Waves in Strongly Nonlinear Gardner-like Equations on a Lattice
WWaves in Strongly Nonlinear Gardner-like Equations on a Lattice
Philip Rosenau
School of Mathematics,Tel-Aviv University,Tel-Aviv 69978, Israel
Arkady Pikovsky
Institute of Physics and Astronomy,University of Potsdam, Karl-Liebknecht-Str. 24/25,14476 Potsdam-Golm, Germany andNational Research University,Higher School of Economics,25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia (Dated: August 27, 2020)
Abstract
We introduce and study a family of lattice equations which may be viewed either as a stronglynonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterrachain. Their deceptively simple form supports a very rich family of complex solitary patterns. Someof these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, likeinterlaced pairs of solitary waves, or waves which may reverse their direction either spontaneouslyor due a collision, are an intrinsic feature of the discrete realm. a r X i v : . [ n li n . PS ] A ug . INTRODUCTION With a progressing realization of their importance, strongly nonlinear lattices started toattract in the recent decades an ever growing attention [1–12]. This may be seen as a naturalscientific evolution following in the foot steps of what may arguably be considered as a dawnof a non-linear science: the Fermi-Pasta-Ulam problem which, as envisioned by J. von Neu-man [13], used for the first time the newly born computer to carry a Gedankenexperiment,to test the long standing hypothesis of energy equipartition of modes in solids due to non-linearity. Though, given the primordial state of affairs, the studied chain, as solids model,was relatively very simple and short, their results after few years in hibernation have led tothe birth of the soliton and the chaos theories and, as computational facilities have advancedand spread around the globe, to ever more evolved theories and applications. Chains of cou-pled elements were introduced and used to simulate more realistic molecular interactions,micro-mechanical arrays, electrical transmission lines, optical lattices or photonic crystals,to name a few. These are well known facts which need no further elaboration, a succinctoverview is presented in Ref. [14], and references therein.A common feature of the aforementioned systems is the presence of a weakly nonlinearregime wherein the dynamics is determined by both linear and nonlinear parts. For suchstates one may employ simple nonlinear expressions, though no one thought in earnestthat the quadratic force assumed by FPU could, apart of very small excitations, reproducerealistic molecular interactions. In fact, the numerical experiments carried a few years laterhave very clearly revealed that as the employed nonlinearity gained in strength, the idyllicrecovery of the initial excitation by the FPU had to be replaced by a far more complexstructures. If further arguments were ever needed, those results have clearly pointed to thenecessity to probe deeper into the nonlinear regime. In the spring-mass context this amountsto probing the truly anharmonic domain which in its ultimate rendition results in a systemwithout linear waves (the so-called sonic vacuum), with all perturbations of the ground statebeing nonlinear as well. Differently stated, a true understanding of nonlinearity’s impactnecessitates to engage the nonlinear regime in its full glory, rather then as an extension ofits weakly nonlinear limit. In an analogy, to the extent that the infra-red and ultra-violet2egime represent opposite ends of the visible spectrum, similarly the linear/weakly-linearand the genuinely nonlinear regimes may be looked upon as conceptual opposites, each ofwhich should be addressed on its own terms. As a typical signature of a genuinely nonlinearsystem one may consider the absence of a non-scalable intrinsically small parameter to serveas a reference for a weakly nonlinear dynamics.Let us recall that aside of the conceptual importance of the genuinely nonlinear limit,many systems by their very nature are ab initio essentially nonlinear. The Hertz-likeinteraction of elastic beads is perhaps the simplest mechanical realization of such sys-tem [1, 2, 15, 16]. And whereas in mechanics one typically deals with lattices describedby coupled second-order differential equations, strongly nonlinear systems have been alsoexplored within the framework of amplitude equations given via discrete Schr¨odinger-typeequations which are first-order in time [4, 9, 12].In yet another set up, closer to our present studies, a genuinely nonlinear conservativesystem emerges in a chain of coupled dissipative oscillators with a limit cycle. In a stronglydissipative regime, where the amplitudes of oscillators are truly stable, one may neglecttheir variations and derive a closed conservative system for their phases only [3, 6]. Suchsystems fall under what could be referred to as a Kuramoto type set-ups and in a cleardistinction from the Newton-like chains, are systems of first order in time . In a similar veinthe Lotka-Volterra chain [11] which originates in prey-predator system is another first orderin time strongly nonlinear lattice.The problems to be addressed here are a direct off spring of the Kuramoto family ofstrongly nonlinear first-order in time lattices [3, 6, 17], but instead of considering latticesof phase variables, with 2 π -periodic nonlinearities, we focus on lattices with polynomialnonlinearities. For the present purpose we shall restrict our attention to nonlinearities whichare sum of two monomials of opposite signs. In this form they are a discrete analog/extensionof the Gardner equation, which is a mixture of the K-dV and the mK-dV equations [18–20].The lattice we address may be viewed as a simplified version of a generic phase lattice,allowing us to cope with some of the difficulties we have encountered there, and sure enoughsimilarly to their original antecedents did not fail to produce new and surprising featureswhich we believe are of independent interest.Paper’s plan is as follows. In Section II we introduce the basic model. In Section IIIwe present an analytical treatment of the traveling waves akin to the quasi-continuous, QC,3endition. In Sections IV and V we return to the full lattice set-up and explore numericallysimple and interlaced travelling waves. Simulations of the traveling solitary waves in a latticeare presented in Section VI and reveal a number of fascinating phenomena not seen in theQC. We conclude with a discussion in Section VII. II. THE BASIC MODEL
Consider a genuinely nonlinear conservative lattice with nearest-neighbor interactions: du k dt = F ( u k +1 ) − F ( u k − ) , k = . . . , − , , , , . . . (1)where F ( u ) is a smooth nonlinear function, to be specified in Eqs. (3-5) below. To put theaddressed problems in a perspective we note that our studies were motivated by chains ofconservatively coupled self-sustained (autonomous) oscillators wherein the state variable u is the phase difference between neighboring oscillators reducing in the simplest case [3, 6, 21]to F ( u ) = cos( u ). With that particular choice the lattice (1) becomes genuinely nonlinearwhich is to say that there are linear waves with the resulting solitary waves, the compactonsand the kovatons, being almost compact (they decay at a doubly exponential rate) andessentially nonlinear entities. Notably, the model (1) has been recently successfully used todescribe complex states in a network of nano-electro-mechanical oscillators [22].In a previous work [17] using a more general setting for phase waves in a chain of au-tonomous oscillators, it was assumed that F ( u ) = sin( α − u ). Though numerical simulationsposed no particular difficulty, their direct analysis turned to be challenging forcing us to turnto their quasi-continuous rendition and the resulting partial differential equation. Yet eventhere we had to further restrict our self to a small amplitude regime which turned to be givenby the Gardner equation which surprisingly enough provided a remarkably good qualitativedescription of the α ≤ π/ F ( u ) seemed to be the source of the encountered difficulties, we have adopted asimpler set-up of a polynomial F ( u ), which replicate two of the vital singular transitions inthe periodic case.In passing we also note that in a lattice of a finite length N and open boundaries, system(1) is Hamiltonian (the proof, due to H. Dullin, follows Ref. [6]). Since for odd N system (1)has an additional integral K = (cid:80) ( N +1) / i =1 u i − , defining Q ( u ) = (cid:82) F ( u (cid:48) ) du (cid:48) , the Hamiltonian4f the lattice reads H ( q i , p i ) = Q ( q ) + (cid:80) mi =1 Q ( p i ) + (cid:80) m − i =1 Q ( q i +1 − q i ) + Q ( K − s m ) N = 2 m + 1 ,Q ( q ) + (cid:80) mi =1 Q ( p i ) + (cid:80) m − i =1 Q ( q i +1 − q i ) N = 2 m . (2)The used canonical variables are related to u i via p i = u i , q i = (cid:80) ij =1 u j − . In the originalvariables the conserved energy can be thus expressed as E = (cid:80) i Q ( u i ).Returning to our specific problem, we assume a strongly nonlinear Gardner-type latticeas F ( u ) = mu n − nu m , < m < n, (3)with integer m, n (the values of the coefficients which are arbitrary and were chosen tosimplify the analysis). Two typical cases will be analyzed:G23: du k dt = 2 u k +1 − u k +1 − u k − + 3 u k − , F ( u ) = 2 u − u , (4)and G35: du k dt = 3 u k +1 − u k +1 − u k − + 5 u k − , F ( u ) = 3 u − u . (5)The G23 model is a direct replica of the original Gardner equation. The G35 model hasdifferent symmetries, with its resulting properties being quite different from those of theG23.At this point we pause to note that since F ( u ) vanishes at u ∗ = (cid:0) nm (cid:1) / ( n − m ) , any se-quence of zeros and u ∗ forms a stationary solution on the lattice. In particular, one canhave stationary ‘pulse’ solutions . . . , , u ∗ , u ∗ , . . . , u ∗ , , . . . of an arbitrary length. While theconstruction of these solution is trivial, their stability properties are rather complex anddepend on both the total length of the lattice and the boundary conditions at its ends.Since we focus on traveling entities we shall not pursue further these solutions.In addition to the standard conservation features, the G23 model is invariant under u k → − u k , (6)whereas G35, and any Gmn if both m and n are odd, is invariant under u k → − u k (7)and under u k → ( − k u k and t → − t, (8)5hich generates from any given solution an alternating sign solution, referred to as a stag-gering solution in Section V A below. Note that whereas the first two invariant featuresextend to the quasi-continuum, the third (Eq. (8)) is obviously an exclusive feature of thediscrete realm.We stress that the novel features of the presented problems are due to the non monotonic nature of the assumed F ( u ). A widely studied version with F = exp( u ), also referred to asthe ladder equation, is integrable (and equivalent to the Toda lattice) [23, 24]. More relevantto our problem is the discrete K ( α, α ) equation with the monomial F = γu α (cf Ref. [25]): du k dt = γ (cid:0) u αk +1 − u αk − (cid:1) , where γ is a const. (9)which may be viewed as a limiting case of the generic lattice (1),(3), either for large u , where γ = m , α = n , or for small u , where γ = − n , α = m .Similarly to our previous studies, Eqs. (1),(3) have a deceptively simple appearance withtheir numerical integration being straightforward, but the understanding of the underlyingdynamics is a very different matter. For starters, our ability to analyze nonlinear discretesystems, or better yet, to unfold their coherent structures, is very limited, and far moreinferior to our ability to analyze systems represented by partial differential equations (PDEs).This naturally leads us to represent the discrete system via the Quasi-Continuum [25, 26],QC (Section III). Yet this approach has its difficulties, for whereas in weakly nonlinearsystems the stationary solution provides a reference level with respect to which one carriesthe asymptotic expansion, as aforementioned, the present problem lacks an essential smallparameter (the distance between the discrete nodes may be scaled out). Therefore theadopted QC route is not asymptotic and its value can be judged only by its utility . Luckily,and one could say: very luckily, we find that as in a number of our previous works [3–6], incertain regimes QC provides an excellent approximation of its discrete antecedent, thoughelsewhere it turned to be of far more limited use. III. TRAVELING WAVES IN A QUASI-CONTINUUMA. The Quasi-Continuum Rendition
Replacing the finite differences of the discrete lattice equations with a continuous spatialderivatives up to a third order yields the QC rendition of the original problem (see Refs. [25]6r [26] for a detailed exposition of the desired QC approach and its limitations) of Eqs.(1),(3): 12 ∂∂t u = ∂∂x L F ( u ) where L = 1 + 16 ∂ ∂x . (10)Similarly to their discrete antecedents, we have the following conservation laws of Eq. (10): I = (cid:90) udx and I = (cid:90) Q ( u ) dx where Q ( u ) = (cid:90) u F ( u (cid:48) ) du (cid:48) , (11)and an unusual Lagrangian structure L agrange = (cid:90) (cid:90) (cid:104) ψ x ψ t + Q (cid:16) L ψ x (cid:17)(cid:105) dxdt, (12)where u = L ψ x , which begets the conservation of the momentum (cid:82) ψ x dx , and in the originalvariables I = (cid:90) u L − u dx. (13) B. Solitary Waves in QC
We now turn our attention to solitary waves u ( x, t ) = u ( s = x − λt ). Since F (0) = 0,then upon one integration we have12 λu + (cid:18) d ds (cid:19) F ( u ) = 0 , (14)or 12 λu + F ( u ) + 16 dds F (cid:48) ( u ) duds = 0 . (15)A crucial role is played by the zeros of F (cid:48) ( u ) wherein equation (15) becomes singular. Also,since F (cid:48) ( u ) = mn ( u n − − u m − ), it vanishes at both u = 0 and u = 1.Multiplying (15) with F (cid:48) ( u ) u s and integrating once yields the energy integral (since weare after solitary waves the integration constant was discarded) F (cid:48) ( u ) u s + 6 P ( u ) = 0 where P = − λnm (cid:16) u m +1 m − u n +1 n (cid:17) + (cid:16) u m m − u n n (cid:17) , (16)and P ( u ) is the potential. Cancelling u m − on both sides begets m n − u n − m ) u s − (cid:16) nλm ( n + 1) (cid:17)(cid:16) n + 1 m + 1 − u n − m (cid:17) u − m + (cid:16) nm − u n − m (cid:17) u = 0 . (17)The singularity at u = 0 which causes degeneracy of the highest order operator and a localloss of uniqueness, allows us to construct compactons - solitary solutions with a compact7upport. Indeed, since the uniqueness of solutions of (16) is violated at u = 0, one may’glue’ there the nontrivial solution of (17) with the trivial u = 0 solution. This begets thecompactons depicted in Figs. 1,2.Now, for u = 1 to be an admissible solution we need the ’total force’ in (15) f . = λ u + F to vanish there. This defines the critical velocity λ n − m . (18)Turning to Eq. (17), which is also singular at u = 1, for u = 1 to be admissible as asolution the potential has also to vanish at this point. This imposes an additional constrainton the velocity λ n − m )( m + 1)( n + 1)2 nm . (19)Consistency demands that λ = λ , which constrains the admissible powers in F ( u ): n = m + 1 m − . (20)Thus, since 1 < m < n , ( m, n ) = (2 ,
3) emerge as the only pair of integers for which theconstraint (20) holds and thus supports formation of solutions incorporating singularities atboth u = 0 and u = 1 (the so-called kovatons [3, 6]).In fact, since P (cid:48) = F (cid:48) f and P (cid:48)(cid:48) = F (cid:48)(cid:48) f + F (cid:48) f (cid:48) , then at u = 1 where F (cid:48) = 0 and P (cid:48) = 0, for P (cid:48)(cid:48) to vanish as well, we need that both f and P vanish there with the samevelocity. When these conditions are satisfied, u s vanishes, and both compact kink/anti-kinkand u = 1 become admissible solutions, which underlines the formation of kovatons (theirexplicit construction will be provided shortly). Formally, − u s (cid:12)(cid:12) u =1 = P ( u ) F (cid:48) ( u ) (cid:12)(cid:12)(cid:12)(cid:12) u =1 = P (cid:48) ( u ) = F (cid:48) f F (cid:48) F (cid:48)(cid:48) (cid:12)(cid:12)(cid:12)(cid:12) u =1 = f F (cid:48)(cid:48) (cid:12)(cid:12)(cid:12)(cid:12) u =1 . Since F (cid:48)(cid:48) = − nm ( n − m ) (cid:54) = 0 at u = 1, therefore if both f and P vanish at the same velocity, u s will vanish as well.On the other hand, if the potential and the force vanish at u = 1 at different speeds, then u s (cid:54) = 0, u = 1 is not a solution and kovatons cannot form.The existence of kovatons may be embedded into a bit more general framework via theinvariance of the equations of motion, whether discrete or QC, under u → − u and F ( u ) → − F (1 − u ) + const., (21)8r simply into the condition F (cid:48) ( u ) = F (cid:48) (1 − u ). If F is a polynomial of degree n , one maydeduce the constraints for this condition to hold. For n = 3 we obtain the G23 model.Among quintic choices the used F ( u ) in the G35 model provides a counterexample whereas F (cid:48) = u (1 − u ) supports kovatons. A more general class admitting kovatons is afforded by F (cid:48) = g ( u ) g (1 − u ), g ( u = 0) = 0, where g may be any smooth function, or by F = sin n ( πu ),etc. C. The G23 case
Let m = 2 and n = 3. After cancelling of the common u factor, Eq. (17) reads32 (1 − u ) u s + P ( u ) = 0 where P ( u ) = − λ u + 38 (6 + λ ) u − u + u . (22)For λ <
2, Eq. (22) enables to calculate numerically the shape of the solitary traveling wave(compacton) and, due to the singularity at u = 0, match the periodic solution with thetrivial state. Several such solutions are displayed in Fig. 1. -0.3 0 0.3 0.6 -0.5 0 0.5 1(a) P ( u ) u -0.5 0 0.5 1 -6 -3 0 3 6(b) u ( s ) s FIG. 1. Panel (a): The effective potential P ( u ) / (1 − u ) for six values of λ (cid:48) s (from left to right: λ = − . , − . , , . ,
2) Note that only in the last case the effective potential remains bounded.Panel (b): compacton profiles for λ = − . , , . , . s = 3. For large and small amplitudes we may derive explicit expressions for the solitary waves.For small amplitudes, neglecting the cubic part in the equation of motion, the resultingcompacton takes a simple form u = 2 λ (cid:32) √ s (cid:33) H (cid:16) π − √ | s | (cid:17) ; , (23)9here H ( · ) is the Heaviside function. The invariance under u → − u , begets also compactdrops hanging from the u = 1 “ceiling”. Assuming 0 ≤ v = 1 − u to be small, we have u = 1 − λ (cid:32) √ s (cid:33) H (cid:16) π − √ | s | (cid:17) . (24)In the limiting λ = 2 case P (1) = P (cid:48) (1) = P (cid:48)(cid:48) (1) = 0, the problem simplifies(1 − u ) (cid:104) u s − u (1 − u ) (cid:105) = 0 where s = x − t. (25)Clearly, the singularity at u = 1 is now accessible and we obtain a kink and/or anti-kink of a finite span . Tied together back-to-back they form the basic kovaton (see Fig. 1) u = cos (cid:18) s √ (cid:19) H (cid:16)(cid:114) π − | s | (cid:17) . (26)Better yet, since u = 1 is now solution as well, we may form a combined three-some entity,a flat hat kovaton (see the right panel of Fig. 1): u kov ( s ) = A for − (cid:113) π ≤ s + s ≤
01 for | s | ≤ s A for 0 ≤ s − s ≤ (cid:113) π (27)where A = cos (cid:16) s −| s |√ (cid:17) H (cid:16)(cid:113) π + s − | s | (cid:17) and 0 < s is an arbitrarily chosen constant,determining top’s width. Again, due to problem’s invariance under u → − u , we also havean anti-kovaton u anti − kov = 1 − u kov . (28)Comparing basic kovaton’s (26) width with its small amplitude sibling in (23), we noticethat the kovaton is ∼
50 percent wider. Compacton’s widening with the amplitude indicatesthat the dispersion-convection balance tilts with amplitude toward the dispersion.For a later use we record the regime of large amplitude waves, wherein F has alreadychanged its sign and its quadratic part may be ignored, leaving us with only the cubic piece.In the resulting problem du k dt = 2( u k +1 − u k − ) , the solitary waves propagate to the left ( λ <
0) and their shape is easily derived via thecorresponding QC equation u ( s ) = ± (cid:18) | λ | (cid:19) / cos (cid:0)(cid:114) s + (cid:1) H (cid:0) π − (cid:114) | s + | (cid:1) where s + = x + | λ | t . (29)The ± sign expresses the reduced equation’s invariance under u k → − u k .10 . The G35 case We now consider the m = 3 and n = 5 case. Following the reduction by a common u factor in (17), we have(1 − u ) u s + P ( u ) = 0 where P ( u ) = − λ
10 (1 − u ) + 2 u − u ) . (30)Its compact solutions are displayed in Fig. 2. As in the previous case the singularity at u = 0, and the associated local loss of uniqueness, enable to glue the periodic solution withthe trivial state to form a compacton which though continuous has its first derivative jumpat u = 0. -0.3 0 0.3 0.6 -1 -0.5 0 0.5 1(a) P ( u ) u 0 0.5 1 -3 -2 -1 0 1 2 3(b) u ( s ) s FIG. 2. Panel (a): The effective potential P ( u ) / (1 − u ) for different values of λ (from topto bottom: λ = 1 , , , , . , . λ = 32 /
5, thus enabling the corresponding solution’s trajectory to cross it. Panel(b):compacton profiles for λ = 0 . , . , , . , . As in the other case, for small amplitudes one keeps only the lower order part in F ( u )with the explicit form of compactons, up to a normalization, given via (29). In the oppositecase of solitary waves with large amplitudes, keeping only the quintic part we have u ( s ) = ± (cid:18) | λ | (cid:19) / cos / (cid:16) √ s + (cid:17) H (2 π √ − | s + | ) where s + = x + | λ | t . (31)As in (29), the resulting compactons propagate to left, λ < u = 1 singularity . From (30) we have P ( u =1) = (16 / − λ ) /
30. At the critical velocity λ = 16 / P (1) = P (cid:48) (1) = 0 (since f (cid:54) = 0,11 (cid:48)(cid:48) (1) (cid:54) = 0). At the limiting velocity we thus have(1 − u ) (cid:104) u s −
6( 43 − u ) (cid:105) = 0 where s = x − t . (32)Though in the present case kovatons cannot form , at the limiting velocity, and only at thisvelocity, the singularity at u = 1 allows the regular solution trajectory to ’sneak through’and cross it (in u = 1 vicinity u ∼ as + bs + ... where a = ± √ and b is a constant) andthe limiting solution assumes a simple form u ( x,
0) = 2 √ (cid:16) √ x (cid:17) H (cid:16) π − √ | x | (cid:17) . (33)Notably, in addition to this exceptional solution, the singularity at u = 1 admits also asolution which is non-smooth at its top u ( x,
0) = 2 √ (cid:16) π √ | x | (cid:17) H (cid:16) π − √ | x | (cid:17) , (34)attained at u = 1 and is thus a compact peakon : it has a finite support, is everywherecontinuous, but its first derivative at the peak switches its sign ± / √
3. From Fig. 2 onealso notes that whereas the exceptional solution’s support undergoes a sizeable jump withrespect the the support of its speed-wise close neighbors, peakon’s support is a continuousextension of the support of its velocity-wise close neighbors and may thus be consideredtheir natural extension.
E. Relevance of a Quasi-Continuum based analysis
Before leaving the QC realm we need to clarify the role of the QC in elucidating thediscrete patterns, but first some basic facts. Whereas the singularity at u = 1 bounds thedomain accessible by the PDEs, as we shall shortly see, the discrete antecedents have no suchbarrier and u = 1 is merely a ’sign road’ of things to change. In fact, both the G23 and theG35 discrete problems have large amplitude solutions which are far beyond the access of theirrespective QC PDEs renditions. Yet though those PDEs failed to describe the dynamicsbeyond the u = 1 barrier, the QC approach may be still of use if applied separately to thesmall and large amplitude domains, see Eqs. (9) and solutions (23), (29),(31). There is nocontradiction here because the singularity is a barrier of the PDE’s and not of the originalproblem . We may bypass the ’mine field’ at u = 1 by splitting the PDE representation into12 ’sub-critical’ domain, valid up to the singularity, and a ’super-critical’ QC description,applied at large amplitudes, Eq. (9), where both F and F (cid:48) have already changed theirsigns, with the corresponding large amplitude solitary solutions recorded in (29) and (31),respectively.And yet, though the discrete problems appear formally to be oblivious of the singularbarriers, nonetheless those barriers appear to be somehow implicitly imprinted in the system(similar phenomenon was also observed in [5]). We shall find again and again that the moreinteresting action in the discrete realm takes place in a close vicinity of those singulartransitions unfolded by PDEs which per se are no longer valid there!Existence of a thin layer where the discrete effects are essential , is analogous to the emer-gence of shock waves in an ideal gas in a boundary layer where, whenever Euler equationsbreak down, one has to restore the viscosity, or better yet, to evoke the original gas-kineticdescription. Yet away from the breakdown zone Euler equations work well. Notably, in bothgas dynamics and in our problem, the more complex and intriguing phenomena occur in thetransition zone, marked by the ideal PDEs, yet described only via the original kinetic or thediscrete set-up to be unfolded next. IV. BASIC TRAVELING WAVES, TWS, ON A LATTICE.A. Integral Formulation
Returning to the original discrete problem we seek solitary traveling waves on the lat-tice (1) of the standard form u k ( t ) = U ( t − ka ) where a = 1 /λ . Substitution in (1) yields a delayed-advanced equation dUdt = F [ U ( t − a )] − F [ U ( t + a )] . Assuming that U ( ±∞ ) = 0, we integrate the last equation and set t = as to simplify theresulting nonlinear integral equation λU ( s ) + (cid:90) − F [ U ( s + s (cid:48) )] ds (cid:48) = 0 (35)which will be explored next. 13 . The Newton-Raphson Algorithm. Equation (35) is in a form which enables to apply the Newton-Raphson algorithm usingthe standard continuation approach [27]. The free parameter λ , was used to search forsolution’s branches. Assuming a given ( λ , U ) solution, to find a new one we appendEq. (35) with an auxiliary equation( λ − λ ) + ( N [ U ] − N [ U ]) = ∆ , where N [ U ] = (cid:90) U ( s ) ds is the norm of the solution, and ∆ a (small) shift parameter. A solution pair ( λ, U ) of theextended system is then sought via the Newton-Raphson algorithm. In practice we haveused ∆ = 10 − − − , and the integral in (35) was evaluated using the Simpson formula,with a typical step of ∆ s = 0 .
02. Solving the joint system for U ( s ) and λ we advancealong the solutions branch. Since the TW tails decay at a doubly-exponential rate [2, 6, 8],for all practical purpose their span may be considered finite. Therefore in choosing theintegration domain we take | s | ≤ L ; with L assumed to be an integer and chosen such that | U ( L − | < − (if this condition was violated the range was extended to L → L + 1).Once solution was found (only symmetric solutions, U ( − s ) = U ( s ), were explored), it wastested for stability via a direct numerical simulation of the original lattice equation on asmall lattice with its size being twice the size of the TW. If after 10 rotations the solutionremained unchanged, it was declared stable. C. Basic TW branches in the G23 model.
In Fig. 3 we present the branches of the TWs found in the G23 lattice (4). Their featuresmay be summarized as follows: • There is solutions branch which is very well described by the QC, Sec. III C (markedwith brown crosses in Fig. 3). The displayed solutions are noted at lettered pointsd,e and f in Fig. 4.
All waves along this branch are stable.
For small amplitudesEq. (4) may be approximated via (9) with α = 3, Being at this regime invariant under u → − u , t → − t , implies that small-amplitude compactons are nearly symmetric. • At large amplitudes wherein F ( u ) has already changed its sign, the unfolded branchhas negative velocities . The (29) may be viewed as its QC rendition. This branch relies14 N o r m [ U ( t ) ] velocitydddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa bccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 U ( ) FIG. 3. Basic traveling wave branches in the G23 model, Eq. (4). Green: stable waves, red:unstable waves. Brown crosses: values predicted by the QC theory. Note the green color whichhides behind the red in the upper branch of the upper plate and the solutions a and c which havethe same amplitude and velocity, but different norms (mass). on F ( u ) being negative and cannot be extended to small amplitudes/velocities. Thepatterns corresponding to lettered points a,b and c are depicted in Fig. 4. Notably,only waves with a larger norm are stable (in the upper amplitude-velocity diagramthe stable and the unstable branches nearly overlap, a much clearer distinction be-tween these branches is provided by the lower norm-velocity panel). Note also thaton the presented large amplitude branch, the quadratic part of F though small is notcompletely negligible which affects the proximity between the analytical and the nu-merical results. It may also effect wave’s stability, for unless the pulse is truly large,for a sizeable part of its profile F ( u ) is positive with this part’s tendency to propagateto the right. u has to be considerably over F (cid:48) s transition value 3 / -1 0 -2 0 2d 0 1 -2 0 2e 00.51 -4 0 4f FIG. 4. Basic traveling waves,TW, profiles U ( t ) at the corresponding lettered points on the diagramin Fig. 3. Red lines: profiles as function of time. Blue circles: snapshots on the lattice.Note thatTW velocity in cases (a), (c) and (d )is the same, though with case (c) being unstable, see Fig.(3),one may not be able to observe its propagation. D. Basic TW branches in the G35 model.
We now address the basic solitary TWs in the G35 model (5). Due to its u → − u symmetry, it suffices to display in Figs. 5,6 only waves with positive amplitudes, and skiptheir symmetric counterparts with negative amplitudes. The solitary waves exhibit thefollowing properties: • As in the G23 case, the G35 small amplitudes regime is very well described by the QC(Sec. III D). However, at larger amplitudes as one approaches the transition zone thefeatures of the found waves diverge considerably from their QC rendition. To recall,the QC begets a continuous branch of solutions with amplitudes ranging from zeroto one and the corresponding velocities in the [0 , /
5] range, with one exceptionalsolution which attains the maximal velocity 16 / / √ N o r m [ U ( t ) ] velocityaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeefffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 0 0.5 1 1.5 2 U ( ) FIG. 5. The basic TW branches in the G35 model, Eq. (5). Green: stable TWs, red: unstableTWs. Brown crosses show the corresponding values of TW in the QC rendition. The upper panelclearly shows an amplitude gap which may cause TW located close to the right edge of the leftbranch to ’hop’ to the right branch, with a consequent switch of direction, whether due to innerinstability or collision, if the resulting amplitude falls below the minimal admissible value of theleft branch. Such scenarios are displayed on Figs. 12 and 13. Fig. 14 shows a reverse scenario:collision between waves on the right branch causes one wave to hop to the left branch and thenhop back. And whereas depending on the direction of hoping wave’s amplitude may increase ordecrease, their mass hardly changes. of which the QC is completely oblivious, with amplitudes hovering slightly belowthe exceptional QC amplitude 2 / √ /
5, with widths which maybe chosen at will (see panel f in Fig 6). An enlarged display of the layer in thevicinity of point e in Fig. 5 is shown in Fig. 9. 2 Clearly, in this layer there is avery strong interaction between the non-linearity and the discreteness which has anessential impact on the resulting dynamics which the QC does not seem to be ableto reproduce. Moreover, whereas in the G23 model both the discrete problem and its17
FIG. 6. Display of the basic G35 TWs located at the corresponding lettered points in Fig. 5.Red lines: amplitude as a function of time. Blue circles: amplitude snapshots on the lattice.Surprisingly enough the amplitude of the flat-like case (f) is not (cid:112) / F ( u ) vanishes, butmuch closer to 2 / √
3, the amplitude of the only QC compacton solution, see Fig. 2, that sneaksthrough the u = 1 barrier, though its velocity 25 / / QC rendition yield kovatons residing on the singular manifold, as a flat-top solutionsof arbitrary width, the presented almost flat-top solutions on the G35 lattice (thoughthey appear to be unstable ), occur only in the discrete model. Notably, their amplitudeis close to 2 / √
3, the maximal QC amplitude and their speed to a high accuracy is25 / • As before, at large amplitudes there is another branch of discrete solutions (Fig 6,panels a,b,c) which may be approximated by the large amplitude QC solutions (31).We were able to extend these solutions down to smaller amplitudes and velocitiesclose to the c-labeled point(though their shape differs considerably from the simplecos form), but failed beyond it probably because, see panel c in Fig 6, at the branch’sedge the solution becomes non-smooth. • Similarly to the G23 case one notes the almost inverse relations between the norm(which up to a sign is wave’s mass) and amplitude’s response to changes in waves18elocity. This comes out in Figs. 12-15 where we display waves hoping from onebranch to the other. The jump from right (left) branch to the left (right) results inamplitude increase (decrease), but as Fig 5 clearly shows, hardly in any changes in itsmass.
V. INTERLACED TRAVELING WAVES, ITW
We now proceed to unfold a more evolved class of solitary traveling waves which havedistinct profiles at odd and even sites and appear as two interlaced solitary waves and maybe considered as a simple form of moving breathers; they will be referred to as an interlacedtraveling waves, ITWs. Denoting u k ( t ) = U (cid:18) t − k λ (cid:19) , u k +1 ( t ) = V (cid:18) t − k + 12 λ (cid:19) , begets a system of two coupled equations dUdt = F [ V ( t − λ − )] − F [ V ( t + λ − )] ,dVdt = F ( U ( t − λ − )) − F [ U ( t + λ − )] . Setting t = λ − s and following the same procedure as in Sec. IV A, yields a system of twocoupled integral equations λU ( s ) + (cid:90) − F [ V ( s + s (cid:48) )] ds (cid:48) = 0 ,λV ( s ) + (cid:90) − F [ U ( s + s (cid:48) )] ds (cid:48) = 0 . (36) A. Staggered Compactons
In systems with a F ( − x ) = − F ( x ) symmetry, a simple interlaced solution may be derivedfrom the basic TW. Setting V = − U reduces system (36) into one equation( − λ ) U ( s ) + (cid:90) − F [ U ( s + s (cid:48) )] ds (cid:48) = 0 , which coincides with (35). Thus any basic TW in a symmetric system produces an ITWwith an opposite velocity. Because the profiles at the odd and even sites are equal andopposite in sign, we shall refer to such wave as a ”staggered compacton” in analogy withstaggered solitons, cf. [28]. 19 . Interlaced traveling waves, ITW Applying the same numerical procedure, including stability analysis, as in the basic TWcase, we now proceed to unfold a more evolved branches of the interlaced traveling waves, -2 0 2 4 6 8 2 4 6 8 10 12 N o r m [ U ( t ) ], N o r m [ V ( t ) ] velocity abc-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 U ( ) , V ( ) FIG. 7. Branches of the ITW in the G23 model (4). Green: stable waves, red: unstable waves.The basic TW branch is also shown (in cyan).
The G23 model.
Figures 7,8 display the ITWs in the G23 lattice. Our understandingof these waves is based on their large amplitude domain wherein the waves have negativevelocities (thus propagate to the left) residing on a corresponding branch in Fig. 3. Inaddition, since at large amplitudes the system is anti-symmetric in u , in this limit it alsosupports staggered compactons which propagate to the right. This corresponds to the ITWsolution in Fig. 7, where at large positive velocities the solutions are seen to be nearlystaggered (panel a in Fig. 8). At smaller velocities/amplitudes, return of the quadratic partin F ( u ) to the game ruins the symmetry between odd and even sites (panel b in Fig. 8).Formally, since the staggering Ansatz leaves the ( − k factor in front of the quadratic part,its sign and thus its impact changes with the parity of k , causing the resulting amplitudesof odd and even sites to be different. Notably, this branch of solutions extends to velocities20 FIG. 8. Display of the G23 ITWs at the corresponding lettered points on the diagram in Fig. 7.Blue and magenta lines: profiles at even and odd sites as functions of the time. Green squares andred circles: snapshots on the lattice (when the center passes even and odd sites). Note the moreintriguing patterns in a and c which emerge in the vicinity of the QC critical speed. slightly under 2, where it co-exists with the basic TW branch, also depicted in Fig. 7. Thereseems to be some sort of “resonance” between these waves which induces large-norm solutionsat a velocity of ≈
2, with the resulting pattern having a remarkable flat-like appearance witha quasi-staggered compacton riding on its top (panel c in Fig. 8).
The G35 model.
Figures 9,10 display the branches of the ITWs in the G35 lattice. Thefirst thing to note is that the whole action takes place in a strip close to the critical speed16 / F ( u ) is still positive. As already aforementioned few times, in this domain thediscrete effects play an essential role. However, this ITW branch does not appear to be astaggered version of the basic TW. Instead, it seems to be a result of a certain symmetry-breaking in (36). At its tip this branch touches the basic TW’s branch. Close to the tipthe profiles at even and odd sites appear to be similar (panel a in Fig. 10). Along the ITWbranch the difference between the even and odd sites profiles increases, but never becomeslarge: they look like mirror cases of the flat-top solitons; one amplitude is above 1 whereasthe other under it (panels b and c in Fig. 10). Though it is tempting to relate the lastpair with the basic TW in Fig. 6, the later attains a higher velocity on the threshold of thecritical speed.Finally, we note that due to the anti-symmetry of the G35 lattice, any interlaced solu-tion ( U, V, λ ), like the ones in Figs. 9,10, always has an associated staggered counterpart( U, − V, − λ ). 21 N o r m [ U ( t ) ], N o r m [ V ( t ) ] velocityaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 0.8 1 1.2 1.4 U ( ) , V ( ) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc FIG. 9. The interlaced TW branches of the G35 model. Note that the main action takes placein the vicinity of the critical transition. Green: stable waves, red: unstable waves. The presentedwaves do not seem to be related to the symmetrically staggered compactons for, unlike the G23model, the amplitudes of U and V , being both are positive, are far from being reflection of eachother, see fig 10. The basic TW branch is also displayed for comparison (stable TWs in cyan,unstable TWs in magenta). VI. DIRECT NUMERICAL SIMULATIONS
Though in the numerical simulations of the lattice (3) we had no difficulty to observe theunfolded TW yet, rephrasing Tolstoy’s maxim, whereas all stable waves are similar, eachunstable wave is unstable in its own way, with this own way being of great interest. Casein point; though most of the unstable TW decompose within few time units, in direct nu-merical simulations of the interlaced TW of the G23 model (Fig. 7), several weakly unstablewaves persisted for a long time. For instance, a wave with the maximal velocity of λ ≈ t ≈
400 traversing for about 4800 sites while preserving its shape.The ITWs at the left side of the diagram in Fig. 7, with velocity λ = 2 and large norms,22 FIG. 10. Profiles of the interlaced TW corresponding to the lettered points in Fig. 9. Blue andmagenta lines: profiles at even and odd sites, respectively, as functions of the time. Green squaresand red circles: snapshots on the lattice (when wave’s center passes the even and the odd sites).
0 10 20 30 40 50 60 70 80 90 100time (in periods) 0 5 10 15 20 25 30 s pa c e -1-0.5 0 0.5 1 1.5 2 2.5 FIG. 11. G23 lattice. Space-time diagram of the evolution of an interlaced TW (case c in Fig. 8).It preserves its shape for about 75 periods(about 2250 sites). are also relatively stable or, if you will, only weakly unstable. This is demonstrated inFig. 11, where the waveform (displayed on a lattice of size 30 at each period, 30 /λ = 15)remains intact for at least 75 periods (i.e. up to t ≈ A. Reversal of propagation direction
A far more intriguing manifestation of the weak instability in action emerges in thenumerical simulations of TWs in the G35 lattice (Fig. 5) where we have noticed a weaklyunstable left-propagating wave, λ = − . reverses its direction turning nto a stable right-propagating TW . The same phenomenon is observed in the correspondingstaggered case (see the bottom panel of Fig. 12).
0 2 4 6 8 10 12 14 16time 0 5 10 15 20 25 30 s pa c e -0.5 0 0.5 1 1.5 2
0 2 4 6 8 10 12 14 16time 0 5 10 15 20 25 30 s pa c e -2-1.5-1-0.5 0 0.5 1 1.5 2 FIG. 12. Reversal of soliton’s path in the G35 model. Top panel: space-time diagram of a weaklyunstable TW to the left. Due to its instability and proximity to branch’s edge, when wave’samplitude becomes too low to be sustained by the left branch, it hops to the other branch with acorresponding change of propagation direction. The lower panel displays the corresponding scenariofor its staggered twin with its propagation in the opposite direction.
The key to the understanding of direction reversal is implicitly imprinted in Fig. 5 with itstwo panels complimenting each other. The upper panel displays an amplitude/velocity gapbetween the two branches wherein no propagation is admissible, with the waves propagatingin opposite directions on each side of the gap. However, the lower plate reveals that theamplitude/velocity jump is a non-event from the norm point of view, for the norm hardly24hanges after the event (note the almost opposite amplitude-velocity and norm-velocityrelations). The jump merely reshapes the wave’s profile, and consequently its amplitude,with the resulting direction of propagation being a byproduct of the branch on which it haslanded.With this understanding we may readily follow Fig. 13 where the two collisions betweenthe right and the left moving waves suffices to relegate the left-moving wave into the otherbranch. Fig. 14 is a natural continuation of this scenario: here we have a back and forthbouncing between the opposite branches, starting from the right branch on Fig. 5 and beingpushed after one collision to the other branch. Following the second collision both wavesare back at their original, right, branch and the game continues. This could have goneindefinitely would it not be for the fact that the collisions are not entirely elastic with somedebris created after each event.
0 2 4 6 8 10 12 14 16 18 20time 0 5 10 15 20 25 30 35 40 45 50 s pa c e -0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 FIG. 13. Collisions of a left-propagating (with amplitude 1 . . N = 50. After two collisions the left-propagating wave reverses its direction resulting in two right-propagating waves and some debrisdue to the collisions. Unfortunately, as Fig. 15 attests, the interaction is very sensitive to the collision phaseof the TW and its staggered twin. A shift by merely one lattice site in the initial positionof one of the waves causes the bouncing effect to disappear.In the G23 model the story is much simpler; no reversal of direction was ever observed.A weakly unstable TW located close to the right edge of the left branch, propagates for a25
0 5 10 15 20 25 30 35 40time 0 10 20 30 40 50 60 70 80 s pa c e -2-1.5-1-0.5 0 0.5 1 1.5 2(a) -1.5-1-0.5 0 0.5 1 1.5 0 5 10 15 20 25 30(b) u k ( t ) time site 10site 11site 50site 51 FIG. 14. G35 periodic lattice of length N = 80. Collisions of a right-propagating wave (withamplitude 1 . t ≈
10 transforms the right moving wave into a left-propagating wave on the left branch. With an opposite effect on its staggered twin. The nextcollision at t ≈
20 brings them back to their original branch. To visualize the ”bouncing” impacton waves shape, we display in panel (b) time series at two neighboring points pairs; k = 10 (red)with k = 11 (green) and k = 50 (blue) with k = 51 (brown). The red and green profiles atneighboring sites k = 10 ,
11 have opposite signs; as befits a staggered wave, whereas the blue andbrown profiles at sites k = 50 ,
51, being merely shifted, corroborate a basic traveling wave. Thechosen times cluster prior to the first, the second and the third collision. Though each encounteris accompanied by amplitude change - the mass (norm) itself hardly changes. while but then decomposes into disordered pieces with no observable coherent structure.26
0 5 10 15 20 25 30 35 40time 0 10 20 30 40 50 60 70 80 s pa c e -2-1.5-1-0.5 0 0.5 1 1.5 2 FIG. 15. Set-up similar to Fig. 14, but with the initial separation between the waves shifted by onesite. Now after one collision the right-propagating wave turns into a fast left-propagating wave,but its staggered. left-propagating, twin stays the course and turns into another left-propagatingstaggered wave.
VII. SUMMARY AND CLOSING COMMENTS
In the presented paper we have studied the traveling patterns of our strongly nonlinearlattice versions of the Gardner-like model equations G23 and G35, Eq. (4) and Eq. (5),respectively. Three distinctive classes of traveling and non-traveling solitary patterns wereunfolded1) Basic Traveling Waves,TW.2) Interlaced Traveling Waves, ITW.3) Stationary Solitary States.Of the three classes, the second and the third have no quasi-continuum, QC, counterpart, which is to say that space-wise they are essentially discrete phenomena, unlike thefirst class which may be and was very well replicated in part via the quasi-continuum. Yetin spite of what may seem on its face as a limited success of the QC, it does play a vital rolein delineating the road map without which most of the first and the second class solutionswould not have been unfolded.The utility of the QC extends well beyond its direct applicability, for though formally theoriginal discrete system seem oblivious of the singularities, a hall mark of the QC, it appears27hat those singular manifolds are implicitly embedded in the system and both control anddetermine the crucial transition zone, though the PDEs which beget them are not valid.Consider, for instance, Figs. 3 and 7 of the G23 model and Figs. 5 and 9 of G35 model,respectively. The role of the critical velocity λ = 2 which emerges in the QC is clearly seenin Fig. 3, but its role is even more impressive in Fig. 7 that describes the quintessentiallydiscrete ITWs, which do not have a corresponding QC counterpart (and thus are unrelatedto breathers found in the original Gardner equation), with the more interesting phenomenabeing clustered around the critical values. The same effect is seen in the G35 model with λ = 16 / λ = 2 line from both sides, with a respectivezoo of patterns, see Fig. 10. All in all we have an amalgam of essentially discrete patternsconcentrated around the macro constrains set by the QC.As a motivation for the present work we have stated that the polynomial force wasadopted because of the difficulty to treat the periodic case. Indeed, we were greatly helpedby the existence of large amplitude regimes in the polynomial cases which helped us to un-fold new wave branches and new types of waves. Unfortunately, in the periodic cases thereare no large amplitude regimes and with everything being eternally coupled the presentedresults offer only a limited help.The closing comment are reserved for the discrete stationary solutions briefly outlinedin Sec. II. They span a finite plateau defined by the the roots of F ( u ) = 0 and vanishinitially elsewhere (unlike the TWs and the ITWs which are associated with the roots of F (cid:48) ( u ) = 0). However, studies of their stability have revealed a very sensitive dependenceon their initial width, parity and lattice’s width as well, which did not yield itself to amanageable characterization. This topic is left for future studies. In passing we also noteanother class of kink-like excitations, not discussed in the paper, for which the roots of F (cid:48)(cid:48) ( u ) = 0 play a key role. 28 CKNOWLEDGMENTS
A.P. is supported in part by the Laboratory of Dynamical Systems and ApplicationsNRU HSE of the Ministry of Science and Higher Education of Russian Federation (GrantNo. 075-15-2019-1931). A. P. thanks A. Slunyaev for useful discussions. [1] S. Flach, Phys. Rev. E , 1503 (1995).[2] B. Dey, M. Eleftheriou, S. Flach, and G. P. Tsironis, Phys. Rev. E , 017601 (2001).[3] P. Rosenau and A. Pikovsky, Phys. Rev. Lett. , 174102 (2005).[4] P. Rosenau and S. Schochet, Phys. Rev. Lett. , 045503 (2005).[5] P. Rosenau and S. Schochet, Chaos: An Interdisciplinary Journal of Nonlinear Science ,015111 (2005).[6] A. Pikovsky and P. Rosenau, Physica D , 56 (2006).[7] E. B. Herbold and V. F. Nesterenko, Phys. Rev. E , 021304 (2007).[8] K. Ahnert and A. Pikovsky, Phys. Rev. E , 026209 (2009).[9] P. Rosenau and A. Pikovsky, Phys. Rev. E , 022924 (2014).[10] P. Rosenau and A. Zilburg, Phys. Lett. A , 2811 (2016).[11] A. Zilburg and P. Rosenau, J. Phys. A , 095101 (2015).[12] G. James, Phil. Trans. Roy. Soc. A: Mathematical, Physical and Engineering Sciences ,20170138 (2018).[13] P. Lax, personal communication.[14] D. Campbell, S. Flach, and Y. Kivshar, Physics Today , 43 (2004).[15] S. Flach and A. V. Gorbach, Physics Reports , 1 (2008).[16] S. Sen, J. Hong, J. Bang, E. Avalos, and R. Doney, Physics Reports , 21 (2008).[17] P. Rosenau and A. Pikovsky, Chaos , 053119 (2020).[18] A. Slunyaev, Sov. Physics JETP , 529 (2001).[19] R. Grimshaw, D. Pelinovsky, E. Pelinovsky, and A. Slunyaev, CHAOS , 1070 (2010).[20] A. M. Kamchatnov, Y.-H. Kuo, T.-C. Lin, T.-L. Horng, S.-C. Gou, R. Clift, G. A. El, andR. H. J. Grimshaw, Phys. Rev. E , 036605 (2012).[21] K. Ahnert and A. Pikovsky, CHAOS , 037118 (2008).
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