Collision of ϕ 4 kinks free of the Peierls-Nabarro barrier in the regime of strong discreteness
Alidad Askari, Aliakbar Moradi Marjaneh, Zhanna G. Rakhmatullina, Mahdy Ebrahimi-Loushab, Danial Saadatmand, Vakhid A. Gani, Panayotis G. Kevrekidis, Sergey V. Dmitriev
CCollision of φ kinks free of the Peierls–Nabarro barrierin the regime of strong discreteness Alidad Askari, ∗ Aliakbar Moradi Marjaneh, † Zhanna G. Rakhmatullina, ‡ Mahdy Ebrahimi-Loushab, § Danial Saadatmand, ¶ Vakhid A. Gani,
6, 7, ∗∗ Panayotis G. Kevrekidis,
8, 9, †† and Sergey V. Dmitriev
10, 11, ‡‡ Department of Physics, Faculty of Science,University of Hormozgan, P.O.Box 3995, Bandar Abbas, Iran Young Researchers and Elite Club, Quchan Branch,Islamic Azad University, Quchan, Iran Institute for Metals Superplasticity Problems,Russian Academy of Sciences, Ufa 450001, Russia Department of Physics, Faculty of Montazeri Technicaland Vocational University (TVU), Khorasan Razavi, IRAN Department of Physics, University of Sistan and Baluchestan, Zahedan, Iran Department of Mathematics, National Research Nuclear UniversityMEPhI (Moscow Engineering Physics Institute), Moscow 115409, Russia Theory Department, Institute for Theoretical and Experimental Physics ofNational Research Centre “Kurchatov Institute”, Moscow 117218, Russia Department of Mathematics and Statistics,University of Massachusetts, Amherst, Massachusetts 01003, USA Mathematical Institute, University of Oxford, OX26GG, UK Institute of Molecule and Crystal Physics,Ufa Federal Research Center of Russian Academy of Sciences, Ufa 450075, Russia Institute of Mathematics with Computing Centre,Ufa Federal Research Centre of Russian Academy of Sciences, Ufa 450000, Russia a r X i v : . [ n li n . PS ] J un bstract The two major effects observed in collisions of the continuum φ kinks are (i) the existenceof critical collision velocity above which the kinks always emerge from the collision and (ii) theexistence of the escape windows for multi-bounce collisions with the velocity below the criticalone, associated with the energy exchange between the kink’s internal and translational modes.The potential merger (for sufficiently low collision speeds) of the kink and antikink produces abion with oscillation frequency ω B , which constantly radiates energy, since its higher harmonicsare always within the phonon spectrum. Similar effects have been observed in the discrete φ kink-antikink collisions for relatively weak discreteness. Here we analyze kinks colliding with theirmirror image antikinks in the regime of strong discreteness considering an exceptional discretizationof the φ field equation where the static Peierls–Nabarro potential is precisely zero and the not-too-fast kinks can propagate practically radiating no energy. Several new effects are observedin this case, originating from the fact that the phonon band width is small for strongly discretelattices and for even higher discreteness an inversion of the phonon spectrum takes place with theshort waves becoming low-frequency waves. When the phonon band is narrow, not a bion but adiscrete breather with frequency ω DB and all higher harmonics outside the phonon band is formed.When the phonon spectrum is inverted, the kink and antikink become mutually repulsive solitarywaves with oscillatory tails, and their collision is possible only for velocities above a threshold valuesufficient to overcome their repulsion. PACS numbers: 11.10.Lm, 11.27.+d, 05.45.Yv, 03.50.-z ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] †† [email protected] ‡‡ [email protected] . INTRODUCTION Continuum and discrete Klein–Gordon type equations contribute to the understanding ofmany physical phenomena [1–4]. In discrete models the Lorentz invariance is lost and severalnew important effects are observed, such as for example, the appearance of the Peierls–Nabarro potential, the associated reduction of soliton mobility, the radiation produced bymoving solitary waves, etc. [2–6]. The case of strong discreteness is of particular interestand it is encountered in many applications, e.g., in the description of arrays of Josephsonjunctions [7], dissipative nonlinear discrete systems [8], dynamics of crowdions [9–13] anddislocations [14–16] in crystals, propagation of domain walls in magnetic materials [17],motion of spring-mass chains [18], in the discussion of electric charge transport in molecularchains [19]. The consideration of the strongly discrete (anti-continuum) limit is a well-knownapproach aiming towards the analytical treatment of discrete breathers [20, 21].One particularly central problem in the dynamics of solitary waves is the analysis oftheir collision outcomes, especially beyond the merely-phase-shifting elastic wave interac-tions within integrable models [22–39]. Continuum Klein–Gordon equations, apart from thefamous integrable sine-Gordon example [4], support exact solutions in the form of movingkinks which interact inelastically. Collisions between a kink and an antikink moving towardseach other with initial velocities ± v c have been extensively studied. It has been found thatif v c > v ∗ , where v ∗ is a critical collision velocity, then the kink and antikink separate afterthe first collision [22–24, 26–28]. On the other hand, collisions with v c < v ∗ produce a set ofescape windows having fractal structure [24, 26–28]. When the collision velocity is withinsuch a window, the kink and antikink move away from each other after multiple collisions.The kinks’ internal vibrational modes [40] have been extensively argued to be responsiblefor this effect: they store some energy of the kinks’ translational motion which can be re-turned in the subsequent collisions, leading the kink and antikink to overcome their mutualattraction. If the kink and antikink do not split after a few collisions, they lose a substan-tial amount of energy to small-amplitude radiative wavepackets (emitted from the collisionlocation) and, being unable to overcome the mutual attraction, create a bound oscillatorystate called bion, whose main frequency lies below the phonon band but higher harmonicswithin the band. Resonating with the phonons, the bion constantly radiates energy and itsamplitude gradually decreases. 3he effect of weak discreteness on the solitary wave collisions was studied in Refs. [26, 29]and radiationless energy exchange between colliding quasi-particles was described. On theother hand, the collision of solitary waves in strongly discrete Klein–Gordon systems hasnot been studied so far. The reason is the above mentioned immobility of kinks in thepresence of Peierls–Nabarro potential induced by discreteness. However this difficulty canbe overcome by considering exceptional discretizations [41] of the Klein–Gordon equationswhere the static Peierls–Nabarro potential is precisely zero [41–53]. In Ref. [54], kink-antikink collisions have been analyzed in various models free of the static Peierls–Nabarropotential for the case of weak discreteness (lattice spacing h ∼ . h because the critical collision velocity v ∗ ,above which kinks separate after the first collision, reduces for larger h .The absence of the static Peierls–Nabarro potential implies that the kink can move alongthe chain practically radiating no energy if its velocity is very small and its profile is notaffected by the dynamical effects. Some discrete Klein–Gordon models support kinks ornanopterons moving with a permanent profile [55–57], but such motion is observed only atso-called “transparent points”, i.e., at isolated velocities at which kinks can propagate ina discrete system as traveling waves. In particular, in Ref. [57] three discrete φ equationsfree of the static Peierls–Nabarro potential were considered. The authors have shown that,in addition to the vanishing velocity, the discrete φ equation proposed by Speight andWard [42, 43] supports a single isolated velocity, the model of Kevrekidis [44] supports threevelocities, and the model of Bender and Tovbis [52] supports none such velocities.In the present work, for the φ equation discretized according to the method proposedin Refs. [42, 43], we analyze kink-antikink collisions in the regime of strong discreteness( h ∼
1) in the absence of the static Peierls–Nabarro potential. We find that this setting isconducive to the emergence of multiple new features including the formation of persistentdiscrete breathers (rather than bions) in the strongly discrete regime. Another key findingis the potential fundamental modification of the nature of the interaction from an attractiveto a repulsive one upon the inversion of the phonon band for strong discreteness in suchmodels.Our presentation is structured as follows. In Section II we present the (exceptionaldiscretization) models and some of their principal static properties. Then in Section III weexamine the collisions of kinks and antikinks in these models. Finally, in Section IV we4ummarize our findings and present our conclusions, as well as some challenges for futurework.
II. THE φ FIELD MODEL AND ITS EXCEPTIONAL DISCRETIZATIONA. Continuum φ equation and its conventional discretization The Klein–Gordon field-theoretic model can be defined by the Hamiltonian H = (cid:90) + ∞−∞ (cid:20) φ t + 12 φ x + V ( φ ) (cid:21) dx, (1)where φ ( x, t ) is a real scalar function of spatial and temporal coordinates x and t , respec-tively. The subscripts x and t denote differentiation with respect to the correspondingcoordinate. The function V ( φ ) defines the on-site potential, which for the φ model reads: V ( φ ) = 12 (cid:0) − φ (cid:1) . (2)The Hamiltonian (1) with the potential (2) gives the following equation of motion: φ tt − φ xx − φ (cid:0) − φ (cid:1) = 0 , (3)which can be transformed to the form of the φ model considered in Ref. [42] by rescalingthe spatial and temporal coordinates by a factor of 1/2. This equation has an exact solutionin the form of a moving kink (antikink), φ = ± tanh x − x − vt √ − v , (4)for the upper (lower) sign, which propagates with the velocity v starting at t = 0 from theinitial position x = x . Substituting Eq. (4) into Eq. (1) one finds for the total energy ofthe continuum kink E cK = 43 √ − v . (5)The discrete φ equation is introduced on the lattice x = nh with spacing h >
0, where n is integer. The conventional discretization of the φ equation (3) reads [22]:¨ φ n = 1 h (cid:0) φ n − − φ n + φ n +1 (cid:1) + 2 φ n (cid:0) − φ n (cid:1) , (6)5here φ n ( t ) = φ ( nh, t ) and differentiation with respect to time is denoted by overdot. Thisdiscretization conserves the following Hamiltonian: H = h (cid:88) n (cid:34) ˙ φ n + (cid:18) φ n +1 − φ n h (cid:19) + (cid:0) − φ n (cid:1) (cid:35) . (7)Collision of highly discrete kinks cannot be studied within this model because the kinks aretrapped by the Peierls–Nabarro potential and cannot propagate freely. B. Exceptional discretization of the φ equation Speight has derived the following discrete φ model [42, 43]:¨ φ n = (cid:18) h + 13 (cid:19) (cid:0) φ n − − φ n + φ n +1 (cid:1) +2 φ n − (cid:104) φ n + (cid:0) φ n + φ n − (cid:1) + (cid:0) φ n + φ n +1 (cid:1) (cid:105) ≡ f n , (8)which possesses the Hamiltonian H = h (cid:88) n (cid:16) ˙ φ n + u (cid:17) , (9)where u ≡ ± φ n − φ n − h − φ n − + φ n − φ n + φ n . (10)For equilibrium states, we have that u = 0, which can be used for obtaining the kink profile(see below). Note that we have denoted the right-hand side of Eq. (8) as f n , which isthe force acting on n -th particle from the two neighboring particles and from the on-sitepotential.It can be proved [42, 43] that static kinks of the model (8) can be derived iterativelyfrom the two-point map by setting (10) equal to 0, which is a quadratic algebraic equationhaving the roots φ n ± = − φ n ∓ h ± √ (cid:114) − φ n ± h φ n + 3 h + 4 . (11)One can take in Eq. (11) either the upper or the lower signs. The iterations can be startedfrom any initial value | φ n | < φ n = 0 and the inter-site kink for φ n = 3 /h − (cid:112) /h .All such kinks have exactly the same potential energy and thus they do not experience a6tatic Peierls–Nabarro potential. Kinks exist also in the particular case of h = 1 and theycan be found from the iterative formula (11).Examples of the inter-site static kink profiles, constructed by iterating Eq. (11), are shownin Fig. 1(a) for h = 0 . h = 1 . h < h > φ n = ± h > h and thus, it is equal to 4/3, as it follows from Eq. (5), which is the result for the continuumkink ( h → E dK , for different values of h and compared it to the prediction of the continuum theory, E cK , Eq. (5). The result is presented in Fig. 2(a), where the solid line shows the prediction ofthe continuum theory, Eq. (5), and the dots stand for various h according to the legend. Tobetter see the effect of h on the energy of the moving discrete kink, in Fig. 2(b) the difference E dK − E cK is given as a function of v c . It can be concluded that the energy of the movingdiscrete kink exceeds the energy predicted by the continuum theory and upon increasing h the difference increases. The difference also increases with growing kink velocity v c , but forvanishing kink velocity, kink energy is indeed equal to 4 / ≈ . h [42]. C. Spectrum of vacuum and small-amplitude vibrations localized on the kink
Let φ n be an equilibrium static solution of Eq. (8). Small-amplitude oscillations aroundthis solution can be studied by inserting φ n ( t ) = φ n + ε n ( t ) into the equation of motion (8),where ε n (cid:28)
1, and obtaining the linearized equation in the form¨ ε n = 1 h (cid:0) ε n − − ε n + ε n +1 (cid:1) + 13 (cid:104) − (cid:0) φ n + φ n − (cid:1) (cid:105) ε n − + 13 (cid:104) − (cid:0) φ n + φ n +1 (cid:1) (cid:105) ε n +1 + 13 (cid:104) − (cid:0) φ n (cid:1) − (cid:0) φ n + φ n − (cid:1) − (cid:0) φ n + φ n +1 (cid:1) (cid:105) ε n . (12)Substituting into Eq. (12) the ansatz ε n = exp( iqn − iωt ), where ω is the frequency and q is the wavenumber, one obtains the eigenvalue problem for finding the spectrum of small-7 IG. 1. (a) Inter-site static kink profiles for h = 0 . h = 1 . h < h > φ n = ± ω and ω , as functions of h , shown by the dashed and solidlines, respectively. The lines cross at h = 1. The scattered data shows the frequencies of the small-amplitude vibrational modes localized on the on-site (dots) and inter-site (circles) static kinks.The zero-frequency mode e is the Goldstone translational mode, which is used for kink boosting.The second lowest frequency mode e is the kink’s internal mode. (c), (d) Profiles of the Goldstonetranslational mode and kink’s internal mode, respectively, for the kinks shown in (a). amplitude vibrations around the static solution φ n .8 a) (b) FIG. 2. (a) Total (potential plus kinetic) energy of the discrete kink, E dK , as a function of itsvelocity for different values of the discreteness parameter h . The solid line shows the prediction ofthe continuum theory, E cK , Eq. (5). (b) The difference E dK − E cK as a function of kink velocity forvarious values of the discreteness parameter h . In particular, the spectrum of the vacuum solution φ n = ± ω = 4 + 4 1 − h h sin (cid:16) q (cid:17) . (13)This spectrum lies in between ω = 4 and ω = 4 + 4 1 − h h . (14)At h = 1 one has ω = ω , i.e., the width of the spectrum vanishes and hence linear modesarise at a single frequency, namely ω = 2. This situation where the frequency is independentof wavenumber is referred to as a flat band and is of particular interest in recent studies [58].For h < | q | ≈ π ) have frequencies higher than the long waves ( | q | ≈ h > φ n the eigenvalue problem is solved numerically. For this, akink is placed in the middle of the lattice of N = 200 particles with boundary conditions φ = − φ N = 1. The solution of the eigenvalue problem gives N − ω k andthe same number of eigenvectors, ( e k ) n . Most of the eigenfrequencies lie within the phononspectrum of vacuum but a few of them are below the relevant band. This is demonstratedin Fig. 1(b). The borders of the phonon spectrum as functions of h , Eq. (14), are shown9y the dashed and solid lines. Dots and circles indicate frequencies of the modes localizedon the on-site and inter-site kink, respectively. For any h there exists the zero frequencymode e , which is the translational (Goldstone) mode. The mode e , which has the lowestnon-zero frequency, is nothing but the kink’s internal mode, which in the continuum limit( h →
0) has frequency √
3. In Fig. 1(c) we plot the Goldstone modes and in Fig. 1(d) thekink’s internal modes for the inter-site kinks shown in (a), i.e., for the discreteness parameter h = 0 . h = 1 . D. Interaction of well separated kink and mirror image antikink
For further discussion it is instructive to understand how the kink and antikink interactwith each other at large distances and how this interaction depends on the lattice spacing h .In the continuum models, the force acting between the kinks is usually calculated as minusgradient of the potential energy of their interaction. In the regime of strong discreteness( h ∼ φ n be thestatic kink solution and φ tn be the antikink tail. We assume the tail solution to be of theform (as will be justified later) φ tn = 1 + (cid:15) tn , (15)where | (cid:15) tn | (cid:28)
1. The force acting on the n -th site, which is within the kink, can be calculatedby substituting the linear superposition φ n = φ n + (cid:15) tn into Eq. (8) and linearizing with respectto (cid:15) tn . The result is f n = 1 h (cid:0) (cid:15) tn − − (cid:15) tn + (cid:15) tn +1 (cid:1) + 13 (cid:104) − (cid:0) φ n + φ n − (cid:1) (cid:105) (cid:15) tn − + 13 (cid:104) − (cid:0) φ n + φ n +1 (cid:1) (cid:105) (cid:15) tn +1 + 13 (cid:104) − (cid:0) φ n (cid:1) − (cid:0) φ n + φ n − (cid:1) − (cid:0) φ n + φ n +1 (cid:1) (cid:105) (cid:15) tn . (16)Notice that linear combinations of the kink solution and the antikink tail solution consideredhere can be used for kinks with short-range tails, as in our case, but this approximation maynot work for kinks with long-range tails, see, e.g., Refs. [39, 59–62].10 IG. 3. (a) The setup of the calculation of the force acting on the central particle of the on-sitekink from the antikink’s tail. The exact on-site static kink solution φ n (large circles) and theantikink (small circles) are shown. Here h = 2 and hence the kinks have oscillatory tails. Thekink is located at n = n = 0. (b) Normalized theoretically predicted force, f n /(cid:15) tn , acting on thecentral particle of the on-site kink from the antikink’s tail as calculated from Eq. (20). An approximate kink tail solution can be derived by substituting Eq. (15) into Eq. (11)and expanding with respect to (cid:15) tn up to the second power. The resulting iterative formulareads (cid:15) tn +1 = 1 − h h (cid:15) tn − h (3 + h )3(1 + h ) ( (cid:15) tn ) . (17)It can be seen from the linear term of Eq. (17) that for h < (cid:15) tn have the same sign, sothat the kink’s tail monotonously approaches the value φ n = 1. For h > φ n = 1 since (cid:15) tn and (cid:15) tn +1 have opposite signs. If h = 1, the linear term in the expansion(17) vanishes and we have a purely anharmonic chain with a corresponding short-range kinktail. Note that the antikink tail is a mirror image of the kink tail (17), that is why theantikink tail can be described using (cid:15) tn − = 1 − h h (cid:15) tn − h (3 + h )3(1 + h ) ( (cid:15) tn ) . (18)The ansatz (15) can be justified by considering the vacuum solution φ n = 1 for all n and getting from Eq. (12) the following static equation for the small deviation from thevacuum: (1 /h − ε n − − ε n + ε n +1 ) − ε n = 0. Looking for the solution to this equation11n the form ε n = q n one obtains the quadratic characteristic equation having the roots q , = (1 ∓ h ) / (1 ± h ), which is equivalent to the linear part of kink tail solution (17).The force f n acting on any site within the kink can be now calculated for any given valueof (cid:15) tn (cid:28) (cid:15) tn ± from Eq. (18) and substituting into Eq. (16). Recall that thestatic kink solution can be found iteratively for any given − < φ n < f n can be simplified for the case of the highlysymmetric on-site kink. Let the on-site kink be located at n = n , i.e., φ n = 0. Then fromEq. (11) one finds the displacements for the neighboring sites: φ n ± = ∓ h ± √ (cid:114) h + 4 . (19)For a given small value (cid:15) tn we find (cid:15) tn ± from the linear part of Eq. (18) and substitutingthese values together with φ n = 0 and φ n +1 = − φ n − into Eq. (16), we obtain the forceacting from the antikink’s tail on the central particle of the on-site kink: f n = − (cid:15) tn (1 − h )(1 + h ) − (cid:15) tn (cid:26) h )(1 − h )(1 + h ) (cid:20) − (cid:16) φ n +1 (cid:17) (cid:21) − (cid:0) φ n +1 (cid:1) + 4 (cid:27) , (20)where φ n +1 is given by Eq. (19). Note that for the considered here kink and mirror imageantikink the sign of (cid:15) tn in Eq. (20) does not change. There are two possible interpretationsof the displacements of particles φ n , i.e., one can think of either longitudinal or transverseparticle motion. In Fig. 3(a) the force acting on the n -th particle is shown in verticaldirection because the particle displacements φ n are also shown in transverse direction. Forpositive such forces, the particle is moving upwards which means that a preceding particle ismoving upward as well, leading the coherent structure to move to the left. By a symmetricargument, when the relevant force is negative, the waves will move toward each other. Inthis way, one can connect the force on individual particles to the force resulting on thenonlinear wave.From Eq. (20) it can be seen that f n is proportional to (cid:15) tn . The dependence of f n /(cid:15) tn on h is shown in Fig. 3(b). It is interesting to note that the sign of the force changes at h = 1 so that the kink and antikink attract each other for h < h >
1. This fact will be confirmed numerically in Sec. III.Note that the ratio f n /(cid:15) tn diverges at h = 1, see Fig. 3(b). The values of the force f n remain finite even for h = 1. In Fig. 4 the values of the force f n acting on the central12 IG. 4. Force acting on the central particle of the on-site kink from the tail of the on-site antikinkat the distance d = nh from the kink: (a) the case of attractive interaction when h < h >
1. The values of the lattice spacing h are given for each curve. particle of the on-site kink from the tail of the mirror image antikink are presented for thelattice spacing close to 1 as a function of the distance between the kink and antikink d normalized by the lattice spacing h . In (a) h < h > h closer to 1. III. KINK-ANTIKINK COLLISIONSA. Simulation setup
The set of equations of motion (8) was integrated numerically using the St¨ormer methodof the sixth order [63] with the time step τ = 0 . N , so that the13adiation emitted by the colliding kinks does not reach the ends of the chain by the end of thesimulation run. This way the effect of the radiation on the kink dynamics is avoided. Fixedboundary conditions were employed, φ L = φ R = −
1, where φ L and φ R are the coordinatesof the particles at the left and right ends of the chain, respectively (though the type ofboundary conditions is not important for such sufficiently long chains). In typical runs wetook the number of lattice points N = 2000, which was sufficient to avoid the effect of theradiation reflected from the fixed ends of the chain on the kink-antikink collisions.The moving kink can be obtained by using the Goldstone translational mode e , seeFig. 1(c). The initial conditions were formulated as follows. At t = 0 we set φ n = φ n , where φ n is the static kink solution, and at t = τ the Goldstone mode is added, φ n = φ n + δ ( e ) n ,with a small coefficient δ which defines the speed of the boosted kink. The eigenvector e isassumed to be normalized, || e || = 1. The resulting kink velocity is measured numericallyand it is called collision velocity v c .The obtained kink moving with the velocity v c collides with its mirror image antikinkhaving velocity − v c in the middle of the chain. The initial distance between the kink andantikink is taken sufficiently large for their exponential tails to not overlap. B. Numerical results
Examples of the kink-antikink collisions are presented in Fig. 5: in (a)–(a (cid:48)(cid:48) ) for h = 0 . (cid:48)(cid:48) ) for h = 0 .
9, in (c)–(c (cid:48)(cid:48) ) for h = 1 .
1, and in (d)–(d (cid:48)(cid:48) ) for h = 1 .
5, by plotting theparticles with the maximal energy. The collision velocity in each row increases from the leftto the right. Corresponding 3D plots are given in Fig. 6.It can be clearly inferred that the collisions are qualitatively different for h < h > h = 1, as itwas demonstrated in Sec. II D. Indeed, in line with the suggestive theoretical analysis of theprevious section for h < h >
1) the kink and antikink attract (repel) each other.At h <
IG. 5. Trajectories of colliding kinks and antikinks in the time-space plane for h = 0 . (cid:48)(cid:48) ),for h = 0 . (cid:48)(cid:48) ), for h = 1 . (cid:48)(cid:48) ), and for h = 1 . (cid:48)(cid:48) ), shown by plotting theparticles with the maximal energy. The origin of the temporal coordinate is chosen at the collisiontime moment t c . The collision velocity v c increases in each row from the left to the right having thevalues: (a) 0.06062, (a (cid:48) ) 0.08237, (a (cid:48)(cid:48) ) 0.09069; (b) 0.1, (b (cid:48) ) 0.15, (b (cid:48)(cid:48) ) 0.2; (c) 0.01751, (c (cid:48) ) 0.1958,(c (cid:48)(cid:48) ) 0.2501; (d) 0.01994, (d (cid:48) ) 0.1481, (d (cid:48)(cid:48) ) 0.1486. while a smaller portion of energy is radiated in the form of small-amplitude waves. If thecollision velocity is above a threshold value v ∗ , the coherent structures pass through eachother and continue their motion with a reduced velocity v < v c , as exemplified in (a (cid:48)(cid:48) ) and15 a) h = 0 . v c = 0 . h = 0 . v c = 0 . h = 0 . v c = 0 .
15, cf. Fig. 5(b’). (d) h = 1 . v c = 0 . h = 1 . v c = 0 . h = 1 . v c = 0 . FIG. 6. 3D plots showing kink-antikink collisions. For each panel the values of the discretenessparameter h and collision velocity v c , as well as the link to corresponding panel of Fig. 5 areprovided. (b (cid:48)(cid:48) ). Plots (a) and (a (cid:48) ) show collisions with velocity v c < v ∗ . Here the waves after thecollision cannot overcome their mutual attraction and collide again. In Fig. 5(a) a bion isformed [see also 3D plot in Fig. 6(a)], which is a kink-antikink bound state. The bion’sfrequency is below the phonon band, ω B < ω = 2. From Eq. (14), for h = 0 . ω = 4. This means that bion’s higher harmonics are alwayswithin the phonon band, and thus the bion radiates its energy due to the relevant resonancemechanism. In Fig. 5(a (cid:48) ), we have a three-bounce collision after which the kinks separate andcontinue their motion with a velocity v < v c [also shown in Fig. 6(b)]. Such multi-bouncecollisions with v c < v ∗ are possible because the energy stored by the kinks’ internal modescan be transformed back into energy of the kinks’ translational motion. This suggests thathere we are still in a regime proximal to the continuum limit where such phenomenologyis well-known [24, 27, 28]. For a discrete breather (DB) to exist on top of the nonzerobackground, it is necessary that its frequency and all higher harmonics are outside of thephonon spectrum [20]. In Fig. 5(b) and (b (cid:48) ) the collision velocity is also below v ∗ . Howeverin this case the width of the phonon band is small and a DB is formed with its frequencyand all higher harmonics outside the phonon band. Formation of DBs can also be seen inthe 3D plots, Fig. 6(c,d). Thus, there is no mechanism for the breather to radiate its energyand it can be expected to persist over a long-time evolution.For the analysis of the kink-antikink collisions in the case of h > (cid:48) ), where the collision velocityis below a threshold value v ∗∗ sufficient to overcome their repulsion. In Fig. 5(c (cid:48) ), (c (cid:48)(cid:48) ) and(d (cid:48)(cid:48) ) the collision velocity is above v ∗∗ and the kinks indeed collide. A 3D picture of kink-antikink repulsion is presented in Fig. 6(e). In Fig. 5(c (cid:48)(cid:48) ) the kink and antikink emerge afterthe collision with a velocity smaller than v c . In Fig. 5(d (cid:48)(cid:48) ) [also in Fig. 6(f)], as a result ofthe collision, a bion is produced with the main frequency within the relatively wide phononband. The bion disappears after a few oscillations producing a burst of radiation. On theother hand, in Fig. 5(c (cid:48) ) [also in Fig. 6(d)] the phonon band is narrow and a DB is formedwith frequency ω DB < ω , i.e., below the phonon band with all higher harmonics above thephonon band, hence the relevant waveform is expected to persist, as is indeed observed inthe corresponding evolution dynamics.Two critical velocities were defined above, v ∗ for h < v ∗∗ for h >
1. In Fig. 7 thecritical velocities are plotted as functions of h . The critical velocity v ∗ has a minimum at h = 0 .
6; here, collisions are most elastic. v ∗ denotes the threshold above which separationof the kink-antikink pair occurs after a single collision. The critical velocity v ∗∗ increaseslinearly with h . v c < v ∗∗ denotes the scenario of repulsion without collision in the case of17 IG. 7. Critical collision velocities v ∗ and v ∗∗ as functions of h , for h < h >
1, respectively.For v c > v ∗ kink and antikink separate after the first collision, as shown in Fig. 5(a (cid:48)(cid:48) ). For v c < v ∗ and h < . (cid:48) ). For v c < v ∗ and 0 . < h < (cid:48) ). When v c > v ∗∗ thekink and antikink collide forming a DB (1 < h < .
5, Fig. 5(c (cid:48) )) or a burst of radiation ( h > . (cid:48)(cid:48) )). For sufficiently large v c and h close to 1, the kink and antikink separate after the firstcollision, Fig. 5(c (cid:48)(cid:48) ). For v c < v ∗∗ they repel each other, Fig. 5(c) or (d), or (d (cid:48) ). h >
1. Depending on h and the collision velocity, Fig. 7 is divided into parts where differentcollision scenarios are observed, as linked to the panels of Fig. 5.In Fig. 8 the kinetic energy of the chain is plotted as a function of time for the cases (a) h = 0 . h = 1 .
3. In both cases kink and antikink collide with the velocity v c = 0 . t = 0 with the formation of DBs. Each collision of the kinks forming a DB results in a sharpdouble-peak of the kinetic energy. We estimated the oscillation period and then frequencyof bions and DBs numerically by averaging over ten oscillations starting from the sixth one,see Fig. 8. The first five oscillations are dropped because in some cases they have very longperiod, e.g., in multi-bounce collisions. For example, in Fig. 8(a) ten oscillations take 53time units, then T = 5 .
3, and ω DB = 2 π/T = 1 .
18, which is below the lower edge of thephonon band ω = 2. The second harmonic has frequency 2 ω DB = 2 .
36, which is above the18
IG. 8. Kinetic energy of the chain with colliding kink and antikink as a function of time. In(a) h = 0 . h = 1 .
3, and in both cases the initial velocity of the kink and antikink is v c = 0 .
9. Collision takes place at t = 0 and in both cases it results in formation of a DB. upper edge of the phonon band ω = 2 .
22. Note that the period of the bions in some casesvaries in time noticeably [e.g., in the case presented in Fig. 5(a)] due to the energy exchangebetween the kink’s internal and translational modes and due to the losses to radiation. Theperiod of the DBs is more robust since they radiate much less energy having no interactionwith the phonons.More information on the effect of the collision velocity on the collision outcome is pre-sented in Fig. 9. For instance, for h = 0 . v c .The circles show the frequency of the bion’s second harmonic. Horizontal dashed lines showthe borders of the phonon spectrum frequencies, ω and ω . It can be seen that the bion’sfrequency is always below the phonon spectrum but its second or third harmonic is withinthe spectrum. For v c > .
08 = v ∗ the kink and antikink separate after the first collision. Thebottom panel shows the bion frequency as a function of its amplitude for all the cases wherea bion was formed for h = 1 . h = 1 . h = 0 .
9, but in this case not a bion but a DB is formed with the main frequency below19 a) (b)(c) (d)
FIG. 9. Top panels: effect of the collision velocity v c on the collision outcome for (a) h = 0 .
5, (b) h = 0 .
9, (c) h = 1 .
2, and (d) h = 1 .
3. Horizontal dashed lines show the borders of the phononspectrum, ω and ω . Bion frequency ω B , or DB frequency ω DB , are shown by dots, and thesecond harmonics by circles. Bottom panels: (a) frequency of a bion formed in the collision with v c < .
07 as a function of its amplitude; (b)–(d) frequency of DB, formed as a result of collisionwith sufficiently large velocity as a function of its amplitude. the phonon band and all higher harmonics lie above the phonon band, hence suggesting thelong time persistence of the structure.In the particular case of h = 1 the kink-antikink collisions result in the formation of DBsif the collision velocity is roughly below 0.25 and the kinks separate after the first collisionfor higher collision velocities, see Fig. 7. Formation of DBs in the case of h = 1 is not20urprising because in this case, as it has been mentioned above, the width of the phononband vanishes, hence DBs should generically survive. IV. CONCLUSIONS AND FUTURE WORK
Kink-antikink collisions were analyzed numerically in the discrete φ model, Eq. (8),which is free of the static Peierls–Nabarro potential. The lattice spacing h of order unitywas considered, which corresponds to a high discreteness with the kink spanning only justa few lattice sites, see Fig. 1(a). Exact static kink solutions were derived iteratively fromthe two-point map (10) starting from any admissible initial value φ n . The existence of aone-parameter set of static kinks positioned arbitrarily with respect to the lattice ensuresthe existence of the zero frequency Goldstone translational mode, see Fig. 1(c). The profileof this mode can be found by solving the eigenvalue problem for the equations of motionlinearized near the static kink solution, see Eq. (12). The moving kink can be obtained byusing the Goldstone mode as described in Sec. III A. Only symmetric collisions between thekink and its mirror image antikink moving with velocities v c and − v c , respectively, wereaddressed.The borders of the phonon spectrum of the considered lattice, ω and ω , cross at h = 1,see Fig. 1(b). Around h = 1, the width of the relevant band is small (and it vanishes at h = 1). The crossing of the phonon band edges changes the kink profile: for h < h > ±
1, asfollows from Eq. (17) and as can be seen in Fig. 1(a). More importantly, for h < h > h <
1) and repulsive ( h >
1) kinkand antikink, as summarized below.For h < v c > v ∗ result in separation of the kink and antikinkafter the first collision, see Fig. 5(a (cid:48)(cid:48) ) or (b (cid:48)(cid:48) ). The critical velocity v ∗ decreases with increas-ing h in the range h < . h , see Fig. 7. Collisions with a velocitysmaller than v ∗ can result in either separation of the kink and antikink after a multi-bouncecollision, see Fig. 5(a (cid:48) ), or in the formation of an oscillatory mode. For h < .
7, when thephonon band is wide, the oscillatory mode is a bion, see Fig. 5(a), with frequency below21he phonon band and higher harmonics within the band, see the upper panel of Fig. 9(a).Due to the interaction with the phonons, the bion constantly radiates energy, its amplitudedecreases and frequency increases, see lower panel of Fig. 9(a). For 0 . < h < (cid:48) ), with the mainfrequency lying below the phonon band and all higher harmonics above the band as shownin the upper panel of Fig. 9(b). The DB does not interact with the phonons and has a verylong lifetime.For h > v c < v ∗∗ cannotovercome the repulsion and they actually do not collide, see Fig. 5(c), or (d), or (d (cid:48) ). Thecritical velocity v ∗∗ increases linearly with increasing h , see Fig. 7. Collision with a velocityabove v ∗∗ can result in either separation of the kink and antikink after a single collision, seeFig. 5(c (cid:48)(cid:48) ), or in formation of a DB, see Fig. 5(c (cid:48) ), or in a burst of radiation producing arapidly decaying bion, see Fig. 5(d (cid:48)(cid:48) ), where the radiation burst itself is not shown. A DB isformed for 1 < h < . h > . h , see Fig. 9. In a forthcomingstudy properties of DBs will be addressed in detail regardless of the mechanism of theirgeneration (generated not only in kink-antikink collisions) in the whole range of possiblefrequencies. So far DBs have not been analyzed in the discrete systems free of the staticPeierls–Nabarro potential. One of the most intriguing features here is the DB mobility: itis relevant to understand whether it is enhanced or not due to the absence of the staticPeierls–Nabarro potential. Contrary to more standard models, the present scenario hasthe potential of mobile breathers even for regimes of high discreteness, a feature that isuncommon for Klein–Gordon models outside the realm of integrable systems.Another direction for future studies is the analysis of kink collisions in the φ and φ models [30, 31, 34, 39, 64, 65]. Interestingly, kinks in the φ model are asymmetric and haveshort-range tails [30, 34, 64]. In the φ and a number of suitable higher-order models thekinks can have long-range tails with power-law decay [39, 59–62]. It would be interesting22o study kink collisions in these models in the regime of high discreteness. In the work[66] two discrete φ models free of the static Peierls–Nabarro potential have been derived.Generalizing such a derivation to φ , φ and φ would enable the consideration of theintriguing interplay of discreteness and long-range interactions. A discrete realm may bemore straightforward of a place to consider such long-range interactions given that extendedlattices would be easier to consider than the considerably more computationally expensivecontinuum analogues thereof.The third natural continuation of this work is the analysis of the multi-bounce collisionsand the related analysis of the kink’s internal modes. As it can be seen from Fig. 1(b), inthe case of small h there is only one kink’s internal mode with the frequency ω ≈ √
3, butfor h ∼ ACKNOWLEDGMENTS
The work of the MEPhI group was supported by the MEPhI Academic Excellence Project(Contract No. 02.a03.21.0005, 27.08.2013). V.A.G. and S.V.D. acknowledge the support ofthe Russian Foundation for Basic Research, Grant No. 19-02-00971. The work was partlysupported by the State assignment of IMSP RAS. PGK gratefully acknowledges the hos-pitality of the Mathematical Institute of the University of Oxford and the support of theLeverhulme Trust during the final stages of this work. This material is based upon worksupported by the US National Science Foundation under Grant DMS-1809074 (PGK). [1] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations,Academic Press, London, 1982.[2] O.M. Braun, Yu. Kivshar,
The Frenkel–Kontorova Model: Concepts, Methods, and Applica-tions , Springer-Verlag, Berlin, 2004.[3] P.G. Kevrekidis, J. Cuevas-Maraver,
A dynamical perspective on the φ model: past, presentand future , in Nonlinear Systems and Complexity, Springer Nature Switzerland, 2019.
4] J. Cuevas-Maraver, P.G. Kevrekidis, F. Williams (Eds.),
The sine-Gordon model and itsapplications: From pendula and Josephson Junctions to Gravity and High-Energy Physics ,Springer-Verlag, Heidelberg, 2014.[5] Yu.S. Kivshar, D.K. Campbell,
Peierls–Nabarro potential barrier for highly localized nonlinearmodes , Phys. Rev. E , 3077 (1993).[6] G.L. Alfimov, E.V. Medvedeva, D.E. Pelinovsky, Wave Systems with an Infinite Number ofLocalized Traveling Waves , Phys. Rev. Lett. , 054103 (2014) [ arXiv:1309.0183 ].[7] Ya. Zolotaryuk, I.O. Starodub,
Moving embedded solitons in the discrete double sine-Gordonequation , In: Nonlinear Systems, Vol. 2, p. 315 (2018). Understanding Complex Systems.Springer, Cham.[8] D.J. Frantzeskakis, N.I. Karachalios, P.G. Kevrekidis, V. Koukouloyannis, K. Vetas,
Dynam-ical transitions between equilibria in a dissipative Klein–Gordon lattice , J. Math. Anal. Appl. , 546 (2019) [ arXiv:1809.07995 ].[9] S.V. Dmitriev, N.N. Medvedev, A.P. Chetverikov, K. Zhou, M.G. Velarde,
Highly EnhancedTransport by Supersonic N -Crowdions , Phys. Status Solidi Rapid Res. Lett. , 1700298(2017).[10] R.I. Babicheva, I. Evazzade, E.A. Korznikova, I.A. Shepelev, K. Zhou, S.V. Dmitriev, Low-energy channel for mass transfer in Pt crystal initiated by molecule impact, Comput. Mater.Sci. , 248 (2019).[11] A. Moradi Marjaneh, D. Saadatmand, I. Evazzade, R.I. Babicheva, E.G. Soboleva, N. Srikanth,K. Zhou, E.A. Korznikova, S.V. Dmitriev, Mass transfer in the Frenkel–Kontorova chaininitiated by molecule impact , Phys. Rev. E , 023003 (2018) [ arXiv:1805.07200 ].[12] E.A. Korznikova, I.A. Shepelev, A.P. Chetverikov, S.V. Dmitriev, S.Yu. Fomin, K. Zhou, Dynamics and Stability of Subsonic Crowdion Clusters in 2D Morse Crystal
J. Exp. Theor.Phys. , 1009 (2018).[13] J.F.R. Archilla, Yu.A. Kosevich, N. Jimenez, V.J. Sanchez-Morcillo, L.M. Garcia-Raffi,
Ultra-discrete kinks with supersonic speed in a layered crystal with realistic potentials , Phys. Rev. E arXiv:1406.4085 ].[14] T.D. Swinburne, S.L. Dudarev, S.P. Fitzgerald, M.R. Gilbert, A.P. Sutton, Theory and sim-ulation of the diffusion of kinks on dislocations in bcc metals , Phys. Rev. B arXiv:1210.8327 ].
15] L. Huang, R. Wang, S. Wang,
A new reconstruction core of the o partial dislocation insilicon , Philos. Mag. , 347 (2019).[16] L. Huang, S. Wang, A theoretical investigation of the glide dislocations in the sphalerite ZnS ,J. Appl. Phys. , 175702 (2018).[17] F.J. Buijnsters, A. Fasolino, M.I. Katsnelson,
Motion of Domain Walls and the Dynam-ics of Kinks in the Magnetic Peierls Potential , Phys. Rev. Lett. , 217202 (2014)[ arXiv:1407.7754 ].[18] M. Hwang, A.F. Arrieta,
Solitary waves in bistable lattices with stiffness grading: Augmentingpropagation control , Phys. Rev. E , 042205 (2018).[19] Yu.A. Kosevich, Charged ultradiscrete supersonic kinks and discrete breathers in nonlinearmolecular chains with realistic interatomic potentials and electron-phonon interactions , J.Phys.: Conf. Ser. , 012021 (2017).[20] S. Flach, A.V. Gorbach,
Discrete breathers — Advances in theory and applications , Phys. Rep. , 1 (2008).[21] P.J. Martinez, L.M. Floria, F. Falo, J.J. Mazo,
Intrinsically localized chaos in discrete nonlin-ear extended systems , Europhys. Lett. , 444 (1999) [ arXiv:chao-dyn/9901030 ].[22] D.K. Campbell, J.F. Schonfeld, C.A. Wingate, Resonance structure in kink-antikink interac-tions in ϕ theory , Physica D , 1 (1983).[23] D.K. Campbell, M. Peyrard, Solitary wave collisions revisited , Physica D , 47 (1986).[24] P. Anninos, S. Oliveira, R.A. Matzner, Fractal structure in the scalar λ ( ϕ − theory , Phys.Rev. D , 1147 (1991).[25] V.A. Gani, A.E. Kudryavtsev, Kink-antikink interactions in the double sine-Gordon equationand the problem of resonance frequencies , Phys. Rev. E , 3305 (1999) [cond-mat/9809015].[26] S.V. Dmitriev, Yu.S. Kivshar, T. Shigenari, Fractal structures and multiparticle effects insoliton scattering , Phys. Rev. E , 056613 (2001).[27] R.H. Goodman, R. Haberman, Kink-Antikink Collisions in the φ Equation: The n-BounceResonance and the Separatrix Map , SIAM J. Appl. Dyn. Syst. , 1195 (2005).[28] R.H. Goodman, R. Haberman, Chaotic Scattering and the n-Bounce Resonance in Solitary-Wave Interactions , Phys. Rev. Lett. , 104103 (2007) [ arXiv:nlin/0702048 ].[29] S.V. Dmitriev, P.G. Kevrekidis, Yu.S. Kivshar, Radiationless energy exchange in three-solitoncollisions , Phys. Rev. E , 046604 (2008) [ arXiv:0806.1152 ].
30] V.A. Gani, A.E. Kudryavtsev, M.A. Lizunova,
Kink interactions in the (1+1)-dimensional ϕ model , Phys. Rev. D , 125009 (2014) [arXiv:1402.5903].[31] V.A. Gani, V. Lensky, M.A. Lizunova, Kink excitation spectra in the (1+1)-dimensional ϕ model , JHEP , 147 (2015) [arXiv:1506.02313].[32] E.G. Ekomasov, A.M. Gumerov, R.R. Murtazin, Interaction of sine-Gordon solitons in themodel with attracting impurities , Math. Meth. Appl. Sci. , 6178 (2017).[33] A. Moradi Marjaneh, D. Saadatmand, K. Zhou, S.V. Dmitriev, M.E. Zomorrodian, Highenergy density in the collision of N kinks in the φ model , Comm. Nonlinear Sci. Numer.Simulat. , 30 (2017) [ arXiv:1605.09767 ].[34] A. Moradi Marjaneh, V.A. Gani, D. Saadatmand, S.V. Dmitriev, K. Javidan, Multi-kinkcollisions in the φ model , JHEP , 028 (2017) [ arXiv:1704.08353 ].[35] A. Moradi Marjaneh, A. Askari, D. Saadatmand, S.V. Dmitriev, Extreme values of elas-tic strain and energy in sine-Gordon multi-kink collisions , Eur. Phys. J. B , 22 (2018)[ arXiv:1710.10159 ].[36] D. Bazeia, E. Belendryasova, V.A. Gani, Scattering of kinks of the sinh-deformed ϕ model ,Eur. Phys. J. C , 340 (2018) [ arXiv:1710.04993 ].[37] V.A. Gani, A. Moradi Marjaneh, A. Askari, E. Belendryasova, D. Saadatmand, Scattering ofthe double sine-Gordon kinks , Eur. Phys. J. C , 345 (2018) [ arXiv:1711.01918 ].[38] V.A. Gani, A. Moradi Marjaneh, D. Saadatmand, Multi-kink scattering in the double sine-Gordon model , Eur. Phys. J. C , 620 (2019) [ arXiv:1901.07966 ].[39] E. Belendryasova, V.A. Gani, Scattering of the ϕ kinks with power-law asymptotics , Comm.Nonlinear Sci. Numer. Simulat. , 414 (2019) [ arXiv:1708.00403 ].[40] O.M. Braun, Yu.S. Kivshar, M. Peyrard, Kink’s internal modes in the Frenkel–Kontorovamodel , Phys. Rev. E , 6050 (1997).[41] I.V. Barashenkov, O.F. Oxtoby, D.E. Pelinovsky, Translationally invariant discrete kinks fromone-dimensional maps , Phys. Rev. E , 035602 (2005) [ arXiv:nlin/0506007 ].[42] J.M. Speight, A discrete φ system without a Peierls–Nabarro barrier , Nonlinearity , 1615(1997) [ arXiv:patt-sol/9703005 ].[43] J.M. Speight, Topological discrete kinks , Nonlinearity , 1373 (1999)[ arXiv:hep-th/9812064 ].
44] P.G. Kevrekidis,
On a class of discretizations of Hamiltonian nonlinear partial differentialequations , Physica D , 68 (2003).[45] P.G. Kevrekidis, A. Khare, A. Saxena, I. Bena, A.R. Bishop,
Asymptotic calculation of discretenon-linear wave interactions , Math. Comp. Simul. , 405 (2007).[46] S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, Standard nearest-neighbour discretizations ofKlein–Gordon models cannot preserve both energy and linear momentum , J. Phys. A: Math.Gen. , 7217 (2006) [ arXiv:nlin/0506002 ].[47] S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, Discrete Klein–Gordon models with statickinks free of the Peierls–Nabarro potential , J. Phys. A: Math. Gen. , 7617 (2005)[ arXiv:nlin/0506001 ].[48] F. Cooper, A. Khare, B. Mihaila, A. Saxena, Exact solitary wave solutions for a discrete λφ field theory in dimensions , Phys. Rev. E , 036605 (2005) [ arXiv:nlin/0502054 ].[49] J.M. Speight, Y. Zolotaryuk, Kinks in dipole chains , Nonlinearity , 1365 (2006)[ arXiv:nlin/0509047 ].[50] S.V. Dmitriev, P.G. Kevrekidis, A. Khare, A. Saxena, Exact static solutions to a translationallyinvariant discrete φ model , J. Phys. A: Math. Theor. , 6267 (2007) [ arXiv:nlin/0703043 ].[51] A. Khare, S.V. Dmitriev, A. Saxena, Exact static solutions of a generalized discrete φ model including short-periodic solutions , J. Phys. A: Math. Theor. , 145204 (2009)[ arXiv:0710.1460 ].[52] C.M. Bender, A. Tovbis, Continuum limit of lattice approximation schemes , J. Math. Phys. , 3700 (1997).[53] S.V. Dmitriev, A. Khare, P.G. Kevrekidis, A. Saxena, L. Hadzievski, High-speed kinks in ageneralized discrete φ model , Phys. Rev. E , 056603 (2008) [ arXiv:0802.2375 ].[54] I. Roy, S.V. Dmitriev, P.G. Kevrekidis, A. Saxena, Comparative study of different discretiza-tions of the φ model , Phys. Rev. E , 026601 (2007) [ arXiv:nlin/0608046 ].[55] A.V. Savin, Y. Zolotaryuk, J.C. Eilbeck, Moving kinks and nanopterons in the nonlinearKlein-Gordon lattice , Physica D , 267 (2000)[56] J.F.R. Archilla, Y. Doi, M. Kimura,
Pterobreathers in a model for a layered crystal withrealistic potentials: Exact moving breathers in a moving frame , Phys. Rev. E , 022206(2019)
57] O.F. Oxtoby, D.E. Pelinovsky, I.V. Barashenkov,
Travelling kinks in discrete φ models , Non-linearity, , 217 (2006)[58] D. Leykam, A. Andreanov, S. Flach, Artificial flat band systems: from lattice models to ex-periments , Adv. Phys.: X , 1473052 (2018) [ arXiv:1801.09378 ].[59] I.C. Christov, R.J. Decker, A. Demirkaya, V.A. Gani, P.G. Kevrekidis, R.V. Radomskiy, Long-range interactions of kinks , Phys. Rev. D , 016010 (2019) [ arXiv:1810.03590 ].[60] I.C. Christov, R.J. Decker, A. Demirkaya, V.A. Gani, P.G. Kevrekidis, A. Khare, A. Saxena, Kink-kink and kink-antikink interactions with long-range tails , Phys. Rev. Lett. , 171601(2019) [ arXiv:1811.07872 ].[61] N.S. Manton,
Forces between kinks and antikinks with long-range tails , J. Phys. A: Math.Theor. , 065401 (2019) [ arXiv:1810.03557 ].[62] A. Khare and A. Saxena, Family of potentials with power law kink tails , J. Phys. A: Math.Theor. , 365401 (2019) [ arXiv:1810.12907 ].[63] N.S. Bakhvalov, Numerical methods: analysis, algebra, ordinary differential equations , MIRPublishers, Moscow, 1977.[64] A. Demirkaya, R. Decker, P.G. Kevrekidis, I.C. Christov, A. Saxena,
Kink dynamics in aparametric φ system: a model with controllably many internal modes , JHEP , 071 (2017)[ arXiv:1706.01193 ].[65] V.A. Gani, A. Moradi Marjaneh, P.A. Blinov Explicit kinks in higher-order field theories ,Phys. Rev. D , 125017 (2020) [ arXiv:2002.09981 ].[66] Zh.G. Rakhmatullina, P.G. Kevrekidis, S.V. Dmitriev,
Non-symmetric kinks in Klein–Gordonchains free of the Peierls–Nabarro potential , IOP Conf. Ser.: Mater. Sci. Eng. , 012057(2018)., 012057(2018).