constituting the ”breather” are ( x 0 ) 1 = − 16, ( x 0 ) 2 = − 13 . 5, and the third particle initial pos ition was v aried. In Fig. 11 the r esults a r e shown for (a) ( x 0 ) 3 = 30 . 51, (b) ( x 0 ) 3 = 23 . 398 , and (c) ( x 0 ) 3 = 23 . 391 . The top pa ne ls show the three-par ticle dynamics in the ( x, t ) space, while the b ottom panels show the corresp onding tra jectory of the particle in the ( α, β )-plane (red line). Collisions in (a) a nd (b) a r e elas tic but the difference is that while in (a’) the particle do es not move alo ng the separatr ix line α = 0, in (b’) it do e s, and a v ery small change in the initial conditions is sufficient to hav e a qual- itatively different r esult of the collision, as presented in (c),(c’), where the breather reflects from the kink [com- pare (c) with a ctual kink- breather co llis ion in the FK mo del shown in Fig. 3(b’)]. E. F ractal kink-breather scattering T o repro duce the kink-breather fractal scattering de- scrib ed in Sec. I I B for the FK mo del we set the fol- lowing para meters for the pa rticles in the three-particle mo del: q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = ( v 0 ) 3 = 0, ( x 0 ) 1 = − ( x 3 ) 2 = − 5, and v aria ble ( x 0 ) 2 . In Fig. 12 w e pres ent the velo cities o f particles after collision ˜ v j as the function of ( x 0 ) 2 . In (b) a blowup of the self-similar reg ion indicated in (a) is pres e nted. Com- parison of the panels (a ) and (b) with the corr esp onding - 2 0 - 1 0 0 1 0 2 0 0 2 0 4 0 6 0 8 0 - 2 0 - 1 0 0 1 0 2 0 0 2 0 4 0 6 0 8 0 t x ( a ) x ( b) - 2 0 - 1 0 0 1 0 2 0 0 2 0 4 0 6 0 8 0 x ( c ) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 4 0 4 0 3 2 4 5 4 5 6 0 5 0 ( a ' ) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 ( b' ) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 ( c ' ) FIG. 11: (Color online) Three-particle mo del simulating t he kink-breather collisions. The top panels sho w the three- particle dy namics in the ( x, t ) space, while th e b ottom pan- els show the correspond ing tra jectory of the particle in the ( α, β )-p lane (red line). On ly the initial p osition of th e third particle is v aried: (a) ( x 0 ) 3 = 30 . 51, (b) ( x 0 ) 3 = 23 . 398, and (c) ( x 0 ) 3 = 23 . 391. Collisions in (a),(a’) and (b),(b’) are elas- tic but in the latter case it is close t o the separatrix (see Fig. 8) resulting in a great sensitivity to vari ations in initial condi- tions, as d emonstrated in (c),(c’). The rest of the parameters are c hosen as q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = 0 . 3, ( v 0 ) 3 = − 0 . 6, ( x 0 ) 1 = − 16, ( x 0 ) 2 = − 13 . 5. 0 . 4 1 0 . 4 2 0 . 4 3 0 . 4 4 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 B A ( x 0 ) 2 ~ ( b ) v j - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 i n ( b) ~ ( a ) v j FIG. 12: F ractal three-particle scattering. Panels (a) and (b) should b e compared with the corresponding p anels of Fig. 4. P arameters: q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = ( v 0 ) 3 = 0, ( x 0 ) 1 = − ( x 3 ) 2 = − 5, and v ariable ( x 0 ) 2 . panels of Fig. 4 reveals the qua litative similarity in the K B co llis ion o utcome in the FK mo del and in the three- particle mo del. W e no te that while we exp ect this par ti- cle mo de l to b ear the essential qualitative characteristics of the three-par ticle c ollisions, their details depend s en- sitively on the precise initial conditions; for this r eason, we exp ect Fig. 12 to matc h qualitativel y the results of Fig. 4. One of the impo rtant elements of the usefulness of the 9 - 5 0 5 0 1 0 0 2 0 0 - 5 0 5 0 1 0 0 2 0 0 t x ( a ) t x ( c ) - 5 0 5 0 1 0 0 2 0 0 t x ( b) - 5 0 5 - 5 0 5 ( d ) FIG. 13: (Color online) (a) The ex act K B solution to the SGE, (b) the same solution in th e wea kly discrete FK mo del, (c) the th ree-particle dynamics in the ( x, t ) space and (d) the correspondin g dynamics in the ( α, β ) space. In (a), ( b ), kink and b reath er hav e zero in itial velocities, breather frequency is ω = 0 . 3, and separation betw een the kink and the breather is equal to 1 . 2. In (c), (d) , q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = ( v 0 ) 3 = 0, ( x 0 ) 1 = − 6 . 18, ( x 0 ) 2 = 0, ( x 0 ) 3 = 3 . 3. three-particle mo del is that it gives the p ossibility to a n- alyze the K B fractal scattering fro m a different p oint of view, na mely , by lo oking at the corresp o nding dynamics of the pa rticle in the scattering p otential in the ( α, β ) space. In Fig. 13 we show (a) the exact K B solutio n to the SGE, (b) the same s olution in the weakly discrete FK mo del, (c) the three-particle dynamics in the ( x, t ) space and (d) the corres po nding dynamics in the ( α, β ) space. In (a), (b), kink and breather ha ve zero initial velocities, brea ther frequency is ω = 0 . 3, and se pa ration betw een the kink and the breather is equal to 1 . 2 (we refer to the for m of the K B so lution giv en in [1 2]). In (c), (d), q 1 = − q 2 = q 3 = 1, ( v 0 ) 1 = ( v 0 ) 2 = ( v 0 ) 3 = 0, ( x 0 ) 1 = − 6 . 18, ( x 0 ) 2 = 0, ( x 0 ) 3 = 3 . 3. One can see from Fig. 13 that the dista nce b etw een the kink and the breather does not change in time in the integrable sys- tem [shown in (a )] but in the nonin tegra ble o nes the dis- tance b etw een them gr adually dec reases and they even- tually collide [see (b) and (c)]. The separa ted k ink and breather having zer o velo cities are presented in the ( α, β ) space by the particle os cillating alo ng the line nor ma l to the trough with orientation β = √ 3 α (kink is to the left of the breather) o r β = − √ 3 α (kink is to the rig ht of the breather). Ho wev er , the troughs hav e a slop e tow ard the origin of the ( α, β ) plane and the oscillating particle gradually approaches the orig in, i.e., the collision p oint of three particles. After the particle has appro ached the origin [this s ituation is shown in Fig. 13 (d)], tw o qualita- tively different scenario s giving different fractal patterns are po ssible. The first sc enario is shown in Fig. 14. Here the pa rti- cle after making a few oscillations normally to the trough β = √ 3 α ca n cross the separatrix line α = 0 and make a few oscillations no rmally to the trough β = − √ 3 α and then again cross the separ a trix line changing the trough. While the particle p erforms such crossings of the sepa- ratrix it r emains close to the origin of the ( α, β ) plane - 1 0 0 1 0 0 5 0 1 0 0 1 5 0 2 0 0 - 1 0 0 1 0 0 5 0 1 0 0 1 5 0 2 0 0 t x ( a ) t x ( b) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 ( a ' ) - 1 0 - 5 0 5 1 0 - 1 0 - 5 0 5 1 0 ( b' ) FIG. 14: (Colo r online) Illustration of one of the tw o p ossible scenarios of fractal kink -breather scattering. (a),(b) show the dynamics of three particles in t h e ( x, t ) plane while (a’),(b’) the corresp onding dy namics in the ( α, β ) plane. The parti- cle in (a’),(b’) after making a few oscillatio ns normally to the trough β = √ 3 α can cross the separatrix line α = 0 and mak e a few oscillations normally to the t rou gh β = − √ 3 α and th en again cross the separatrix line changing the trough. While the particle p erforms such crossings of the separatrix it re- mains close to the origin and thus, all three p articles are close to each oth er. This defines the lifetime of the multi-soliton b ound state discussed in [14, 15, 24]. Eventually , the p arti- cle will mov e a wa y from th e origin along one of t h e troughs. P arameters: ( v 0 ) i = 0, i = 1 , 2 , 3, ( x 0 ) 1 = − ( x 0 ) 3 = − 5, (a) ( x 0 ) 2 = 0 . 433 and (b) ( x 0 ) 2 = 0 . 4275. and th us, all three particles ar e clo se to ea ch other. This defines the lifetime of the mult i-so liton b ound state dis- cussed in [14, 15, 24]. The probability to hav e a three- particle bound state with a long lifetime is small (see Fig . 6) meaning that even tually the par ticle will mo ve awa y from the or igin along one of the troug hs remaining in the half-plane β > 0 (co mpare Fig. 1 4 with Fig. 5 wher e the K B dyna mics in the FK mo del is presented). The se c ond sc enario is more obvious beca use it is di- rectly related to the sepa r atrix solution Eq. (1 1) pre- sented in Fig. 8. After making a few oscillatio ns nor- mally to the tr ough β = √ 3 α a s shown in Fig. 13 (d), the par ticle c an b e sent b y the scattering potential a l- most exactly along the se pa ratrix line α = 0. Then the particle will mak e several o scillations along the ridge of the p otential, as shown in Fig . 8 (c), b efore the inher ent instability of this tra jectory “e jects” the par ticle a wa y from the or igin in one of the four directio ns a long the troughs β = ± √ 3 α . This con trast to the first sc e nario where the particle can b e sca ttered b y the p otential in the t wo of the four directions, namely , in the ones with β > 0. The fir s t scena rio is asso ciated with the self-similar re - gions connecting the t wo “ butterflies” [see Fig. 4 (b) 10 and (c)] while the second one is asso ciated with the self- similar regio ns connecting the “wings” of a ”butterfly ” [see Fig. 4 (d)]. How ever, the whole fractal patter n is the res ult of the combination of b oth mechanisms. E a ch scenario is r elated to a perio dic orbit of the particle in the scattering potential [28]. In the second scena rio the per i- o dic orbit is the separa trix kink-breather solution shown in Fig. 8, while in the fir st scenario there exists an in- finite se t of perio dic orbits. O ne particular o rbit can b e describ ed as follows: the particle in the scattering p oten- tial makes N os cillations a lmost nor mally to the troug h β = √ 3 α and then jumps to the tr ough β = − √ 3 α where it also makes N os cillations a nd then returns to the trough β = √ 3 α completing one per io d o f the per io dic o r- bit [Fig. 14 (a’) and (b’) give exa mples when the particle makes such jumps betw een the troughs β = ± √ 3 α but in these cases the tra jectories are ap er io dic]. If the particle follows a p erio dic orbit exactly , the three-pa r ticle sys- tem never exp er iences a breakup; how ever, the even tual separatio n o f the str uctures is a result of the dynamica l instability of such per io dic orbits. It is well-kno wn that the probability p of the time delay T for the par ticle interacting with the scattering p oten- tial without the perio dic orbits decreases exponentially with T while in the presenc e of the p erio dic or bits it de- creases algebr a ically [28]. The s cattering p otential in our case do es hav e the p erio dic orbits and the pro ba bilit y p to obser ve a b ound state with the lifetime L (analogous to the time delay T ) decreases algebraic a lly , p ∼ L − α . This was found in [15] for the tw o-so liton c o llisions in the weakly p erturb ed NLSE, and in the present study this w as also confirmed for the kink-breather sy stem in the FK mo del, a s pres e nted in Fig. 6. IV. DISCUSSION AND CONCLUSIONS Through direct n umerical simulations, we have pre- sented so me of the striking effects g enerated by even a weak brea king of in tegra bilit y (via discr etization) in the sine-Gordon mo del. W e hav e indicated that alternative mechanisms such a s the excita tio n of int erna l modes a nd the emission of phonon ra dia tion are to o weak to ex- plain the phenomena observed in numerical simulations, and we have therefor e attributed them to the ra diation- less energ y exchange betw een the so litons. Indeed, these effects hav e b een sys tema tically explained in a qualita- tive fashion in the framework of the three-particle model suggested in Sec. I I I A, lending direct suppo rt to the conclusion that all the nontrivial effects are due to the r adiationless ener gy exchange b etw een co lliding solitons [2, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The follo wing is known ab out the REE effects: (i) Manifestatio ns of the REE effect grow prop ortio na lly to the perturba tio n pa- rameter ǫ while r adiation a nd excita tio n of soliton inter- nal mo des grow as ǫ 2 . (ii) In the sine-Gor don mo del the REE effect can happ en only if a t least three solitons col- lide s im ultaneously . E nergy exchange in the tw o-soliton collision is suppress ed by the tw o co nserv atio n laws that remain exa ctly or approximately preserved in the weakly per turb ed system. (iii) The REE effect is rela ted to the existence of the sepa ratrix multi-soliton solutions to the in teg rable equations. Near- separatr ix motion is ex- tremely sensitive to the per turbations [2 9]. (iv) T he REE effect c an b e resp ons ible for the fractal soliton sc attering. The REE effect is g eneric and some o f the ab ove con- clusions ca n be also extended to other nearly integrable mo dels [2]. F or instance, the REE effect is observed in the w eakly p erturb ed NLSE alr eady in t wo-soliton colli- sions b ecause here each soliton has tw o parameter s and the total num b e r of parameters des cribing the tw o-so liton solution (four) exceeds the n umber of the remaining co n- serv atio n laws. O n the other ha nd, the REE is not p o s - sible in the weakly per tur be d KdV equation or w eakly per turb ed T o da lattice [30] be c ause in these cases the soliton’s cores ne ver mer ge during collisio ns and thus, the m ulti-particle effects ar e a bsent. Interestingly , the fractal pattern of different nature (not related to REE) is p os sible in K dV sy s tems [3 1]. The three- particle mo del o ffer ed in the present study can b e reduced to the study of the dyna mics o f a par- ticle interacting with the tw o -dimensional scatter ing p o- ten tial. Such a reductio n gives a clea r in terpretatio n of the ab ovemen tioned fea tur es o f the REE effect obser ved in the three-s oliton collis ions. P articular ly , the fo llowing features hav e bee n identified: 1. The three-par ticle mo del gives a visual ima ge of the separatr ix three-kink and k ink-breather solutions to the integrable SGE, see Fig. 7 and Fig. 8. The separatr ix solution cor resp onds to motion of the particle along a ridge of the scatter ing p otential. 2. K K K collisions and K K K co llis ions ar e al- wa y s pra ctically elastic while K K K collisions are strongly inelastic in the vicinity o f ( x 0 ) 2 . Only in the latter case the p oint mov es along the ridge of the scattering p otential, which is the motio n along a separatrix , s e e Fig. 7. Also only in K K K colli- sions the p oint passes through the orig in o f the scat- tering p otential which means that the three kinks collide at one p oint sim ultaneously . 3. The three-par ticle mo del ex pla ins why the REE effect is more pronounced for the solitons collid- ing with a small relative velocity . F o r the particle moving in the ( α, β )-plane along the separ atrix [red line in Fig. 7 (c)], any p erturbation res ults in e xp o - nent ial in time deviation from the p otential ridg e. High-sp eed collision res ults in faster pa ssing of the scattering p otential and the tra jecto r y of the par ti- cle cannot b e co ns iderably changed. The situation is opp o site fo r the slow pa r ticle, which corresp onds to the collision o f solitons with a small relative ve- lo city . 4. The fractal solito n sca ttering is e x plained b y the presence o f the p erio dic orbits of the particle in the 11 scattering p otential. Periodic orbits of t wo types were found, each of them is res po nsible for a par- ticular scenario of the par ticle dyna mics, and each scenario yields a self-similar pattern for the co lli- sion outcome as a function of a parameter, such as the locatio n of the ce ntral effectiv e particle (Sec. II I E). 5. P erio dic orbits ar e a lso resp onsible fo r the algebra ic law p ∼ L − α , wher e p is the proba bility to observe the three-soliton bound state with the lifetime L (see Fig . 6 and Sec. II I E). In the weakly p erturb ed sys tems the REE is the dom- inant effect. How ever, if the per turbation is not sma ll, the energy exchange effect is mixed with ra diation and po ssibly with excitation of in ternal mo des . W e thus be- lieve that the net effect of inela sticity o f solito n collis ions can b e deco mpo sed into thre e ma jor parts: the radia tion- less energy exchange, excitation of the soliton’s internal mo des, and emission of ra diation. This highlights the need for a sys tematic study as a function of increasing deviations from the in tegra ble r e gime of the r elative r o le of these three complement ary mec hanisms. Suc h a study would be o f par ticular in terest for future in vestigations. Ackno wl edgments The authors thank A.A. Sukhoruko v for useful discus- sions. SVD acknowledges a financial supp ort of the Rus- sian F oundation for Basic Research, grant 07- 0 8-12 1 52. PGK ac knowledges a suppo rt from NSF-DMS-02 0458 5, NSF-DMS-05056 63, NSF-CAREER, and the Alexander von Hum b oldt F oundatio n. [1] Y u .S. Kivshar and B.A. Malomed, Rev. Mo d. Phys. 61 , 763 (1989). [2] P .G. Kevrekid is and S.V. Dmitriev, Soliton Collisions, in Encyclop e di a of Nonline ar Scienc e , Edited by A. Scott (Routledge, New Y ork, 2005), PP . 148-150. [3] M.J. Ablo witz and H. Segur, Solitons and the Inverse Sc attering T r ansform , S IAM (Philadelphia, 1981). [4] Y u .S. Kivshar, D.E. P elinovsky , T. Cretegn y , and M. P eyrard, Phys. Rev. Lett. 80 , 5032 (1998). [5] P . Anninos, S. Oliveira , and R.A. Matzner, Phys. Rev. D 44 , 1147 (1991). [6] D. K. Ca mpb ell, J. F. Sc honfeld, and C. A. Wingate, Physica D 9 , 1 (1983); M. Peyrard and D. K. Campb ell, Physica D 9 , 33, (1983); D.K. Campbell an d M. Peyrard, Physica D 18 , 47 (1986); i bid. 19 , 165 (1986). [7] T. I. Belo v a and A. E. Kudrya vtsev, Ph ys. Usp. 40 , 359 (1997). [8] R.H. Go o dman and R. Hab erman, S IAM J. Appl. D yn. Sys. 4 , 1195 (2005); R.H. Go o dman and R. H ab erman, Phys. Rev. Lett. 98 , 104103 (2007). [9] S.V. Dmitriev, L.V. Nauman, A.A. Ovcharo v, and M.D. Starostenko v , Ru ssian Ph ysics Journal 39 , 164 (1996). [10] H. F rauenk ron, Y u .S . Kivshar, and B.A. Malomed, Phys. Rev. E 54 , R2244 (1996). [11] S.V. Dmitriev, L.V. Nauman, A.M. W usatow sk a-Sarnek, and M.D. Starostenko v , phys. stat. sol. (b ) 201 , 89 (1997). [12] A. E. Miroshnichenko , S. V. Dmitriev, A. A . V asiliev, and T. Shigenari, Nonlinearity 13 , 837 (2000). [13] S.V. Dmitriev, T. Miyauc hi, K. Ab e, and T. Shigenari, Phys. Rev. E 61 , 5880 ( 2000). [14] S.V. D mitriev, Y u.S. Kivshar, and T. Shigenari, Ph ys. Rev. E 64 , 056613 (2001). [15] S . V. Dmitriev and T. Sh igenari, Chaos 12 , 324 (2002). [16] S . V . Dmitriev, D. A. Semagin, A. A. S ukhoruko v, and T. Shigenari, Phys. Rev. E 66 , 046609 (2002). [17] S . V. Dmitriev, P .G. Kevrekidis, B.A. Malomed, and D.J. F rantzesk akis, Phys. Rev. E 68 , 056603 (2003). [18] J. A. Com bs and S . Yip, Phys. R ev. B 28 , 6873 (1983). [19] R . Boesc h, C. R. Willis, and M. El-Batanouny , Ph y s. Rev. B 40 , 2284 (1989). [20] R . Bo esch and M. P eyrard, Phys. Rev. B 43 , 8491 (1991). [21] Y . Ishimori and T. Munak ata, J. Phys. Soc. Jpn. 51 , 3367 (1982). [22] J. Y ang and Y. T an, Phys. Rev. Lett. 85 , 3624 (2000). [23] Y . T an and J. Y ang, Phys. Rev. E 64 , 056616 (2001). [24] Y . Zhu, R. Hab erman, and J. Y ang, Ph y s. Rev . Lett. 100 , 14390 1 ( 2008); Y. Zhu and J. Y ang, Phys. R ev. E 75 , 036605 (2007). [25] V .I. K arpman and V.V. Solov’ev, Physica D 3, 142 (1981). [26] I .E. Papac haralampous, P .G. Kevrekidis, B.A. Malomed, and D .J. F rantzesk akis, Phys. Rev. E 68 , 0466 04 (2003). [27] R . Hirota, J. Phys. S o c. Jpn. 33 , 1459 (1972). [28] E. Ott and T. T el, Chaos 3 , 417 (1993). [29] G. M. Zasla vsk y , R. Z. Sagdeev, D. A. Usiko v , and A. A. Cherniko v We ak Chaos and Quasi-R e gular Patter ns (Cam bridge Univ. Press, Cam bridge, 1991). [30] M. T o da, The ory of Nonli ne ar L attic es , Springer-V erlag (New Y ork, 1989). [31] E. Zamora-Sillero and A.V. Shap ov alo v, Phys. Rev. E 76 , 046612 (2007).