Compacton solutions and (non)integrability for nonlinear evolutionary PDEs associated with a chain of prestressed granules
aa r X i v : . [ n li n . S I] S e p Compacton solutions and (non)integrability of nonlinear evolutionaryPDEs associated with a chain of prestressed granules
A. Sergyeyev a , S. Skurativskyi b , V. Vladimirov ca Mathematical Institute, Silesian University in Opava,Na Rybn´ıˇcku 1, 746 01 Opava, Czech Republic b Subbotin Institute of Geophysics of NAS of Ukraine,Acad. Palladina Ave. 32, 03142 Kyiv, Ukraine, c Faculty of Applied Mathematics,AGH University of Science and Technology,Al. Mickiewicza 30, 30059 Krak´ow, PolandE-mails:
[email protected], [email protected] , [email protected] September 21, 2018
Abstract
We present the results of study of a nonlinear evolutionary PDE (more precisely, a one-parameter family of PDEs) associated with the chain of pre-stressed granules. The PDE inquestion supports solitary waves of compression and rarefaction (bright and dark compactons)and can be written in Hamiltonian form. We investigate inter alia integrability properties ofthis PDE and its generalized symmetries and conservation laws.For the compacton solutions we perform a stability test followed by the numerical study. Inparticular, we simulate the temporal evolution of a single compacton, and the interactions ofcompacton pairs. The results of numerical simulations performed for our model are comparedwith the numerical evolution of corresponding Cauchy data for the discrete model of chain ofpre-stressed elastic granules.
Keywords: chains of pre-stressed granules; compactons; integrable systems; symmetryintegrability; symmetries; conservation laws; stability test; conserved quantities; Hamiltonianstructures; numerical simulation
MSC 2010
This paper deals with nonlinear evolutionary PDEs associated with dynamics of a one-dimensionalchain of pre-stressed granules which arises in quite a number of applications. Since Nesterenko’spioneering works [1, 2] propagation of pulses in such media has been a subject of a great number ofexperimental studies and numerical simulations, see [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and referencestherein. We consider a nonlinear evolutionary PDE which is derived from the infinite system ofODEs describing the dynamics of one-dimensional chain of elastic bodies interacting with each otherby means of a nonlinear force. The PDE in question is obtained through the passage to continuumlimit followed by the formal multi-scale decomposition.1he PDE under study turns out to admit a Hamiltonian representation and possess localizedtraveling wave solutions manifesting some features of solitons. For this reason, it is of interestto investigate its complete integrability. We do this below along with the study of generalizedsymmetries and conservation laws. We show below that the compacton traveling wave (for travelingwaves in general see e.g. [14, 15] and references therein) solutions satisfy the necessary condition forthe extremum of a functional associated with the Hamiltonian. Using this we also perform a stabilitytest followed by the numerical study of the compacton solutions. Somewhat surprisingly, numericalsimulations show that even in a nonintegrable case the compacton solutions recover their shapesafter the collisions, yet the dynamics of interaction slightly differs from that of KdV solitons. In thisconnection note that compactons, i.e. soliton-like solutions with compact support, see [1, 3, 16] andreferences therein, exist for a number of physically relevant models and possess several interestingfeatures making them a subject of intense research, cf. e.g. [17, 18, 19, 20, 21, 22] and referencestherein.The paper is organized as follows. In section 2 we introduce the continual analog of the granularpre-stressed media with the specific interaction of the adjacent blocks which allows for the descriptionof both the waves of compression and rarefaction. In section 3 we present the Hamiltonian structure ofthe equation in question. In section 4 we study the conservation laws admitted by the said equation.In section 5 we perform the integrability test that singled out an exceptional integrable case, whichis studied in more detail in section 6. In section 7 we show that the compacton traveling wave (TW)solutions that satisfy factorized equations also satisfy necessary conditions of extrema for the appro-priate Lagrange functionals. Next we perform stability tests for compacton solutions based on theapproach developed in [23, 24, 25], and show that both dark and bright compactons pass the stabilitytest. The results of qualitative analysis are backed and partly supplemented by the numerical studyperformed in section 8. We also present the results of numerical simulation of the Cauchy problemfor discrete chains and compare the results obtained with the analogous simulations performed forthe continual analogue of these chains. The closing section 9 contains conclusions and discussion.
Amazing features of the solitons associated with the celebrated Korteweg–de Vries (KdV) equation,as well as other completely integrable models [14], are often ascribed to the existence of highersymmetries and infinite sets of conservation laws, cf. e.g. [26, 27, 28]. However, there exist non-integrable equations possessing localized TW solutions with quite similar behavior. A well-knownexample of this is provided by the K ( m, n ) equations [16]: K ( m, n ) : u t + ( u m ) x + ( u n ) xxx = 0 , m ≥ , n ≥ . (1)The members of this hierarchy are not completely integrable at least for generic values of the param-eters m , n , see [17, 20] and references therein, and yet possess compactly-supported TW solutionsexhibiting solitonic features [16, 29].The K ( m, n ) family was introduced in the 1990s as a formal generalization of the KdV hierarchywithout referring to its physical context. Earlier V.F. Nesterenko [1] considered the dynamics of achain of preloaded granules described by the following ODE system:¨ Q k ( t ) = F ( Q k − − Q k ) − F ( Q k − Q k +1 ) , k ∈ { , ± , ± . . . . } (2)where Q k ( t ) is the displacement of the k th granule center-of-mass from its equilibrium position, F ( z ) = Az n , n > . (3)2e has described for the first time the formation of localized wave patterns and evolution within thismodel [1, 2, 3]. In [1, 2, 5] he presented the nonlinear evolutionary PDEs being the quasi-continuallimits of the discrete models; in this connection cf. also [30].The transition to the continual model is achieved via the substitution Q k ( t ) = u ( t, k · a ) ≈ u ( t, x ) , (4)where a is the average distance between granules. Insert this formula, together with the substitutions Q k ± = u ( t, x ± a ) = exp( ± aD x ) u ( t, x )= X j =0 ( ± a ) j j ! ∂ j ∂x j u ( t, x ) + O (cid:0) a (cid:1) , (5)into (2), and observe that the term of lowest order in a on the right-hand side of (2) is proportionalto a n +1 . Now expanding the right-hand side of (2) divided by a n +1 into the (formal) Taylor seriesand then dropping the terms of the order O ( a ) and higher in this expansion yields from (2) theequation u tt = − C n ( − u x ) n + β ( − u x ) n − h ( − u x ) n +12 i xx o x , where C = Aa n +1 , β = na n + 1) . Differentiating the above equation with respect to x and employing the new variable S = ( − u x )corresponding to the strain field, one obtains the Nesterenko equation [5]: S tt = C n S n + βS n − h S n +12 i xx o xx . (6)Eq. (6) was derived using only one small parameter corresponding to the long wave approximation.Thus it can describe the dynamics of “strongly preloaded media” with dynamic amplitude muchsmaller than the preload or the dynamics of “weakly preloaded media” when the dynamic amplitudein the wave is much larger than the preload or even when the preload is equal to zero, in whichcase the propagation of acoustic waves is impossible (the effect of “sonic vacuum” [2]). As it isshown in [5], equation (6) possesses a one-parameter family of compacton TW solutions describingthe propagation of the waves of compression.Unfortunately, the compacton solutions supported by (6) are unstable. This can be verified bya direct numerical calculation, substituting in the corresponding difference scheme as Cauchy dataknown compacton solutions.A similar situation occurs in the case of the Boussinesq equation, obtained as a continuum limit ofthe Fermi–Pasta–Ulam system of coupled oscillators [14]. As is well known, the Boussinesq equationpossesses unstable soliton-like solutions, and the KdV equation, supporting the stable uni-directionalsolitons, is extracted from the Boussinesq equation by means of the asymptotic multi-scale expansion[14], cf. also [31] and references therein.In this connection it should be also noted that the instability caused by the short wavelengthscan be removed using the regularization consisting in replacing the space derivatives of the forceby mixed space and time derivatives. The regularized equation for the case of general power law isnothing but Eq. (1.110) from [5], and its counterpart for a general interaction law is Eq. (1.156) from[5]. It should be further noted that, at least for n = 3 /
2, equation (6) has stationary compactonsolutions which are close to the numerical solutions of the discrete Hertzian chain, see e.g. [8], and3he numerical simulations and experiments strongly suggest that the latter solutions are stable. Forexample, such solutions are generated from various initial conditions on short distances from thedisturbed end and propagate in experiments despite disturbances due to inevitable dissipation andviolation of periodicity, see e.g. experimental results in [5].Another interesting observation is that the conditions for existence of solitary waves in discretechain [32] and in the continuum approximation, see Eq.(1.154) at p. 108 in [5], are identical andbased on the sign of the second derivative of the force, see p. 113 in [5].Our approach to finding a “proper” compacton-supporting equation is as follows. We start fromthe discrete system (2) in which the interaction force has the form F ( z ) = Az n + Bz. (7)In addition, we assume that B = γa n +3 , where | γ | = O ( | A | ).The interaction law in (7) is a special case of general interaction law that results in long waveequation, Eq.(1.154) at p. 108 in [5] or its regularized counterpart, Eq.(1.156) in [5]. The stationarysolutions of the said long wave equation are studied in [5], where, depending on the behavior of thesecond derivative of the interaction law, strongly nonlinear compression or rarefaction solitary wavesare predicted.In this connection also note that for small deformations the interaction law in (7) is a special caseof the situation where the first derivative is nonzero and higher derivatives are zero except for the n -th order one which, in conjunction with the discussion in the preceding paragraph implies, cf. [5],in particular p.110–123, that the linear part of the interaction law is, to an extent, irrelevant for thestudy of qualitative behavior of stationary solutions.Inserting (4), (5) into the formula (2) and assuming that the interaction is described by (7), weobtain, up to the terms of the order O ( a ) and higher in the expansion of the right-hand side of (2)divided by a n +1 , cf. the discussion after (5), the equation u tt = − C n ( − u x ) n + β ( − u x ) n − h ( − u x ) n +12 i xx o x − γa n +3 ( − u x ) x . Differentiating the above equation with respect to x and introducing the new variable S = ( − u x ),we obtain the following equation: S tt = C n S n + βS n − h S n +12 i xx o xx + γa n +3 S xx . (8)Now we use a series of scaling transformations. Employing the scaling τ = p γa n +3 t enables us torewrite the above equation in the form S ττ = Cγa n +3 n S n + βS n − h S n +12 i xx o xx + S xx . Next, the transformation ¯ T = a q τ , ξ = a p ( x − τ ), S = a r W is used. If, for example, we assignthe following values to the parameters q = 1, p = − r = 5 /n, then the higher-order coefficient O ( a ) will be that of the second derivative with respect to ¯ T . So, dropping the terms proportionalto O ( a ), we obtain, after the integration with respect to ξ , the equation: W ¯ T + Aγ (cid:26) W n + n n + 1) W n − h W n +12 i ξξ (cid:27) ξ = 0 . Performing the rescaling and returning to the initial notation t = Aγ L ¯ T , x = Lξ, L = q n +1) n , we finally obtain the sought-for equation W T + n W n + W n − h W n +12 i XX o X = 0 , (9)to which we shall hereinafter refer as to the Nesterenko equation . Note that Eq.(9) appears in [19](see also [22]) as a particular case of the C ( m, a + b ) hierarchy introduced as a generalization of theset of K ( m, n ) equations.The description of waves of rarefaction in the case n = 2 k requires the following modification ofthe interaction force: F ( z ) = − Az k + Bz (10)(for n = 2 k +1 the formula (7) describes automatically both waves of compression and of rarefaction).Applying the above machinery to (2) with the interaction (10), we obtain, in the same notation, theequation W T − n W n + W n − h W n +12 i XX o X = 0 , n = 2 k. (11)Thus, the universal equation describing waves of compression and rarefaction for arbitrary n ∈ N can be written in the form W T + [sgn( W )] n +1 n W n + W n − h W n +12 i XX o X = 0 . (12)In closing note that equations (9), (11) and (12) are obtained by formal application of the multi-scale decomposition method which cannot be substantiated in our case because of negativity of theindex p , cf. [33] where this problem is discussed in a more general fashion. Further study of theseequations is justified by the fact that they possess a set of compacton solutions possessing interestingdynamical features. As will be shown below, these solutions describe well enough propagation ofshort impulses in the chain of pre-stressed blocks. Now return to (9) which we now write in the manifestly evolutionary form, that is, W T = − (cid:0) W n + W ( n − / (cid:2) W ( n +1) / (cid:3) XX (cid:1) X (13)Note that for n = − W T = ( W − ) X which is obviously integrable, and for n = 1 equation (13) becomes linear.Equation (13) can be written (cf. [22]) as W T = D X δ H Nest /δW ≡ F. (14)Thus, (14) is written in Hamiltonian form with the Hamiltonian H Nest and the Hamiltonian structure P = D X .This implies, in particular, that to any nontrivial local conserved density of (14) there correspondsa (generalized, but not necessarily genuinely generalized (see the definition below), and possiblytrivial) symmetry of (14).Here δ/δW is the variational derivative (see below for details) and H Nest = Z h Nest dX with thedensity h Nest = (cid:18)
14 ( n + 1) W n − W X − W n +1 / ( n + 1) (cid:19) for n = − , ln | W | for n = −
1. (15)5ere and below the integrals are understood in the sense of formal calculus of variations, see e.g. [28,34]. Here we put, cf. [27, 28, 34], W j = ∂ j W/∂X j , j = 1 , , . . . , W ≡ W , and define [26, 27, 28, 34]the total derivatives D X = ∂∂X + ∞ X j =0 W j +1 ∂∂W j , D T = ∂∂T + ∞ X j =0 D jX ( F ) ∂∂W j . (16)The variational derivative of a functional F = Z f ( X, T, W, W , . . . , W k ) dX has the form δ F δW = ∞ X j =0 ( − D X ) j (cid:18) ∂f∂W j (cid:19) . (17)For any f = f ( X, T, W, . . . , W k ) we also define, cf. e.g. [27, 28], its linearization f ∗ = k X j =0 ∂f∂W j D jX . Recall, cf. e.g. [26, 27, 28, 34, 35, 36, 37, 38] and references therein, that a local conservation law for(13) is, roughly speaking, a relation of the form D T ( ρ ) = D X ( σ ) , (18)where ρ = ρ ( X, T, W, W , . . . , W r ) and σ = σ ( X, T, W, W , . . . , W s ), which holds by virtue of (13).Here ρ and σ are called a (conserved) density and the flux of our conservation law.Also recall, cf. e.g. [36], that a conservation law (18) is called nontrivial if there exists no function ζ ( X, T, W, W , . . . , W q ) such that ρ = D X ζ , i.e., ρ Im D X .It is well known, see e.g. [28, 34], that a necessary and sufficient condition for a function f = f ( X, T, W, W , . . . , W r ) to not belong to the image of D X is E W f = 0, where E W is the Euler operator E W = ∞ X j =0 ( − D X ) j ◦ ∂∂W j . Hence ρ is a conserved density for (13) if and only if E W D T ( ρ ) = 0, and this density is nontrivialif and only if E W ρ = 0.It is readily checked that we have the following Proposition 1.
For any n equation (13) admits the following three conserved densities: ρ = W, ρ = W / , ρ = h Nest . (19) For n = 0 we have an extra density ρ = X W W X − T W X /W. (20) Moreover, for n = 0 , , − , − (resp. for n = 0 ) the densities (19) (resp. (19 and (20)) exhaust,modulo the addition of trivial ones, the linearly independent conserved densities of order up to five,i.e., of the form ρ = ρ ( X, T, W, W X , . . . , W XXXXX ) .
6t is very likely that for n = 1 , − , − n = 1 , − , − ρ is the density of the Hamiltonian H Nest for (13) with respect to the Hamiltonianstructure P = D X . To the functional C = Z W dX there corresponds a trivial symmetry, i.e., asymmetry with zero characteristic, as D X δ C /δW = 0, so C is a Casimir functional for P . To thefunctional P = Z W dX there corresponds a symmetry with the characteristic W X = D X δ P /δW ,that is, X -translation, and to H Nest there corresponds a symmetry with the characteristic equal tothe r.h.s. F of (14), i.e., the time translation symmetry.For n = 0 to the conserved functional H = Z ρ dX there corresponds a scaling symmetry withthe characteristic 4 T F +2 XW X +2 W = D X δ H /δW . Again, it is very likely that ρ i , i = 0 , . . . ,
3, arethe only local conserved densities (modulo trivial ones) for (13) with n = 0 in view of nonintegrabilityof this special case of (13). Integrable equations of the form (14) with the Hamiltonian of general form H = Z dXh ( W, W X )where the density h = h ( W, W X ) is such that ∂ h/∂W X = 0 were classified (modulo point transfor-mations leaving T invariant) in [39]. Note that in [27, 39] and references therein integrability of anevolution equation W T = K ( X, W, W X , . . . , ∂ k W/∂X k ) (21)with k ≥ T . In order to avoid ambiguity we shall, following the commonusage, refer below to this kind of integrability as to the symmetry integrability .Recall, cf. e.g. [26, 27, 28, 34], that a generalized symmetry of order r for (21) is a function G = G ( X, T, W, W , . . . , W r ) such that ∂G/∂W r = 0 and D T ( G ) = K ∗ ( G ) , (22)where now D T = ∂∂T + ∞ X j =0 D jX ( K ) ∂∂W j .Such a symmetry G is known as genuinely generalized if it cannot be written in the form G = c ( T ) K + b ( X, T, W, W X ) for some functions b and c , that is, it is not equivalent to a pointor contact symmetry. As far as point symmetries of the equations studied in the present paper, and,more broadly, of C ( m, a, b ) equations (see e.g. [19, 22]), cf. e.g. [40] and references therein.Thus, symmetry integrability of (21) means existence of an infinite hierarchy of generalized sym-metries of the form G i ( X, W, W , . . . , W r i ) of increasing orders r i .Now turn to comparison of the density h Nest of our Hamiltonian and the densities h found in [39]for which the equation W T = D X ( δ H /δW ) with the general Hamiltonian H = Z h ( W, W X ) dX issymmetry integrable. Proposition 2.
The only symmetry integrable case of (13) which is genuinely nonlinear and gen-uinely of third order is that of n = − . For the sake of simplicity and without loss of generality we identify here a generalized symmetry with its charac-teristic. roof. It is not difficult to observe (cf. e.g. [41]) that using point transformations leaving t invariantthe density h Nest of our Hamiltonian for n = − h = W X / (2 f ) − P/f, (23)where f = c + c W + c W , P = P i =0 d i W i , and c i and d i are arbitrary constants.Moreover, it is clear that in our case f should actually be a monomial: f = cW α , α = 0 , , W X in (23) and (15) modulo an obvious rescaling of W , wesee that all values of n for which (13) could be integrable should satisfy n − , − , −
6. Thecase of n = 1 is trivially integrable, as then (13) is just a linear equation, so we are left with twopossibilities n = − n = − α = 1 and α = 2.Now upon inspecting the remaining terms in h Nest and in (23) we readily conclude that thepolynomial P should also reduce to a single monomial: P = dW β , where β = 0 , , , ,
4, so we havea system n − − α and n + 1 = β − α , where α = 1 , β = 0 , , , ,
4. An obvious corollaryof this system is − α + 2 = β − α , whence β = 2(1 − α ). However, β ≥ n = −
5, when α = 2 and we should have β = −
2, is not integrable.Thus, the only integrable case of (13) which is genuinely nonlinear and genuinely of third orderis that of n = −
2, and the result follows. (cid:3) .Recall that for n = − n = 1 equation (13) is just linear.In fact, the result of Proposition 2 can be further strengthened so that absence of any generalizedsymmetries, rather than just those that do not depend explicitly on T , can be established.To this end consider, following [27], the so-called canonical density ρ − = ( ∂F/∂W XXX ) − / . Itis readily checked that E W D T ( ρ − ) = 0 for n = − , − , − ,
1. Hence for n = − , − , − , D T ( ρ − ) Im D X , and thus ρ − is not a density of a local conservation law for (13).In turn, by virtue of the results from [42] this immediately implies Proposition 3.
Equation (13) for n = 1 , − , − , − has no generalized symmetries of order greaterthan three. In other words, Proposition 3 means that for n = 1 , − , − , − G = G ( X, T, W, W , . . . , W r )of the equation D T ( G ) = F ∗ ( G ) , (24)where D T and F are given in (16) and (14), in fact depends at most on X, T, W, W X , W XX , W XXX .This implies that (13) for n = − , − , − , n = − , − , − , n = ±
1, turn to the remaining two special cases: n = − n = −
5. We believe that using the technique similar to that of [43] (cf. also [20, 44]) it can beshown that in the case of n = − T and not just the time-independent ones whose nonexistencefollows from the above comparison of (15) with (23), so we are left with just one integrable case of n = − n = − : integrability and beyond The following result is readily checked by straightforward computation:8 roposition 4.
For n = − equation (13) has a Lax pair of the form ψ XX = (1 + W λ ) ψ, ψ T = 2 W ψ XXX − W X W ψ XX + 2 W ψ X − W X W ψ (25) and admits a recursion operator R = 1 W D X − W X W D X + (4 W + 6 W X − W XX ) W − (cid:0) W − + W − / (cid:2) W − / (cid:3) XX (cid:1) X D − X (26)The recursion operator (26) can be found e.g. using the technique from [45] (cf. also [46]). Alsonote that upon passing to a new dependent variable equal to a square of W the first equation of (25)can be identified with a special case of the eigenvalue problem related to the extended Harry Dymsystems, see e.g. [47] and references therein.Equation (13) for n = − P = R ◦ D x , that is, P = 1 W D X − W X W D X + (4 W + 6 W X − W XX ) W D X − (cid:0) W − + W − / (cid:2) W − / (cid:3) XX (cid:1) X which is compatible with P = D X , so the recursion operator R is hereditary and equation (13) for n = − second Hamiltonian form as W T = P ( δ ˜ h/δW ) , (27)where ˜ h = W/ Proposition 5.
Equation (13) for n = − is an integrable bihamiltonian system with two localHamiltonian operators P and P and two local Hamiltonian representations (14) and (27). Using general theory of bihamiltonian systems (see e.g. [28, Ch. 7] and [48]), we also readilyobtain
Corollary 1.
Equation (13) for n = − possesses an infinite hierarchy of commuting generalizedsymmetries of the form R k W X , k = 0 , , , . . . and an infinite hierarchy of local conservation lawswhose densities h j are generated recursively through the relations P ( δh j +1 /δW ) = P ( δh j /δW ) , where j = 0 , , , . . . and h = W/ , and of associated integrals of motion H j = R h j dX in involutionwith respect to the two Poisson brackets associated with P and P . The fact that the generalized symmetries R k W X and the conserved densities h k for k = 0 , , , . . . do not involve any nonlocal terms can be established using the results of [49] or [50] (cf. also [46]).As we have already pointed out above, up to a suitable rescaling of T and obvious change ofnotation equation (14) for n = − S -integrable Calogero–Degasperis–Fokas [51, 52]equation in the manner described therein.Namely, pass first to the potential form of (14) with n = − V T = − V XXX V X + 3 V XX V X + 1 V X , W = V X .The subsequent hodograph transformation interchanging X and V turns the above equation intoa constant separant equation V T = − V XXX V X V XX V − V X (4 + V X )4 V , or, upon a suitable rescaling of T , V T = V XXX − V X V XX V + 3 V X (4 + V X ) V . (28)Finally, putting V = exp( U/
2) turns (28) into a special case of the Calogero–Degasperis–Fokas[51, 52] equation, viz. , U T = U XXX − U X + 6 U X exp( − U ) . (29) Consider the pair of equations (9), (11), which can be represented by the single expression W T + ǫ n W n + W n − h W n +12 i XX o X = 0 , ǫ = ± . (30)As we are interested in the traveling wave (TW) solutions W = W ( z ) ≡ W ( X − cT ), it is convenientto pass to the TW coordinates T → T , X → z = X − cT . This change of variables yields from (30)the equation W T − cW z + ǫ n W n + W n − h W n +12 i zz o z = 0 . (31)It is easy to check that equation (31) admits a Hamiltonian formulation W T = D z δ ( ǫ H Nest + c P ) /δW , (32)where now H Nest = Z h Nest dz, P = Z W dz, and h Nest = (cid:18)
14 ( n + 1) W n − W z − W n +1 / ( n + 1) (cid:19) for n = − , ln | W | for n = − ǫ follows directly from the Hamiltonian form (cf.(14)) of equation (9) after the change of coordinates. Recall that both functionals H Nest and P areconserved in time.Now consider the following functions: W ǫc ( z ) = ǫW c ( z ) = ( ǫM cos γ ( Kz ) , if | Kz | < π , , (33)where ǫ = ± M = (cid:20) c ( n + 1)2 (cid:21) n − , K = n − n + 1 , γ = 2 n − . It is readily checked that we have the following 10 roposition 6. If n = 2 k + 1 , k ∈ N , then the functions W ± c ( z ) are weak solutions to the equation δ ( H Nest + c P ) /δW | W = W ± c = 0 . (34) If n = 2 k , k ∈ N , then the functions W ± c ( z ) are weak solutions to the equation δ ( ±H Nest + c P ) /δW | W = W ± c = 0 . (35)So, the TW solutions (33) are the critical points of either the Lagrange functional Λ = H Nest + β P (the case of n = 2 k + 1) or Λ ǫ = ǫ H Nest + β P (the case of n = 2 k ) with the common Lagrange mul-tiplier β = c . As is well known, necessary and sufficient condition for Λ (resp. Λ ǫ ) to attain theminimum on the compacton solutions can be stated in terms of the positivity of the second variationof the corresponding functional, which, in turn, guarantees the orbital stability of the TW solution[53]. Here we do not touch upon the problem of strict estimating of the signs of the second variations.We follow instead the approach suggested in [23, 24, 25], which enables us to test the possibility ofappearance of the local minimum on selected sets of perturbations of TW solutions.Consider the following family of perturbations W ǫc ( z ) → λ α W ǫc ( λz ) . (36)Upon choosing α = 1 / P [ λ ] = 12 Z π/ − π/ h λ W ǫc ( λz ) i dz = P [1] . (37)Thus, for this choice P [ λ ] keeps its unperturbed value. By imposing this condition we reject “fake”perturbations associated with the translational symmetry T δ [ W ǫc ( z )] = W ǫc ( z + δ ). Indeed, sinceequations (34), (35) are invariant under the shift z → z + δ , T δ W ǫc ( z ) belongs to the set of solutionsas well, while formally the transformation W ǫc ( z ) → W ǫc ( z + δ ) can be treated as a perturbation. Inorder to exclude the perturbations of this sort, the orthogonality condition is imposed. Introducingthe representation for the perturbed solution W ǫc ( z )[ λ ] = W ǫc ( z ) + v ( z, λ ) , and using the condition (37), we find0 = P [ λ ] − P [1] = Z π/ (2 K ) − π/ (2 K ) W ǫc ( z ) v ( z, λ ) dz + O (cid:0) || v ( z, λ ) || (cid:1) , so if P is independent of λ , then, up to O ( || v ( z, λ ) || ) the perturbation created by the scalingtransformation is orthogonal to the TW solution.For α = 1 / n ∈ N , we arrive at the following functions to be tested:Λ ν [ λ ] = ( ν H Nest + c P )[ λ ] = ν n λ n +32 I ǫn − λ n − J ǫn o + c P , (38)where I ǫn = n + 14 Z π/ (2 K ) − π/ (2 K ) [ W ǫc ] n − [( W ǫc ) z ] dz, J ǫn = 1 n + 1 Z π/ (2 K ) − π/ (2 K ) [ W ǫc ] n +1 dz,ν = ǫ n +1 = ( +1 if n = 2 k + 1 ,ǫ if n = 2 k.
11f the functional Λ ν = ν H Nest + c P attains the extremal value on the compacton solution, then thefunction Λ ν [ λ ] has the corresponding extremum in the point λ = 1. The verification of this propertyis employed as a test.A necessary condition for the extremum ddλ Λ ν [ λ ] (cid:12)(cid:12)(cid:12) λ =1 = 0 gives us the equality I ǫn = n − n + 3 J ǫn . (39)Using (39), we obtain the estimate d dλ Λ ν [ λ ] (cid:12)(cid:12)(cid:12) λ =1 = ν ( n − J ǫn = n − n + 1 ǫ n +1) Z [ W ǫc ] n +1 ( z ) dz > , which is valid for both n = 2 k + 1 and n = 2 k . Thus, the generalized solutions (33) pass the test forstability, and we can state the following Conjecture.
For n > weak solutions (33) provide minima of the functional Λ ν . Further information about the properties of the compacton solutions is provided by the numericalsimulations discussed below.
The dynamics of solitary waves is studied by means of direct numerical simulation based on thefinite-difference scheme. a) b)Figure 1: Numerical evolution of a single compacton solution of Eq. (40) characterized by thevelocity c = 1 (a) and a pair of compacton solutions characterized by the velocities c = 1 and c = 1 / εW x , where ε is a small parameter. Thus, instead of (9) wehave for the case of n = 3 the following equation: W t + (cid:8) W (cid:9) x + (cid:8) W (cid:2) W (cid:3) xx (cid:9) x + εW x = 0 . (40)12igure 2: Numerical evolution of a pair of dark compactons characterized by the velocities c = 1 and c = 1 /
4, respectively.Let us approximate the spatial derivatives as follows:1120 ( ˙ W j − + 26 ˙ W j − + 66 ˙ W j + 26 ˙ W j +1 + ˙ W j +2 )++ 124 h ( − W j − − W j − + 10 W j +1 + W j +2 )++ 124 h ( − L j − − L j − + 10 L j +1 + L j +2 )++ ε h ( W j − − W j − + 6 W j − W j +1 + W j +2 ) = 0 , (41)where L j = W j W j − − W j + W j +2 h .To integrate the system (41) in time, we use the midpoint method. Then the quantities W j and˙ W j are represented in the form W j → W n +1 j + W nj , ˙ W j → W n +1 j − W nj τ . The resulting nonlinear algebraic system with respect to W n +1 j can be solved by iterative methods.We test the scheme (41) by considering the movement of a single compacton. Assume that themodel parameters c = 1 and the scheme parameters N = 600, h = 30 /N , τ = 0 . ε = 10 − arefixed. The application of the scheme (41) gives us fig. 1a.The starting profile providing the initial condition for the numerical scheme is chosen accordingto (33) where n = 3, c = 1 and c = 1 /
4, namely, W , = n ǫ p c , cos (( z − z , ) /
2) if | ( z − z , ) / | < π/ , z = 5, z = 13, and ǫ = +1 for fig. 1 while ǫ = − W i corresponds to i -thfigure).To study the interaction of two bright compactons, we combine the compacton having the velocity c = 1 with the slow one characterized by the velocity c = 1 /
10 15 20 25 30 35 x W x W x W x W Figure 3: Evolution of the initial perturbation in the granular media (marked with dots) on thebackground of the corresponding evolution of the compacton (marked with solid lines) obtained atthe following values of the parameters: n = 3 / c = 1 . A = 0 . , B = 0 . . Upper row: left: t = 0; right: t = 4; lower row: left: t = 9; right: t = 14Now we are going to compare the evolution of the compacton solutions with corresponding solu-tions of the finite (but long enough) discrete system. Since the average distance a between adjacentblocks does not play the role of a small parameter anymore, we assume from now on that it is equalto one. With this assumption in mind, we can write equation (12) in the initial variables t, x asfollows: W t + Q [sgn( W )] n +1 n W n + ˆ βW n − h W n +12 i xx o x = 0 , (42)where Q = Aγ , ˆ β = n n + 1) . It is easy to verify that equation (42) possesses the following compacton solutions: W ǫc ( z ) = ǫW c ( z ) = ( ǫ ˜ M cos γ (cid:16) ˜ Bz (cid:17) , if | ˜ Bz | < π , , (43)where ǫ = ± , z = x − ct, ˜ M = (cid:20) c ( n + 1)2 Q (cid:21) n − , ˜ B = n − n + 1) √ β , γ = 2 n − . We introduce the functions R k = Q k − − Q k being the discrete analogs to the strain field W ( t, x ).These functions are assumed to satisfy the system¨ R ( t ) = 0 , ¨ R k ( t ) = A [ R k − | R k − | n − − R k | R k | n − + R k +1 | R k +1 | n − ]+ γ [ R k − | R k − | n − − R k | R k | n − + R k +1 | R k +1 | n − ] ,k = 2 , . . . , m − , ¨ R m ( t ) = 0 (44)14 X - W
10 20 30 40 50 X - W
10 20 30 40 50 X - W
10 20 30 40 50 X - W Figure 4: Evolution of two initially separated compacton perturbation in the granular media (markedwith dots) on the background of the corresponding compacton solutions of the continual model(marked with solid lines), obtained at the following values of the parameters: n = 2, c = 1 . c = 1 . A = B = 1 . Upper row: left: t = 0; right: t = 12; lower row: left: t = 18 .
25; right: t = 26We solve this system with the following initial conditions induced by the compacton solution (43) inthe respective nodes: R k (0) = ( ǫ ˜ M cos γ [ ˜ Bk − I ] if | ˜ Bk − I | < π/
20 otherwise , (45)˙ R k (0) = ( ǫ ˜ M cγ ˜ B cos γ − [ ˜ Bk − I ] sin[ ˜ Bk − I ] if | ˜ Bk − I | < π/
20 otherwise , (46) R (0) = ˙ R (0) = R m (0) = ˙ R m (0) = 0 , (47)where I is a constant phase, k = 2 , , . . . , m −
1. Note that A and γ appear in equation (42) inthe form of the ratio Q = A/γ , whereas in the system (44) they appear as independent parameters.Therefore, one should not expect a one-to-one correspondence between the solutions of the discreteand continuous problems for arbitrary values of the parameters. The numerical experiments confirmthis hypothesis by showing that synchronous evolution of the same compacton perturbation withintwo models can be observed for a unique value of the velocity c = c . This value depends strongly onthe parameter γ and depends on the parameter A in a much weaker fashion. It has been observedthat at c < c the discrete compacton moves quicker than its continuous analogue while at c > c the opposite effect occurs. The result of comparison for a single compacton is shown at fig. 3. Onecan see that at the chosen values of the parameters the main perturbations move synchronously anddo not change their form. However, in the tail part of the discrete analogue small nonvanishingoscillations appear after a while.Since for every value of the parameter γ there is a unique value of the wave pack velocity forwhich the discrete and continuous compacton perturbations move synchronously, one should notexpect that the collision of compactons within these two models will proceed in the same way for anyset of values of parameters. However, collision processes display not much of qualitative differencesfor the discrete pulses which interact elastically like their continuous analogues. This is illustratedon fig. 4 showing the evolution of two initially separated discrete compactons. For convenience, the15ontinuous compactons which coincide with the right-hand side of the initial data (45) at t = 0(the leftmost graph in the first row) are also shown in this figure. Continuous curves shown on thefollowing graphs are obtained by appropriate translations. They are presented in order to emphasizethe quasi-elastic nature of interaction of the discrete pulses. In the present paper we have studied compacton solutions supported by the nonlinear evolutionaryPDEs. The equations we considered, (9), (11), and (12), are obtained from the dynamical system(2) describing one-dimensional chain of prestressed elastic bodies. Equation (8) obtained in [5] fromthis model without resorting to the method of multi-scaled decomposition possesses the compactonsolutions which fail the stability test. Numerical simulations show that the compacton solutionssupported by equation (8) are destroyed in a very short time.In contrast with the above, equations (9) (resp. (11)), which are obtained using formal multiscaledecomposition, possess families of bright (resp. dark) compacton solutions which appear to be stable.This is backed both by the stability test and the results of the numerical simulations.As we have shown in Sections 3–5, for generic values of the parameter n equation (9) does notpossess an infinite set of higher symmetries or other signs of complete integrability such as infinitehierarchies of conservation laws. Nevertheless the compacton solutions to this equation possess somefeatures which are characteristic for “genuine” soliton solutions. In this connection it would be in-teresting to compare the traveling wave solutions for the distinguished case n = − n . Qualitative analysis of the factorized equations describingthe TW solutions shows that there are no compacton solutions for the models with the negative n , butnevertheless all of them seem to possess periodic solutions resembling peakons. It would be interestingto find out whether there is any difference in the qualitative behavior of periodic solutions of the onlyintegrable case ( n = −
2) in comparison with the periodic TW solutions supported by the model char-acterized by other values n <
0. Perhaps the differences will be manifested in the stability propertiesas this is the case with the soliton solutions supported by the family of the KdV-type equations.A characteristic feature of equations (9), (11) related to the decomposition we used is that theydescribe processes with “long” temporal and “short” spatial scales. Hence it is rather questionablewhether these equations can adequately describe a localized pulse propagation in discrete media inthe situation when the distance between the adjacent particles is comparable to the compacton width∆ x . In fact, making the “reverse” transformations X → ξ → x we get the following formula for thewidth of the compacton solution (33) in the initial coordinate system:∆ x = πa s n ( n + 1)6( n − ;this is nothing but equation (1.130) from [5]. For n = 3 /
2, corresponding to the Hertzian forcebetween spherical particles, we get ∆ x ≈ . a . It is then interesting to notice that the same resultsfor the particles with the spherical geometry were obtained in the course of numerical simulations,and experimental studies [1, 3, 2, 54, 55]. We wish to stress that results of our analysis as well as themain conclusions are in agreement with the earlier publications by other authors. In particular, P.Rosenau notes, when considering the general models of dense chains [19], that the natural separationof scales leading to an unidirectional PDE of first order in time does not exist.16 cknowledgments VV gratefully acknowledges support from the Polish Ministry of Science and Higher Education. Theresearch of AS was supported in part by the RVO funding for I ˇC47813059, and by the Grant Agencyof the Czech Republic (GA ˇCR) under grant P201/12/G028. AS gratefully acknowledges warmhospitality extended to him in the course of his visits to AGH in Krak´ow.We are pleased to thank the anonymous referee and M.V. Pavlov for useful suggestions.
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