Breathers in oscillator chains with Hertzian interactions
aa r X i v : . [ n li n . PS ] N ov Breathers in oscillator chains with Hertzianinteractions
Guillaume James a , , Panayotis G. Kevrekidis b , Jes´us Cuevas c . a Laboratoire Jean Kuntzmann, Universit´e de Grenoble and CNRS, BP 53, 38041Grenoble Cedex 9, France. b Department of Mathematics and Statistics, University of Massachusetts,Amherst, Massachussets 01003-4515, USA. c Grupo de F´ısica No Lineal. Departamento de F´ısica Aplicada I. EscuelaPolit´ecnica Superior. Universidad de Sevilla. C/ Virgen de ´Africa, 7. 41011Sevilla, Spain.
Abstract
We prove nonexistence of breathers (spatially localized and time-periodic oscilla-tions) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chainof beads interacting via Hertz’s contact forces. We then consider the setting in whichan additional on-site potential is present, motivated by the Newton’s cradle underthe effect of gravity. Using both direct numerical computations and a simplifiedasymptotic model of the oscillator chain, the so-called discrete p -Schr¨odinger (DpS)equation, we show the existence of discrete breathers and study their spectral prop-erties and mobility. Due to the fully nonlinear character of Hertzian interactions,breathers are found to be much more localized than in classical nonlinear latticesand their motion occurs with less dispersion. In addition, we study numerically theexcitation of a traveling breather after an impact at one end of a semi-infinite chain.This case is well described by the DpS equation when local oscillations are fasterthan binary collisions, a situation occuring e.g. in chains of stiff cantilevers decoratedby spherical beads. When a hard anharmonic part is added to the local potential, anew type of traveling breather emerges, showing spontaneous direction-reversing ina spatially homogeneous system. Finally, the interaction of a moving breather witha point defect is also considered in the cradle system. Almost total breather reflec-tions are observed at sufficiently high defect sizes, suggesting potential applicationsof such systems as shock wave reflectors. Key words:
Hamiltonian lattice, Hertzian contact, discrete breathers,direction-reversing waves, discrete p -Schr¨odinger equation, cantilever arrays. Corresponding author. E-mail: [email protected]
Preprint submitted to Physica D 17 June 2018
Introduction
The study of nonlinear waves in granular crystals is the object of intensiveresearch, both from a theoretical perspective and for practical purposes, e.g.for the design of shock absorbers [33,64,26], acoustic lenses [68] or diodes [6].Due to the nonlinear interactions between grains, several interesting types oflocalized waves can be generated in chains of beads in contact. Solitary wavesare the most studied type of excitations and can be easily generated by animpact at one end of a chain [58,21,32,51,1,60,64]. These solitary waves, in theabsence of an original compression in the chain (the so-called precompression),differ from classical ones (i.e. KdV-type solitary waves [27]) due to the fullynonlinear character of Hertzian contact interactions. Indeed, their decay issuper-exponential and their width remains unchanged with amplitude [20,69].Another interesting class of excitations consists of time-periodic and spatiallylocalized oscillations. Such waves may correspond to Anderson modes [34] inthe presence of spatial disorder, or to defect modes localized at an impurityin a granular chain under precompression [70]. A different class of spatiallylocalized oscillations that occur in the absence of defects consists of discretebreathers, which originate from the combined effects of nonlinearity and spa-tial discreteness (see the reviews [23,24]). These waves exist in diatomic gran-ular chains under precompression [7,71,38], with their frequency lying betweenthe acoustic and optic phonon bands and can be generated e.g. through mod-ulational instabilities. However, because precompression suppresses the fullynonlinear character of Hertzian interactions, these excitations inherit the usualproperties of discrete breathers, i.e. their spatial decay is exponential and theirwidth diverges at vanishing amplitude, i.e. for frequencies close to the bottomof the optic band [38].For granular systems without precompression, the above discussion raises thequestion of existence of spatially localized oscillations. Defect modes inducedby a mass impurity have been numerically observed on short transients inunloaded granular chains [31,42], but the existence of permanent localized os-cillations remains an open question. In this paper, we give a partial answer tothis problem by showing the nonexistence of time-periodic spatially localizedoscillations in uncompressed granular chains. This result seems surprising ata first glance, because Hertzian models of granular chains fall within the classof Fermi-Pasta-Ulam (FPU) lattices, which sustain discrete breathers undersome general assumptions on the interaction potentials and particle masses(see [59] and references therein). However these conditions do not hold for un-compressed granular chains. Using a simple averaging argument, we show thatthe non-attracting character of Hertzian interactions between grains (repulsiveunder contact, and vanishing in the absence of contact) precludes the existenceof time-periodic localized oscillations, both for spatially homogeneous systems2nd for inhomogeneous chains.However, in contrast to the above picture, the existence of discrete breathers ina chain of linear oscillators coupled by Hertzian potentials has been recentlyreported [41]. This model with an on-site potential describes e.g. the smallamplitude waves in a Newton’s cradle [35], which consists of a chain of beadsattached to pendula (see figure 1, left). In [41], static and moving breatherswere numerically observed as a result of modulational instabilities of periodictraveling waves, and extremely stable static breathers were generated fromspecific initial conditions. In addition, a reduced model, the so-called discrete p -Schr¨odinger (DpS) equation was derived as an asymptotic model for smallamplitude oscillations in the Newton’s cradle, and successfully reproduced theabove localization phenomena. The discrete breathers possess special proper-ties both in the original cradle model and the simplified DpS system, i.e. theirspatial decay is super-exponential and their width remains nearly constant atsmall amplitude. Fig. 1. Left : prototypical Newton’s cradle. Right : array of clamped cantileversdecorated by spherical beads (displacements are amplified for clarity).
In this paper we extend the above results in three directions. The first oneconcerns traveling breather excitations, i.e. localized waves displaying an inter-nal oscillation in addition to their translational motion. We show numericallythat such waves can be excited very simply in the cradle system by an impactat one end of a semi-infinite chain, and check that this dynamics is also re-produced by the DpS model. We also discuss some unusual properties of themoving breathers obtained in this way. Due to the fully nonlinear Hertzianinteractions, these breathers display a strong localization (super-exponentialdecay) and their dispersion remains very weak during propagation. In contrast,introducing a precompression attenuates spatial localization and enhances dis-persive effects (due to the fact that precompression adds effectively a linearcomponent to Hertzian interactions). We illustrate this idea using both nu-merical simulations and the discrete nonlinear Schr¨odinger equation, whichallows us to approximate small amplitude traveling breathers under the effectof precompression. In addition, we check that the whole phenomenology re-mains valid at small amplitude when the linear local potential is replaced bya smooth anharmonic potential. However, such local nonlinearities yield addi-tional phenomena, such as the excitation of a surface mode after the impactfor soft local potentials, and for hard potentials the occurence of a direction-3eversing traveling breather. The latter is reminiscent of excitations known as“boomerons” (direction-reversing solitons) that were found up to now only inparticular integrable models (see [16] and references therein).Our second contribution concerns the computation of static breathers in theNewton’s cradle and their numerical continuation. Using a modified Gauss-Newton method introduced in [10], we obtain branches of site- and bond-centered breathers parametrized by their frequency ω > ω ( ω being thefrequency of the local linear oscillators). These branches bifurcate from thetrivial equilibrium when ω → ω and can be continued up to a strongly non-linear regime. Moreover, the Floquet spectra of these breathers display (inaddition to the usual double eigenvalue +1) an extra pair of eigenvalues veryclose to unity. As an effect of this near-degeneracy, small perturbations of thebreathers along an associated pinning mode generate a translational motionwith negligible radiation, according to the process analyzed in [2]. This pro-vides a connection between these standing breathers and the traveling onesmentioned above. In addition to these numerical computations, we obtainan analytical quasi-continuum approximation of the breather profiles validat small amplitude. These approximate breathers have a compact support,which provides a reasonable approximation to the super-exponential decayof the exact breathers. This situation is analogous to what is known for theapproximation of solitons in uncompressed granular chains [58,20].Lastly, we examine possible experimental realizations of these kinds of granu-lar lattices and the related observation of moving breathers after an impact, aswell as their potential practical usefulness (e.g., in the form of granular protec-tors; see below). In the usual Newton’s cradle, the period of local oscillations(of the order of a second) is much larger than the collision time between twobeads (typically of the order of 0 . We consider an infinite chain of particles of masses m n >
0, interacting withtheir nearest neighbors via anharmonic potentials V n . This type of system(which can be thought of in general, i.e., for unequal masses m n , as a spatiallyinhomogeneous FPU lattice) corresponds to the Hamiltonian H = X n ∈ Z m n x n + V n ( x n +1 − x n ) (1)where x n denotes the particle displacements from the ground state. We con-sider interaction potentials V n of the form V n ( x ) = W n [( − x ) + ] , where ( a ) + = max(a , W n ∈ C ( R + , R + ), W ′ n (0) = 0 and W ′ n ( x ) > x >
0. The form of V n implies that particle interactions are repulsive undercompression (i.e. for x <
0) and unilateral (interaction forces vanish underextension, i.e. for x > W ′ n ( x ) ≤ f ( x ) ∀ x ∈ [0 , r ] , ∀ n ≥ n , (2)for some real constant r >
0, integer n and a monotone increasing function f ∈ C ([0 , r ]) satisfying f (0) = 0. For example, these assumptions are satisfiedwith f ( x ) = sup n ≥ n W ′ n ( x ) if the functions W n are convex in [0 , r ] and belongto some finite set for n ≥ n (this is the case in particular for spatially periodic5ystems). Another example is given by Hertzian interactions W n ( x ) = 1 α n + 1 γ n x α n +1 , where the coefficients γ n , α n > f ( x ) = γ x α (and r = 1) provided γ n ≤ γ and α n ≥ α > n ≥ n .The Hamiltonian (1) leads to the equations of motion m n ¨ x n = V ′ n ( x n +1 − x n ) − V ′ n − ( x n − x n − ) , n ∈ Z . (3)In what follows we show that under the above assumptions, the only time-periodic breather solutions of (3) are trivial equilibria. Due to the transla-tional invariance of (1), breathers are defined as time-periodic solutions whichconverge (uniformly in time) towards translations x n = c ± ∈ R as n → ±∞ .This implies that relative particle displacements vanish at infinity, i.e. one haslim n →±∞ k x n − x n − k L ∞ (0 ,T ) = 0 (4)for a T -periodic breather. In what follows, we prove in fact a more generalnonexistence result of nontrivial periodic solutions vanishing as n → + ∞ . Theorem 1
All time-periodic solutions of (3) satisfying lim n → + ∞ k x n − x n − k L ∞ = 0 (5) are independent of t and increasing with respect to n . Proof.
Let us consider a T -periodic solution of (3) and integrate (3) over oneperiod. This yields the equality ¯ F n = ¯ F n +1 , where ¯ F n = T R T V ′ n − ( x n ( t ) − x n − ( t )) dt is the average interaction force be-tween masses n − n . Consequently ¯ F n = ¯ F is independent of n .Now let us check that ¯ F vanishes thanks to the bound (2) uniform in n . Wehave for all n | ¯ F | = 1 T Z T W ′ n − [ ( x n − ( t ) − x n ( t )) + ] dt ≤ k W ′ n − [ ( x n − − x n ) + ] k L ∞ . n large enough | ¯ F | ≤ k f [ ( x n − − x n ) + ] k L ∞ = f [ k ( x n − − x n ) + k L ∞ ]since f is increasing. It follows that | ¯ F | ≤ f ( k x n − − x n k L ∞ ) → n → + ∞ hence ¯ F = 0.Now we use the fact that the interactions between particles are repulsive, i.e.we have − V ′ n ( x ) = W ′ n [ ( − x ) + ] ≥
0. Since the T -periodic functions F n ( t ) = V ′ n − ( x n ( t ) − x n − ( t )) are negative, continuous and satisfy R T F n ( t ) dt = 0as shown previously, we have consequently F n ( t ) = 0 for all t and n . Using(3), this implies ¨ x n = 0 and thus x n is an equilibrium solution (due to time-periodicity). Moreover one has x n ≥ x n − since F n = 0.We note that the above arguments do not work if an on-site potential is addedto (1), because the average interaction forces are no more independent of n .In the next section, we numerically show the existence of breathers for suchtype of nonlinear lattices. We consider a nonlinear lattice with the Hamiltonian H = X n
12 ˙ y n + W ( y n ) + V ( y n +1 − y n ) , (6)where V ( r ) = 25 ( − r ) / . (7)The system (6) corresponds to a chain of identical particles in the local poten-tial W , coupled by the classical Hertz potential V describing contacts betweensmooth non-conforming surfaces. Unless explicitly stated, the on-site potential W will be chosen harmonic with W ( y ) = 12 y . (8)In that case, the dynamical equations read¨ y n + y n = ( y n − − y n ) / − ( y n − y n +1 ) / . (9)Figure 1 depicts two examples of such systems. In practical situations, theassumption of a local harmonic potential implies that the model will be valid7or small amplitude waves and suitable time scales on which higher order termscan be neglected. In order to capture higher order effects, different parts ofour analysis will be extended to symmetric anharmonic local potentials W ( y ) = 12 y + s y , (10)where the parameter s measures the degree of anharmonicity.In the work [41], long-lived static and traveling breather solutions of (9) havebeen numerically observed, starting from suitably chosen localized initial con-ditions, or from small perturbations of unstable periodic traveling waves. How-ever, the classical result of MacKay and Aubry [50] proving the existence ofstatic breathers near the anti-continuum limit does not apply in that case.Indeed, if Hertzian interactions forces are cancelled (or equivalently, if oneconsiders breathers in the limit of vanishing amplitude), one obtains an in-finite lattice of identical linear oscillators, and the nonresonance assumptionof reference [50] is not satisfied. Moreover, other existence proofs based onspatial dynamics and the center manifold theorem [40] do not apply, due tothe fully-nonlinear character of interaction forces (the same remark holds truein the case of traveling breathers [36,39,66]). Variational tools [3,59] may besuitable to obtain existence proofs in this context, but this question is outsidethe scope of the present paper, where we shall resort chiefly to numerical andasymptotic methods.In section 3.1, we recall the relation between (9) and the DpS equation derivedin [41]. In section 3.2, we numerically compute breather solutions of (9) bythe Newton method, and compare them to a quasi-continuum approximationdeduced from the DpS equation. Section 3.3 concerns the stability and mobilityof breathers in model (6)-(10) and the DpS equation, and the generation oftraveling breathers by an impact is studied in the same models in section3.4. In section 3.5, we consider a general class of granular chains with localpotentials, and show that the DpS regime occurs in the above impact problemwhen local oscillations are faster than binary collisions. Section 3.6 providesan application of the above results to a chain of stiff cantilevers decorated byspherical beads. p -Schr¨odinger equation Small amplitude solutions of system (6)-(8) can be well approximated by anequation of the nonlinear Schr¨odinger type, namely the discrete p -Schr¨odinger(DpS) equation with p = 5 / i ˙ v n = ( v n +1 − v n ) | v n +1 − v n | p − − ( v n − v n − ) | v n − v n − | p − . (11)8he most standard model reminiscent of this family of equations is the so-called discrete nonlinear Schr¨odinger (DNLS) equation, studied in detail in anumber of different contexts, including nonlinear optics and atomic physicsover the past decade [45,19]. However, the DpS equation is fundamentallydifferent in that it contains a fully nonlinear inter-site coupling term, corre-sponding to a discrete p -Laplacian.To make the connection with the DpS equation more precise, we sum up somebasic elements of the analysis of [41]. Let us consider the lattice model (9) andthe DpS equation2 iτ ˙ A n = ( A n +1 − A n ) | A n +1 − A n | / − ( A n − A n − ) | A n − A n − | / , (12)where the time constant τ reads τ = 5(Γ( )) √ π ≈ . ǫ > y app n ( t ) = 2 ǫ Re [ A n ( ǫ / t ) e it ] . (13)The approximate solution (13) and amplitude equation (12) have been derivedin [41] using a multiple-scale expansion. According to [18], for initial conditionsof the form y n (0) = 2 ǫ Re [ A n (0) ]+ O ( ǫ / ), ˙ y n (0) = − ǫ Im [ A n (0) ]+ O ( ǫ / )with ǫ ≈
0, this approximation is O ( ǫ / )-close to the exact solution of (9) atleast up to times t = O ( ǫ − / ) (see also numerical results of [41] comparingthe DpS approximation and exact solutions of (9)). Moreover, for some familyof periodic traveling wave solutions of the DpS equation, the ansatz (13) is O ( ǫ / )-close to exact small amplitude periodic traveling waves of (9) [41].Lastly, it is interesting to mention that the DpS equation depends on the termsof (9) up to order O ( | y | / ) (see [41], section 2.1). It follows that this equa-tion remains unchanged for smooth anharmonic on-site potentials W ( y ) = y + O ( | y | ), because the associated extra nonlinearity is at least quadratic.Consequently, the addition of a local anharmonicity doesn’t change the dy-namics of (9) for small amplitude waves, on the timescales governed by theDpS equation. The work of [41] illustrated the existence of time-periodic and spatially local-ized solutions of the DpS equation. Figures 2 and 3 (top panels) display the9rofiles of spatially antisymmetric or symmetric breather solutions of the DpSequation (11). These are sought by using the standard stationary ansatz forDNLS type equations of the form v n = exp( iµt ) u n with µ > u n ∈ R .The resulting coupled nonlinear algebraic equations read − µu n = ( u n +1 − u n ) | u n +1 − u n | / − ( u n − u n − ) | u n − u n − | / (14)and are solved via a fixed point iteration of the Newton-Raphson type, for freeend boundary conditions.Note that equation (11) has a scale invariance, since any solution v n ( t ) gener-ates a one-parameter family of solutions a v n ( | a | / t ), a ∈ R . Thanks to thisscale invariance, the whole families of antisymmetric and symmetric breatherscan be reconstructed from the case µ = 1 of (14). In particular, breather am-plitudes are ∝ µ and the breather width remains unchanged when µ →
0, aproperty that strongly differs from the broadening of DNLS breathers at smallamplitude (see e.g. [12], section 3).In what follows we approach the two breather profiles using a quasi-continuumapproximation. Fixing µ = 1 and introducing w n = ( u n +1 − u n ) | u n +1 − u n | / , equation (14) becomes w n +1 − w n + w n − + w n | w n | − / = 0 , (15)where the nonlinear coupling has been linearized (at the expense of havingan on-site nonlinearity non-differentiable at the origin). The spatial profiles offigures 2 and 3 suggest to use the so-called staggering transformation w n =( − n f ( n ), which yields f ( n + 1) − f ( n ) + f ( n −
1) = − f ( n ) + f ( n ) | f ( n ) | − / . (16)Now we look for an approximate solution F of (16). For this purpose we usethe formal approximation F ( n ± ≈ F ( n ) ± F ′ ( n ) + F ′′ ( n ), in same thespirit as the approximations of soliton profiles performed in reference [58](the accuracy of this approximation will be checked a posteriori by numericalcomputations) . This leads to the differential equation F ′′ = − F + F | F | − / , (17) Note that w n corresponds to a spatially modulated binary oscillation, and a con-tinuum approximation is obtained for its envelope, whereas the continuum approx-imation of [58] was performed on the full soliton profiles. F ( x ) = ± g ( x + φ ),where g ( x ) = (cid:16) (cid:17) cos (cid:16) x (cid:17) for | x | ≤ π , g = 0 elsewhere.Replacing f by its approximation F and performing appropriate choices ofsign and spatial shifts in F , one obtains the symmetric approximate solutionsof (15) w (1) n = ( − n +1 g ( n ) , w (2) n = ( − n +1 g ( n + 12 ) . The case µ = 1 of (14) yields u n = w n − − w n , therefore we get the fol-lowing quasi-continuum approximations of the antisymmetric and symmetricbreather profiles u (1) n = ( − n [ g ( n ) + g ( n − , (18) u (2) n = ( − n [ g ( n + 12 ) + g ( n −
12 )] . (19)The first graphs of figures 2 and 3 show the excellent agreement of theseapproximations with the numerical solutions of the stationary DpS equation.Returning to the ansatz (13) and the time-dependent (non-renormalized) DpSequation (12), we obtain approximate breather solutions of (9) taking the form y ( s ) n ( t ) = 2 ǫ u ( s ) n cos ( ω b t ) , ω b = 1 + ǫ / τ , s = 1 , . (20)It is interesting to observe that approximation (20) is unaffected by smoothon-site nonlinear terms for ǫ ≈
0, since we have noticed that the DpS equationremains unchanged.In what follows we compare the above approximations with breather solu-tions of (9) computed numerically for free end boundary conditions. Let usnote Y n = ( y n , ˙ y n ). We use a method described in [10] to compute zeros Y n (0) = ( y n (0) ,
0) of the period map of the flow of (9) (these initial condi-tions correspond to breathers even in time). The method of [10] is based onan adapted Gauss-Newton scheme and path-following.An example of computation of a breather with frequency ω b = 1 . E = k{ Y n ( T b ) − Y n (0) } n k ∞ k{ Y n (0) } n k ∞ , T b = 2 πω b , reaches 1 . . − and the relative variation (measured using sup norms) of par-ticle positions between successive iterates drops to 1 . . − . Figure 4 com-pares the initial breather positions computed by the Newton method andtheir evolution at t = 100 T b , which shows that the breather oscillations are11xtremely stable. The super-exponential spatial decay of the breather is shownin figure 5.Using the above numerical scheme, we obtain two branches of breather solu-tions of (9) with different symmetries, parametrized by their frequency ω b > y − n +1 = − y n (figure 2) and site-centered breathers (figure 3). Thelatter possess subtle symmetry properties. Since the Hertz potential is non-even, equation (9) is not invariant by the symmetry Sy n := y − n . However,the set of T b -periodic solutions of (9) is invariant under the transformation S ′ y n ( t ) = − y − n ( t + T b / S ′ and not by S (their asymmetry under S increases with ω b and becomesvisible in the bottom panel of figure 3). On the contrary, the DpS equationadmits both symmetries S and S ′ , which both leave the site-centered DpSbreathers invariant.These different types of symmetries are illustrated by figures 2 and 3, whichcompare the approximations (20) with breather solutions of (9) computed bythe Newton method (middle and bottom plots). While approximation (20) isexcellent at small amplitude (case ω b = 1 . ω b = 1 . ω b are shown in figure6, which compares the maximal amplitude and the energy of the bond-centeredand site-centered breather solutions of (9) when ω b is varied (the continuationis performed for ω b ∈ (1 , y n = 0 when ω b → + , and their amplitude and energy increases with ω b . While minordifferences between the breather amplitudes are visible, one observes that therespective energy curves are indistinguishable.More generally, considering system (6) with the local anharmonic potential(10) and choosing s ∈ [ − , ω b → + (results notshown). The persistence of both types of symmetries is due to the evenness of W .In what follows we study in more detail the energy barrier separating site-centered and bond-centered breathers. As illustrated below in section 3.3, thisallows us to approximate the so-called Peierls-Nabarro energy barrier, whichcorresponds to the amount of energy required to put a stable static breatherinto motion under a momentum perturbation.A notion of energy barrier separating discrete breathers is usually defined asfollows (cf. also [52]). From (18)-(20), one can deduce a family of approximate PSfrag replacements ny n (0) u ( ) n PSfrag replacements n y n ( ) u (1) n PSfrag replacements n y n ( ) u (1) n Fig. 2. Top panel: spatially antisymmetric solution of the stationary DpS equation(14), computed numerically for µ = 1 (marks). This solution is compared to thequasi-continuum approximation u (1) n defined by equation (18) (continuous line). Theother graphs compare a bond-centered breather solution of (9) computed numer-ically (marks) and its quasi-continuum approximation y (1) n (continuous line). Themiddle plot corresponds to a small amplitude breather ( ω b = 1 . ω b = 1 . static breather solutions of (6)-(10) y n ( t ) = 2 ǫ [ g ( n + 12 − Q ) + g ( n − − Q )]( − n cos ( ω b t ) , (21)where ω b = 1 + ǫ / τ and Q ∈ R (the cases Q = 0 and Q = 1 / PSfrag replacements ny n (0) u ( ) n PSfrag replacements n y n ( ) u (2) n PSfrag replacements n y n ( ) u (2) n Fig. 3. Top panel: spatially symmetric solution of the stationary DpS equation(14), computed numerically for µ = 1 (marks). This solution is compared to thequasi-continuum approximation u (2) n defined by equation (19) (continuous line). Theother graphs compare a site-centered breather solution of (9) computed numerically(marks) and its quasi-continuum approximation y (2) n (continuous line). The middleplot corresponds to a small amplitude breather ( ω b = 1 . ω b = 1 . obtained from (21). Their dynamics is described by an effective Hamiltonian,whose critical points correspond to site-centered and bond-centered breathershaving the same area A = Z π/ω b X n ˙ y n dt. The absolute energy difference ˜ E P N between the two breather solutions pro-vides an approximation of the Peierls-Nabarro barrier. However, because the14
10 15 20 25 30−0.010−0.0050.0000.0050.010 Numerical solution at t=0 (full line), numerical solution at t= 100 breather periods (marks)
PSfrag replacements n y n Fig. 4. Initial breather positions computed by the Newton method for a chain of 31particles (full line), compared to their evolution at t = 100 T b (marks). Computa-tions are performed for a site-centered breather with frequency ω b = 1 .
42 44 46 48 50 52 54 56 581e−161e−141e−121e−101e−81e−61e−41e−21 Moduli of initial positions (logarithmic scale)
PSfrag replacements n | y n ( ) | Fig. 5. Moduli of the initial breather positions computed by the Newton method,plotted in semi-logarithmic scale. Computations are performed for a chain of 99particles and a site-centered breather with frequency ω b = 1 . latter appears to be very small in system (6)-(10) (a phenomenon that will beillustrated in section 3.3), a very precise computation of ˜ E P N would be neces-sary. This yields additional numerical difficulties, due to the fact that the twobreather frequencies have to be retrieved from a given area A . Due to thesedifficulties, we shall use a more straightforward approach and define (followingref. [14]) the approximate Peierls-Nabarro barrier E P N as the absolute energydifference between site-centered and bond-centered breathers having the samefrequency ω b .We obtain extremely small values of E P N both for harmonic and anharmonicon-site potentials, even quite far from the small amplitude regime. This resultis illustrated by figure 7 for s = − / s = 0 and s = 1. For small amplitudebreathers ( ω b ≈ .
01 in our computations), the different values of s yieldcomparable values of E P N , of the order of 10 − − − . We find that E P N ω b s up n | y n | ω b E ne r g y Fig. 6. Maximal amplitude at t = 0 (left plot) and energy (6) (right plot) of breathersolutions of (9) as a function of their frequency ω b . The continuous line correspondsto bond-centered breathers, and the dashed line to site-centered breathers. Noticethat the energy curves are indistinguishable. increases with the breather amplitude but remains very small in our parameterrange (e.g. E P N is close to 10 − for ω b = 1 . s = − / − ω b = 1 . In this section we examine the stability properties of spatially antisymmetricand symmetric breather solutions of (11) and (6), and link these propertieswith the existence of traveling breather solutions. The linear (spectral) sta-bility of breather solutions of (11) is investigated by means of the perturba-tion [45]: v n ( t ) = exp( iµt ) [ u n + ( a n exp( λt ) + b ⋆n exp( λ ⋆ t ))] (22)where u n is a spatially symmetric or antisymmetric solution of (14) homoclinicto 0. The resulting linear problem for the eigenvalue λ and the eigenvector( a n , b n ) T (where T denotes transpose) is solved by standard numerical linearalgebra solvers and the results are depicted by means of the spectral plane( λ r , λ i ) of the eigenvalues λ = λ r + iλ i .From the bottom panels of Fig. 8, we can infer that spatially antisymmetricsolutions are spectrally stable and therefore should be structurally robust (aresult confirmed by our direct numerical simulations –data not shown here–).16 −20 −15 −10 −5 ω b E ne r g i e s −15 −10 −5 ω b E ne r g i e s −15 −10 −5 ω b E ne r g i e s Fig. 7. Approximate Peierls-Nabarro barrier computed as a function of breatherfrequency, for different degrees of anharmonicity of the on-site potential (top leftplot : s = − /
6, top right plot : s = 0, bottom plot : s = 1). The red curves give theenergy E bc of bond-centered breathers defined by (6). The blue curves correspond tothe approximate Peierls-Nabarro barriers E P N (see text), and the black curve to therelative energy ratio E P N /E bc between the energy barrier and the bond-centeredbreather energy. This is due to the absence of eigenvalues of non-vanishing real part in thisHamiltonian system (in which whenever λ is an eigenvalue, so are λ ⋆ , − λ and − λ ⋆ ).On the other hand, the stability and associated dynamical properties are moreinteresting in the case of the site-centered solution of Fig. 9. In this case, wecan observe the presence of a real eigenvalue pair. As can be seen in thebottom panel of Fig. 9, the real part of the relevant eigenvalue pair (whichcorresponds to the instability growth rate) grows linearly with the eigenvalueparameter µ , inducing a progressively stronger instability for larger ampli-tude solutions. The dynamical manifestation of this instability is illustratedin Figure 10. Here we perturb the dynamically unstable solution of the rightpanel of Fig. 9 by a uniformly distributed random perturbation (of amplitude0 . | v n ( t ) | ) of Fig. 10. Clearly, the instability of17
10 0 10−0.0500.05 u n n −10 0 10−505 u n n−1 0 1−202 λ i λ r −1 0 1−10010 λ i λ r Fig. 8. The profiles (top panels) and the linear stability (bottom panels) of thespatially antisymmetric solution of the DpS equation are shown for the values of µ = 1 (left panels) and µ = 10 (right panels). This inter-site solution is linearlystable. the site-centered mode is associated with a “translational” eigenmode of thelinearization problem, whose excitation induces the motion of the localizedmode.In the above analysis, the breather stability properties remain qualitativelyunchanged for all values of µ . This follows from the scale invariance of (11)pointed out in section 3.2, which also implies the linear dependence of theeigenvalues λ on µ . However, we note in passing that this simplification isobviously not valid for the model (9).Having determined the spectral stability of bond-centered and site-centeredbreather solutions in the DpS equation, we now consider the same problem fortheir analogues in the original lattice (6), including in our analysis the effectof a possible addition of a local anharmonic potential (10).We have computed the Floquet spectrum of (6)-(10) linearized at the bond-centered breather and the site-centered breather, for different values of thebreather frequency ω b ∈ (1 ,
2] and the anharmonicity parameter s ∈ [ − , e ± i π/ω b . The spectral proper-ties of these discrete breathers differ from usual ones [54] for several reasons.Firstly, no bands of continuous spectrum are present on the unit circle for theinfinite chain. This is due to the fact that system (9) linearized at y n = 0 (thelimit of a breather solution at infinity) consists of an infinite chain of uncou-pled identical linear oscillators, and thus the phonon band reduces to a singlefrequency, equal to unity in the present case. Secondly, another nonstandardproperty originates from the quadruplet of eigenvalues close to +1. Due tothe Hamiltonian character of (9), +1 is always at least a double eigenvalue ofthe Floquet matrix. In addition, we always find an extra pair of eigenvaluesin the immediate vicinity of +1 corresponding to a pinning mode (see below).18
10 0 10−0.06−0.04−0.0200.02 u n n −10 0 10−6−4−2024 u n n−0.02 0 0.02−101 λ r λ i −0.02 0 0.02−10010 λ r λ i −3 M a x ( λ r ) µ Fig. 9. The top panel is directly analogous to the results of Fig. 8, but for the case ofthe site-centered solution. The presence of a real eigenvalue pair of linearly growingmagnitude as µ increases can be observed in the spectral plane and is more clearlyhighlighted in the figure of the bottom panel. n t Fig. 10. The figure shows the space-time contour plot of the square modulus ofthe field | v n ( t ) | for equation (11). The initial condition is a site-centered localizedmode (from the right panel of Fig. 9), perturbed by a uniformly distributed randomperturbation of amplitude 0 .
01. The perturbation leads to the manifestation of theinstability of the site-centered mode which, in turn, leads to its mobility.
This contrasts with the case of Klein-Gordon lattices, where this situation isa codimension-one phenomenon, occuring near critical values of the couplingconstant and for particular classes of on-site potentials [2,9,4].In what follows we describe the evolution of the quadruplet of eigenvalues closeto +1 for ω b = 1 . s ∈ [ − , λ, λ − emerges from the unit circle after a collisionat +1, for s > s b ≈ .
26. For the site-centered breather, a pair of multipliers λ, λ − (with λ >
1) exists for s < s s ≈ .
05, and enters the unit circle for s > s s after a collision at +1. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−505 x 10 −4 s a r g ( λ ) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.99960.999811.00021.0004 s | λ | Fig. 11. Arguments (upper plot) and moduli (lower plot) of the quadruplet of Flo-quet eigenvalues λ close to +1, corresponding to system (9)-(10) linearized at thebond-centered breather. Computations are performed for ω b = 1 .
1, and eigenvaluesare plotted as a function of the anharmonicity parameter s ∈ [ − , −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.500.51 x 10 −3 s a r g ( λ ) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.99960.999811.00021.00041.0006 s | λ | Fig. 12. Same plot as in figure 11, for the site-centered breather with ω b = 1 . From the above spectral study, one can infer that for harmonic on-site poten-tials (i.e. s = 0) and ω = 1 .
1, the site-centered breather is weakly unstableand the bond-centered breather is spectrally stable. These results agree withthe above results obtained for the DpS equation. This provides a consistentpicture, given that the DpS equation correctly approximates breather profilesof amplitudes ǫ = O (( ω b − ) for ω b ≈ O ( ǫ ) initial data on times of order O ( ǫ − / ) [18]. Hence, we expect a parallel to the instability of site-centered20odes of the DpS dynamics in Eq. (9). Note that these instabilities are ex-tremely small for ω b close to 1, because the instability of the site-centeredbreather is already very weak at the renormalized (slow) time-scale of theDpS equation (see figure 9), and becomes O ( ǫ / ) times weaker at the level of(9) for a breather with amplitude ǫ .The above picture persists for s ≈
0, but the site-centered and bond-centeredbreathers display a change of stability at the two different critical values s = s b,s > s s being quite small), after which their dynamical stability differsfrom the stability of the DpS breathers. It would be interesting to analyze thebifurcations of new types of discrete breathers near these critical values of s ,and this problem will be considered in a future work.In what follows we illustrate the effect of the additional Floquet eigenvaluesclose to +1 on the breather dynamics, again considering the case ω b = 1 . g n = y n +1 (0) − y n − (0) P n | y n +1 (0) − y n − (0) | , which reveals that the two profiles are very close. The associated mode willthus be referred to as a translation mode or pinning mode, and the effectof a perturbation along its direction is to shift the breather center [9]. Theexistence of this mode has the effect of enhancing the breather mobility. Toillustrate this, we perturb at t = 0 the velocity components of a stationarybreather, adding the discrete gradient g n multiplied by a velocity factor c .The kinetic energy imprinted to the lattice is then c /
2. We consider belowthe energy density at the n -th site, which is defined from (6): e n = 12 ˙ y n + W ( y n ) + 25 γ ( y n − y n +1 ) / . (23)Fig. 14 shows the energy density plot in the system of Eqs. (6)-(8), for a site-centered and a bond-centered breather perturbed with c = 2 × − . Thisperturbation results in a translational motion of the breather at an almostconstant velocity with negligible dispersion, which illustrates the strong mo-bility of discrete breathers in the present model. These results are consistentwith the approximation E P N of the Peierls-Nabarro barrier computed previ-ously, since we found E P N ≈ . . − for s = 0 and ω b = 1 . c / × − , which is well-above E P N .To describe the effect of breather perturbations below the Peierls-Nabarro21arrier, it is convenient to consider the breather energy center X = P n ′ + mn = n ′ − m ne n P n ′ + mn = n ′ − m e n (24)with n ′ being the location of the maximum energy density of the breather and m > m = 5). Figure 15 displays X ( t ) for c = 3 . − , i.e. c / . × − lyingbelow E P N . In that case, only the unstable site-centered breather is able tomove along the lattice (it is able to jump 2 sites but gets pinned subsequently).For the stable bond-centered breather, a transition from pinning to mobilityis obtained for c > c c ≈ . . − . The value of the Peierls-Nabarro barrierresulting from dynamical simulations is thus c c / ≈ . . − , which is quiteclose to the approximation E P N computed previously. n g n n g n Fig. 13. Pinning mode (full line) and discrete gradient (dashed line) correspondingto a bond-centered (left plot) and site-centered (right plot) stationary breather insystem (9), for the breather frequency ω b = 1 .
1. The components of the pinningmode correspond to particle positions at t = 0 (initial particle velocities vanish). Having demonstrated the mobility of breather modes in the DpS equation, indirect analogy with the dynamics of the full oscillator model, we attempt theexcitation of the first site of a Newton’s cradle and the associated DpS chain,and observe the ensuing space-time evolution.Consider the equation (12) on a semi-infinite lattice with n ≥ n = 1. We numerically compute the solution of (12)with the initial condition A (0) = − i, A n (0) = 0 for n ≥ . (25)It can be clearly seen in Fig. 16 that the result is the formation of a localized22 t
30 40 50 60 70 80 90012345678910 x 10 −5 n t
30 40 50 60 70 80 90012345678910 x 10 −5 Fig. 14. Energy density plot of a moving breather in system (6)-(8), obtained by per-turbing along the pinning mode a bond-centered (left plot) and site-centered (rightplot) stationary breather with frequency ω b = 1 .
1. The initial velocity perturbationhas a magnitude c = 2 × − . The velocities v of resulting traveling breathers arevery close, i.e. v ≈ . . − for the site-centered case and v ≈ . . − for thebond-centered case, resulting in nearly identical figures. t X Fig. 15. Time-evolution of the breather energy center after a momentum pertur-bation of a bond-centered (full line) and site-centered (dashed line) breather. Allparameters are the same as in figure 14, except the initial velocity perturbation c .We fix c = 3 . − , which corresponds to an increase of kinetic energy below thePeierls-Nabarro barrier. excitation which is traveling robustly through the chain. This is the travelingbreather resulting from the mobility of the discrete breathers that we con-sidered before. In addition to this strongly localized nonlinear excitation, wecan observe a weak residual excitation at the end of the chain (this is some-what reminiscent of the phenomenology described in [32]). Strictly speaking,we cannot consider this mode to be a surface mode of the chain [57], as ourobservations indicate that its profile is fairly extended and non-stationary (orperiodic).For all ǫ > y app n (0) = 0, ˙ y app1 (0) =23 n
10 20 30 40 5000.10.20.30.4 | v n | n Fig. 16. The space-time evolution of the square modulus of the field, similarly toFig. 10, under the DpS equation is shown for an initial excitation of the domainboundary site with A n (0) = − i δ n, ( δ i,j denotes the usual Kronecker symbol). Thecorresponding spatial profile is depicted in the inset for the final shown simulationtime. Notice the robust propagating localized mode (traveling breather), as well asthe presence of a weak residual (fairly extended) excitation near the boundary. ǫ + O ( ǫ / ), ˙ y app2 (0) = O ( ǫ / ) and ˙ y app n (0) = 0 for n ≥
3. Figures 17 and 18compare the corresponding approximate solution (13) and the solution of (9)with initial condition y n (0) = 0 , ˙ y (0) = 2 ǫ, ˙ y n (0) = 0 for n ≥ ǫ . One can see that the DpS equation and the full oscil-lator model give rise to similar dynamics, i.e. the initial impulse splits into atraveling breather and an extended wavetrain emitted from the boundary. Theamplitude of the latter is reasonably well reproduced by the DpS approxima-tion over a long transient, while the traveling breather amplitude and velocityare overestimated (e.g., the breather amplitude is approximately 13 × − with the DpS approximation and 9 × − for the full lattice). In addition, thetraveling breather velocity resulting from the DpS approximation is slightlylarger than in the full oscillator model. The same kind of waves are visible upto moderate initial velocities. The traveling breather remains highly localized(mainly supported by 7 lattice sites), and is followed by a small oscillatorytail reminiscent of the periodic traveling waves analyzed in [41].In what follows we analyze the effect of considering the local anharmonicpotential (10). Due to the smoothness of W , the DpS equation remains un-changed with respect to the harmonic case, as observed in section 3.1. Conse-quently, the dynamics of (9) after the impact is expected to remain unchangedfor small excitations, on the time scales given in section 3.1. However, it isinteresting to examine possible additional effects of anharmonicity occuringon longer time scales or for large amplitude excitations. For example, a trap-ping of large amplitude traveling breathers can occur in Klein-Gordon lattices[14,52], due to the Peierls-Nabarro energy barrier separating site-centered and24
20 40 60 80 100 120 140 160 180 200−4−20246 x 10 −4 n y n −3 n y napp Fig. 17. Comparison between the solution of (9)-(26) (upper plot) and its approxi-mation given by (12)-(13)-(25) (lower plot), at a fixed time t ≈ τ = 250. Computations are performed for ǫ = 0 . . − . For this value of ǫ ,the static breathers given by (20) have frequency 1 .
01, and the DpS equation yieldsvery good approximations of the static breathers.Fig. 18. Energy density plot showing the comparison between the solution of(9)-(26) (left plot) and its approximation given by (12)-(13)-(25) (right plot), for ǫ = 0 . × − . Time t in the bottom plot is related to the slow time τ through t = ǫ − / τ . bond-centered breathers.In order to characterize the breather motion we consider the traveling breatherenergy center X ( t ) defined by (24). The average velocity of the travelingbreather is computed as the slope of the linear least squares approximation ofthe function X ( t ), taking only into account the points for which the travelingbreather is sufficiently far from the boundary in order to eliminate boundaryeffects. Figure 19 displays the traveling breather velocity and maximum energydensity (computed from (23)) as a function of the initial velocity ˙ y (0), for dif-ferent values of the parameter s ≤
0. As expected, the different graphs are veryclose at small initial velocity where the DpS equation drives the dynamics, butdiscrepancies appear at larger velocities. The graphs of figure 19 corresponding25o s < s < n = 1. This phenomenonis illustrated in figure 20, which compares the traveling breather propagationin two chains with s = 0 (left plot) and s = − . y (0) = 0 . dy (0)/dt M a x i m u m ene r g y den s i t y s=0s=−1/6s=−1/3s=−1/2s=−1 dy (0)/dt V e l o c i t y s=0s=−1/6s=−1/3s=−1/2s=−1 Fig. 19. Maximum energy density (left plot) and velocity (right plot) of the travelingbreather generated in system (6)-(10) with initial condition (26), for several valuesof ˙ y (0) and anharmonicity parameter s ≤ Note that the above-mentioned blow-up phenomenon is due to potential (10)with s < W ( y ) = 1 − cos y , which correspondse.g. to the gravitational potential acting on the usual Newton’s cradle. In thelatter case, the dynamics resulting from the impact becomes rather similarto the phenomena studied in [15]. For sufficiently large impact velocities thetraveling breather is replaced by a kink reminiscent of Nesterenko’s soliton[58], resulting in the ejection of a finite number of particles at the end of thechain (result not shown). It would be interesting to analyze how the transitionbetween traveling breather and kink excitations occurs in this system, but thisproblem lies beyond the scope of the present paper.In the case s > n = 1, i.e. it experiencesalternating phases of deceleration, direction-reversing, accelerated backward26 n t n Fig. 20. Space-time diagrams showing the interaction forces f n = − ( y n − y n +1 ) / in system (6)-(10) for the initial condition (26) with ˙ y (0) = 0 .
94. Forces are repre-sented in grey levels, white corresponding to vanishing interactions (i.e. beads not incontact) and black to a minimal negative value of the contact force. Several valuesof the anharmonicity parameter are considered : s = 0 (left plot), s = − . bead velocities0 20 40 60 80 100 120 140 160 180−0.6−0.4−0.20.00.20.40.60.81.0 t=587.1995 PSfrag replacements n ˙ y n ( t ) Fig. 21. Snapshot of particle velocities in system (6)-(10) with anharmonicity pa-rameter s = − /
6. The profile is plotted at t ≈ y (0) ≈ .
87. The excitations of a surface mode and a traveling breather areclearly visible. motion towards the boundary, and rebound at the boundary (bottom panelof figure 22). During a few rebounds, the breather center behaves like a New-tonian particle in an almost constant effective force field. This phenomenonseems to take place as soon as s > s ≥ . s = 0. The effective force field both increases with s and with the imprintedinitial velocity. For moderate initial velocities the traveling breather decelera-tion is quite slow, as shown by the top panel of figure 22. Figure 23 displays atraveling breather profile at the onset of direction-reversing. These travelingbreathers with direction-reversing motion are reminiscent of excitations knownas “boomerons”, consisting of direction-reversing solitons discovered in differ-ent kinds of integrable models (see [16] and references therein), but the linkbetween both phenomena remains quite speculative at this stage. Although we27ave no clear explanation of the origin of direction-reversing for the travelingbreather, one possibility might be its interaction with other nonlinear wavesvisible in figure 23, which are confined between the traveling breather andthe boundary. In addition, the rebound dynamics can be followed by phasesof intermittent trapping or erratic motion of the breather (figure 22, middlepanel). The DpS limit has been previously described for equation (9) written in a nor-malized form. In this section, we consider a general class of granular systemswith on-site potentials and use suitable scalings to rewrite the system in theform (9). Returning to the above impact problem, this allows us to analyze inwhich parameter regime the DpS equation drives the dynamics. As we shallsee, this case occurs when local oscillations are faster than binary collisions.We consider a chain of identical beads of mass m sitting in local harmonicpotentials, described by the Hamiltonian H = X n m x n + k x n + 25 γ ( x n − x n +1 ) / , (27)where γ is the nonlinear stiffness of Hertzian interactions and k the linearstiffness of local potentials.Let us first consider two interacting beads, one being initially at rest and theother having an initial velocity V , and temporarily neglect the local restoringforce of the on-site potentials. After collision, their contact time is approxi-mately equal to 2 . τ h with τ h = [ m / ( γ V )] / , and their maximal compres-sion distance is close to 0 . δ , where δ = ( mV /γ ) / [49,21]. Moreover, thestiffness constant of Hertzian interactions linearized at precompression δ is ofthe order of κ h = γ √ δ .Including back the local restoring forces, the displacement ξ at which Hertzianand local forces equilibrate satisfies γ ξ / = k ξ and is given by ξ = ( k/γ ) .In addition, the period of local oscillations is 2 πτ c with τ c = ( m/k ) / .Now we are ready to perform a suitable rescaling of (27). Setting x n ( t ) = ξ y n ( t/τ c ), the Hamiltonian (27) yields the equations of motion (9) in dimen-sionless form. Moreover, the initial condition x n (0) = 0 , ˙ x n (0) = V δ n, (28)28 ig. 22. Space-time diagram giving the energy density (23) after an initial conditionof the form (26), for the on-site potential (10) with s = 1. The upper plot corre-sponds to ˙ y (0) = 1 . y (0) = 1 .
9. The bottom plot providesa zoom of the middle one. reads in dimensionless form y n (0) = 0 , ˙ y n (0) = λ / δ n, , (29)29 ead velocities0 10 20 30 40 50 60 70 80 90 100−1.5−1.0−0.50.00.51.0 t=238.32 PSfrag replacements n ˙ y n ( t ) bead velocities0 10 20 30 40 50 60 70 80 90 100−1.0−0.8−0.6−0.4−0.20.00.20.40.60.8 t=241.52 PSfrag replacements n ˙ y n ( t ) Fig. 23. Particle velocities for s = 1 and the initial condition (26) with ˙ y (0) ≈ . t ≈ t ≈ where λ = κ h /k (30)measures the relative strengths of the Hertzian interaction at initial velocity V and the local potential. Since κ h = m/τ h , we have equivalently λ = mkτ h , i.e. λ / measures the relative duration of local oscillations and binary colli-sions.From (29) we deduce that the DpS regime takes place when λ / is small.For example, for a Newton’s cradle with strings of length 50 cm and binarycollision time 2 . τ h = 0 .
077 ms (value taken from [49] for an impact velocityof 1 . . s − ), one obtains λ / ≈ . × , hence we are extremely far fromthe DpS regime. In section 3.6 we will introduce a mechanical system for whichlocal oscillators are much stiffer and the DpS dynamics is relevant. Several types of mechanical models have been devised to analyze the propertiesof discrete breathers experimentally, see e.g. [13,63,46,30]. In this section weintroduce a simplified model of the cantilever system sketched in figure 1(right picture). We consider the form (9) analyzed previously, but examinethe more general situation when the lattice is spatially inhomogeneous. Withthis model, we shall observe that a moving breather generated by an impacton the first cantilever can be almost totally reflected by a localized impuritycorresponding to a moderate increase of the bead radii on a single cantilever.30e begin by introducing a simplified model of the cantilever system of (theright panel of) figure 1, where cantilever compression is neglected and beaddeformations are treated quasi-statically. More precisely, each bead is seen asan elastic medium at equilibrium, clamped at a cantilever at one side, andeither free or in contact with one bead of a neighboring cantilever at theopposite side. So any bead deformation is fully determined by two cantileverpositions, and can be approximated by Hertz’s contact law. In addition, eachcantilever decorated by two spherical beads is described by a point-mass modelwhich approximates the dynamics of the slower bending mode, following aclassical approach in the context of atomic force microscope cantilevers [61].Under these approximations, our model incorporates a single degree of freedomper cantilever, namely its maximal deflection.The point-mass model is obtained as follows. Using a rod model and under theassumption of small deflection, a cantilever clamped at both ends and bent by aforce applied to its mid-point can be represented by an equivalent linear springof stiffness k = 192 E Iℓ − , where E is the cantilever’s Young modulus, ℓ itslength and I = w h /
12 its area moment of inertia, w, h denoting the cantileverwidth and thickness respectively (see e.g. [48], pp. 77 and 81). For a cantileverwithout attached beads, the first bending mode frequency satisfies ω min ≈ . EI/ ( ρA )] / ℓ − ([48], p.102) where ρ denotes the cantilever density and A = w h its cross section. A single cantilever is then represented by an effectivemass m ∗ = k/ω ≈ . m c , where m c = ρAℓ is the exact cantilever mass.The effective mass of a cantilever decorated by two beads of masses m b isthen m = m ∗ + 2 m b . For beads of radius R and density d we fix consequently m = 0 . m c + (8 / πdR .Now let us describe the model for a one-dimensinal chain of such cantilevers,where all beads are made of the same material with Young’s modulus E andPoisson coefficient ν . We denote by R n = R ˜ R n the radius of the two beads ofthe n th cantilever ( R being a reference value and ˜ R n an adimensional number), x n ( t ) the maximal cantilever deflections and m n = 0 . m c + (8 / πdR n theireffective masses. The array of decorated cantilevers is then described by theHamiltonian H = X n m n x n + k x n + 25 γ n ( x n − x n +1 ) / , (31)where γ n = γ η n is the nonlinear stiffness constant of Hertzian interactionsbetween two beads on different cantilevers n and n + 1, defined by γ = E √ R − ν ) and η n = [2 ˜ R n ˜ R n +1 / ( ˜ R n + ˜ R n +1 )] / (see e.g. [48]).Setting x n ( t ) = ξ y n ( t/τ c ) as in section 3.5, the Hamiltonian (31) yields thefollowing equations of motion in dimensionless form µ n ¨ y n + y n = − η n ( y n − y n +1 ) / + η n − ( y n − − y n ) / , (32)31here µ n = m n /m . Note that if all beads have radius R (i.e. ˜ R n = 1) then η n = µ n = 1.Our main purpose is to analyze an impact problem in a chain of N cantileverswith free end boundary conditions, where the first cantilever is hit by a strikerat t = 0. For this purpose we consider a simpler initial condition where allcantilevers with index n ≥ V and zero deflection. This corresponds to fixing the initialcondition (28), which yields (29) in rescaled form.Numerical simulations are performed for a chain of N = 200 stainless steelcantilevers with ρ = 8 × kg . m − , E = 193 GPa, ℓ = 25 mm, w = 5 mm, h = 1 mm, decorated by teflon beads with d = 2 . × kg . m − , E = 1 .
46 GPa, ν = 0 .
46 [60]. All beads have radius R = 2 .
38 mm, except at the middle of thechain where ˜ R can be tuned. These values correspond to a cantilever arrayat the macroscopic scale (as in reference [46]), but extensions to the microscalemight be also considered [63,73].We fix the impact velocity V = 1 m . s − , which yields τ h ≈ .
047 ms. Since τ c ≈ .
025 ms, we have λ ≈ .
29 and λ / is small. Consequently, under theabove conditions the DpS approximation is valid in the spatially homogeneouscase, or in sufficiently long homogeneous segments of a chain including defects.The initial impact generates a traveling breather and a fairly extended wave-train emitted from the boundary, as previously analyzed in section 3.4. Thetraveling breather velocity is close to 2030 sites per second. Evaluating thetraveling breather characteristics at n = 80, we find a maximal bead velocityclose to 0 . . s − (i.e. half the impact velocity), a maximal cantilever de-flection close to 11 µ m and a maximal interaction force close to 2 . . .
14 ms ≈ T / (1 . T = 2 πτ c being the period of linear local oscillations.When the breather reaches the defect site, it appears to be almost totallyreflected for a large enough inhomogeneity, whereas it remains significantlytransmitted for a sufficiently small inhomogeneity. This phenomenon is il-lustrated by figure 24, which compares the cases ˜ R = 1 . R = 1 . R = 1 .
6, a small part of the vibrational energy remainsloosely trapped near the defect site. Such phenomena resulting from breather-defect interactions have been already numerically observed in different typesof Klein-Gordon lattices [25,72,11]. In the present model, almost total reflec-tion occurs for physically realistic parameter values, which suggests potentialapplications of such systems as shock wave reflectors.32 ig. 24. Space-time diagrams showing the interaction forces f n = − γ n ( x n − x n +1 ) / in system (31) for the impact problem described in the text (forces are expressedin N ). Forces are represented in grey levels, white corresponding to vanishing in-teractions (i.e. beads not in contact) and black to a minimal negative value of thecontact force. Left plot : ˜ R = 1 .
6. Right plot : ˜ R = 1 . In section 3 we have analyzed the properties of discrete breathers in chains ofoscillators coupled by fully nonlinear Hertzian interactions. We have obtainedhighly-localized static breathers, which display a super-exponential spatialdecay and have an almost constant width in the small amplitude limit. More-over, small perturbations of the static breathers along a pinning mode generatetraveling breathers propagating at an almost constant velocity with very smalldispersion.These properties are largely due to the fully-nonlinear coupling between oscil-lators, which reduces the phonon band to a single frequency. Intuitively, theabsence of linear coupling terms enhances localization, because linear disper-sion tends to disperse localized wave packets. Though this phenomenon canbe compensated by nonlinearity, breathers in nonlinear lattices with phononbands generally have a slow exponential spatial decay in the limit of vanishingamplitude (see e.g. [40,38]). Moreover, due to resonance with phonons, exacttraveling breathers are generally superposed on nondecaying oscillatory tails,a phenomenon mathematically analyzed in [36,39,66,37,5] (see also section 4.5of [24] for more references); only under special choices of the speed (or the sys-tem parameters) can it then be the case that the amplitude of these oscillatorytails exactly vanishes [55,56].Due to these noticeably different breather properties in the presence or absenceof phonon band, it is interesting to consider physical systems possessing atunable phonon band, allowing to pass from one situation to the other. Thisis the case in particular for granular crystals under tunable precompression,since the latter results in a perturbation of the interaction potential inducingan additional harmonic component. In this section, we incorporate this effect33o model (6), formally analyze the existence of discrete breathers through thephenomenon of modulational instability, and numerically demonstrate thatthe existence of a phonon band can drastically modify the outcome of aninitial impact.
We consider the system (6) with the modified interaction potential V ( r ) = 25 ( d − r ) / + d / r − d / , (33)where d > r ≈ V ( r ) = v r v r v r O ( | r | ) , with v = 32 d / , v = − d − / , v = − d − / . This modified potential possesses a harmonic component of size d / in theneighborhood of the origin, and it becomes linear for r ≥ d . The first termof (33) corresponds to the classical Hertzian potential including a precom-pression effect. For example, this type of interaction can be achieved in thecantilever system of figure 1 in the case of a sufficiently long chain. This canbe done by applying a force at both ends of the system when the cantileversare unclamped, which results in a compression of all the beads by a distance d (compression becomes uniform for an infinite system), and by clamping thecantilevers at this new equilibrium state. The second and third terms of (33)do not modify the equations of motion, and just aim at putting the modifiedHertz potential in a standard form with V (0) = 0, V ′ (0) = 0.System (6)-(10)-(33) consists of a mixed Klein-Gordon - FPU lattice, whichadmits a phonon band with a finite width (of order O ( d / ) when d ≈ y n ( t ) = A e i ( qn − ωt ) + c.c. of the system linearized at y n = 0 obey thedispersion relation ω ( q ) = 1 + 2 v (1 − cos q ) , (34)where q ∈ [0 , π ] denotes the wavenumber and ω the phonon frequency.For this class of systems combining anharmonic local and interaction poten-tials, the modulational instability (MI) of small amplitude periodic and stand-ing waves has been studied in a number of references [22,17,43,28,29]. Thisphenomenon has been analyzed in [28,29] through the continuum nonlinear34chr¨odinger (NLS) equation, which describes the slow spatio-temporal mod-ulation of small amplitude phonons under the effects of nonlinearity and dis-persion (see also the basic papers [62,47]). From the general results of [28,29],system (6)-(10)-(33) admits solutions of the form y n ( t ) = ǫ A [ ǫ t, ǫ ( n − c t )] e i ( qn − ωt ) + c.c. + O ( ǫ / ) (35)on time intervals of length O ( ǫ − ) ( ǫ being a small parameter), where A ( τ, ξ )satisfies the NLS equation i ∂ τ A = − ω ′′ ( q ) ∂ ξ A + h | A | A. (36)In the above expressions, ω is given by (34), c = ω ′ ( q ) is the group velocity, h = β/ω , ω ′′ = v ω /ω and β = 16 v (sin q ) (1 − cos q ) v (1 − cos q ) + 3 + 32 [ 4 v (1 − cos q ) + s ] ,ω = cos q − v ω (sin q ) (see [28], equation (2.12) p. 557).The so-called focusing case of the NLS equation occurs for ω ′′ ( q ) h <
0, i.e.under the condition Φ := − β ω > . (37)In that case the spatially homogeneous solutions of (36) are unstable, and(36) admits sech -shaped solutions corresponding (at least on long transients)to small amplitude traveling breather solutions taking the form y n ( t ) = ǫ M e i [ qn − ( ω − ǫ ω ′′ / t + ϕ ] cosh [ ǫ ( n − c t )] + c.c. + O ( ǫ / ) , (38)where M = ( − ω ′′ /h ) / . These solutions decay exponentially in space andbroaden in the small amplitude limit ǫ →
0, in contrast with the travelingbreathers numerically obtained in section 3.4 in the absence of precompres-sion. For Klein-Gordon lattices (i.e. for harmonic interaction potentials), theexistence of exact traveling breather solutions close to (38) has been provedin special cases in [36,39,66] (breathers are superposed on a nondecaying os-cillatory tail, exponentially small in ǫ ). However these results do not directlyapply to our model having anharmonic interaction potentials.The frequency ω ( q ) defined by (34) admits a unique inflection point in theinterval (0 , π ), at the wavenumber q = q c ∈ (0 , π/
2) satisfying cos q c = v (1 − cos q c ) . In the generic case when β ( q c ) = 0, it follows that Φ changes signat q = q c (since w changes sign). Consequently, MI generically occurs forwavenumbers in some interval lying at one side of q c . This interval may extend35r not up to one edge of the phonon band, depending on parameter values.For q = 0 (in-phase mode) the condition Φ > s <
0, and for q = π (out-of-phase mode) it reduces to 16 v + s >
0. These conditions havebeen also obtained in [22] through a Hill’s type analysis.In the next section, we numerically check that MI can lead to the formationof traveling breathers on long transients, and revisit the impact problem ofsection 3.4.
In this section we fix d = 1 /
2, so that v ≈ . v ≈ − .
53 and v ≈ − . s = 1, s = 0 and s = − /
6. For all these values, there exists a band of unstable phonon modescharacterized by Φ( q ) > s = 0 inside the band of unstable modes, due to the smallness of h . In that case, slower MI should occur according to the NLS approximation,but at the same time the applicability of the latter should be restricted tosmaller values of ǫ . PSfrag replacements q Φ Fig. 25. Graphs of the MI coefficient Φ defined by (37) as a function of wavenumber q , for s = − / s = 0 (blue curve), s = 1 (green curve). Modulationalinstability occurs in the bands where Φ > To illustrate the MI phenomenon, we integrate (6)-(10)-(33) numerically forinitial conditions x n (0) = a sin ( qn ) (1 + b cos (2 nπ/N )) , (39)˙ x n (0) = − a ω cos ( qn ) (1 + b cos (2 nπ/N ))corresponding to slowly modulated phonons, with a = 0 . b = 0 .
01, awavenumber q in the band of unstable modes (see fig. 25), and ω determined36
20 40 60 80 100 120 140 160 180 200−0.20−0.15−0.10−0.050.000.050.100.150.20 Initial bead displacements from equilibrium
PSfrag replacements n y n ( t ) bead displacements from equilibrium0 20 40 60 80 100 120 140 160 180 200−0.5−0.4−0.3−0.2−0.10.00.10.20.30.4 t=6619 PSfrag replacements n y n ( t ) Fig. 26. Evolution of particle positions in the system (6)-(10)-(33) with periodicboundary conditions ( N = 200 particles). We consider the case s = − / d = 0 .
5. The initial condition (39) is plotted in the left panel (case q = π/ t = 6619 (right panel) revealthe formation of a traveling breather resulting from a modulational instability (theenvelope propagates rightwise). by (34). We consider a chain of N particles with periodic boundary conditions.Figure 26 displays the results for s = − / q = π/ N = 200. The initialperturbation generates a traveling breather over a long transient (at the endof which a splitting of the pulse occurs). The same phenomenon occurs for s = 1 and s = 0, albeit the latter case results in slower instabilities and lesslocalized traveling breathers (results not shown).According to the previous computations, traveling breathers with profiles rem-iniscent of (38) can be generated from slow modulations of small amplitude un-stable phonons. This raises the question of the nucleation of traveling breatherfrom other types of initial conditions, in particular for a localized impact. Insection 3.4 we observed that this type of excitation systematically generatestraveling breathers. By extending this study to the case of potential (33), wewill show in which way linear dispersion may modify the impact dynamics.In what follows we keep the same values of parameters d, s and integrate (6)-(10)-(33) numerically (for free end boundary conditions), starting from theinitial condition (26) with ˙ y (0) ≈ .
87. Depending on the value of s , theinitial excitation may lead to different dynamical phenomena, and notabledifferences with respect to the case without precompression are observed.The case s = 1 is described in figure 27, which shows the particle velocityprofiles at different times. A traveling breather reminiscent of the sech -typeenvelope solitons (38) appears after the impact. It forms around t = 290(top plot), and remains much less localized than the traveling breathers pre-viously obtained without precompression (compare figures 27 and 23). Thebreather propagates away from the boundary (middle and bottom plots) andthe “boomerang effect” that occurs without precompression disappears. In ad-dition, the initial perturbation generates a dispersive wavetrain of substantial37mplitude, and the traveling breather becomes ultimately superposed on anoscillatory tail at both sides of the central pulse (bottom plot), which yieldsa traveling breather profile reminiscent of the waves computed in [67] (seealso [56]). bead velocities0 50 100 150 200 250−0.4−0.3−0.2−0.10.00.10.20.30.4 t=290.7 PSfrag replacements n ˙ y n ( t ) bead velocities0 100 200 300 400 500 600 700−0.4−0.3−0.2−0.10.00.10.20.30.4 t=724.2 PSfrag replacements n ˙ y n ( t ) bead velocities1600 1650 1700 1750 1800 1850 1900 1950 2000 2050−0.4−0.3−0.2−0.10.00.10.20.30.4 t=2999.7 PSfrag replacements n ˙ y n ( t ) Fig. 27. Snapshot of particle velocities in system (9)-(10)-(33) with anharmonicityparameter s = 1 and precompression d = 0 .
5, for the initial condition (26) with˙ y (0) ≈ .
87. The profile is plotted at three different times t ≈
291 (top panel), t ≈
724 (middle panel) and t ≈ s = − / s = 0 yield a different situation described in figure28. The difference with the case without precompression is striking (comparefigure 28 with figure 21). The initial localized perturbation generates an im-portant dispersive wavetrain, and no traveling breather is excited, at leaston the timescales of the simulation. Moreover, in the present case we do notobserve the formation of a surface mode.As a conclusion, according to our results, introducing a precompression at-tenuates spatial localization and enhances dispersive effects, a phenomenonlinked with an additional linear component embedded within the Hertzianinteractions. bead velocities0 500 1000 1500 2000 2500 3000−0.15−0.10−0.050.000.050.100.15 t=4399.12 PSfrag replacements n ˙ y n ( t ) bead displacements from equilibrium0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.10−0.08−0.06−0.04−0.020.000.020.040.060.080.10 t=1998 PSfrag replacements n ˙ y n ( t ) Fig. 28. Same as in figure 27, for s = 0 (top panel, particle velocities at t ≈ s = − / t ≈ We have analyzed the properties of discrete breathers in FPU lattices andmixed FPU-Klein-Gordon lattices with Hertzian interactions. While staticbreathers don’t exist in the absence of precompression and of onsite potentials,39he addition of the latter creates highly localized breathers, which display aparticularly strong mobility, a phenomenon well-described by the DpS equa-tion in the small amplitude regime and associated with the spectral properties(i.e., the pinning mode) of such states. Beyond the DpS limit, we have identi-fied different phenomena depending on the softening or hardening character ofthe local potential, namely the generation of a surface mode after an impactor the existence of direction-reversing traveling breathers. Importantly alsothe stability of both the on-site and inter-site breather states obtained wascritically dependent on the strength (and sign) of the anharmonicity.We have also introduced a mechanical model consisting of a chain of stiffcantilevers decorated by spherical beads, which may allow to realize the abovelocalized excitations. According to our study, an impact at one end of thecantilever chain should generate a highly-localized traveling breather. In thisregime, contrary to what is the case for a regular cradle under gravity, theranges of parameters of the problem (e.g., beads of about 1cm diameter, loadsof about 1N, and cantilever width of about 1cm) are deemed relevant for theobservation of such breathers and for the description of the system by the DpSapproximation examined herein.Obviously, one has to stress that the lattice model (32) is simplified and im-portant corrections may apply. For example, a finite-element modeling wouldbe helpful to validate the model or improve its calibration. In addition, itwould be important to take dissipation into account, following e.g. the ap-proach of [8]. Since many sources of dissipation are present (friction, plasticityeffects, transmission of vibrational energy through the walls), one can won-der if dissipation may overdamp the dynamics and completely destroy thebreathers. However, recent experimental results [7] have demonstated thatstatic breathers with lifetimes of the order of 10 ms could be generated indiatomic granular chains. During this time, the moving breather computed insection 3.6 would travel over approximately 20 sites (performing roughly 70 in-ternal oscillations), which would allow for an experimental detection, providedthis excitation persists in the presence of dissipation, with moderate changesin velocity and frequency. Although the setting of decorated cantilevers pro-posed herein would have the additional source of dissipation through radiatingenergy into the ground (through the clamping of the cantilevers), it is certainlydeemed worthwhile to consider such experimental setups and to examine sys-tematically the resulting dynamics.A different approach which may allow to generate static breathers is linkedwith modulational instability. Indeed, static breathers have been excited bymodulational instabilities in experiments performed on diatomic granular chains[7], a phenomenon also numerically illustrated in the Newton’s cradle [41]. Inthis respect, an extensive study of MI in the cradle model (with the help of theDpS equation) would be of interest. A related aspect concerns the actuation40f the system through the driving of a bead with a particular frequency. Infact, the experiments of [7] were realized based on such actuation of the chainat modulationally unstable frequencies rather than the generation of suitablespatially extended, modulationally unstable states. In that regard, it should benoted that it is not straightforward to experimentally initialize desired spatialprofiles throughout the lattice in this system.As we have seen, static breathers may be deformed by weak instabilities re-sulting in a translational motion and traveling counterparts thereof. However,in an experimental context, these weak instabilities are likely to be irrelevantdue to dissipation. To fix the ideas, let us assume a breather lifetime of theorder of 10 ms in the presence of dissipation, as in the experiments of [7]. Inthe computations of section 3.6, the breather periods at small amplitude were(roughly) close to 0 .
15 ms, therefore unstable Floquet eigenvalues 1 + ǫ wouldhave an effect over times of order 0 . ǫ − ms. Consequently, dissipation shoulddestroy the breather well before the instability becomes observable as soon as ǫ < . ǫ < − )would be largely dominated by dissipative effects.From a numerical point of view, an interesting open problem concerns thecomputation of traveling breathers. In the above computations, approximatetraveling breathers were generated by the dynamics after an impact at oneend of the chain. It would be interesting to compute exact traveling breathersolutions using the Newton method, as in references [2,67,56]. Moreover, theexistence (and physical explanation) of direction-reversing traveling breathersremains to be elucidated. Furthermore, it would be relevant to understand inmore detail the nature of interactions of these traveling breather with staticdefects. Studies in these directions are currently in progress and will be re-ported in future publications. Acknowledgements.
Part of this work has been carried out during a visitof P.G.K. to laboratoire Jean Kuntzmann, supported by the CNRS, France,to which P.G.K. is grateful for the hospitality. The authors are grateful toNicholas Boechler for helpful inputs on experimental issues. G.J. acknowl-edges stimulating discussions with E. Dumas, B. Bidegaray, M. Peyrard andG. Theocharis. J.C. acknowledges financial support from the MICINN projectFIS2008-04848. P.G.K. also acknowledges support from the US National Sci-ence Foundation through grant CMMI-1000337 and also from the AlexanderS. Onassis Public Benefit Foundation through the grant RZG 003/2010-2011.41 eferences [1] K. Ahnert and A. Pikovsky. Compactons and chaos in strongly nonlinearlattices,
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