Breathers in oscillator chains with Hertzian interactions

Breathers in oscillator chains with Hertzian interactions Guillaume James a,1, Panayotis G. Kevrekidis b, Jes´us Cuevas c. aLaboratoire Jean Kuntzmann, Universit´e de Grenoble and CNRS, BP 53, 38041 Grenoble Cedex 9, France. bDepartment of Mathematics and Statistics, University of Massachusetts, Amherst, Massachussets 01003-4515, USA. cGrupo de F´ısica No Lineal. Departamento de F´ısica Aplicada I. Escuela Polit´ecnica Superior. Universidad de Sevilla. C/ Virgen de ´ Africa, 7. 41011 Sevilla, Spain. Abstract We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz’s contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton’s cradle under the effect of gravity. We show the existence of breathers in such systems, using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr¨odinger (DpS) equation. From a spectral analysis, we determine breather stability and explain their translational motion under very weak perturbations. Numerical simulations demonstrate the excitation of traveling breathers from simple initial conditions corresponding to small perturbations at the first site of the chain. This regime is well described by the DpS equation, and is found to occur for physical parameter values in granular chains with stiff local oscillators. In addition, traveling breather propagation can be hindered or even suppressed in other parameter regimes. For soft on-site potentials, a part of the energy remains trapped near the boundary and forms a surface mode. For hard on-site potentials and large to moderate initial excitations, one observes a “boomeron”, i.e. a traveling breather displaying spontaneous direction-reversing motion. In addition, dispersion is significantly enhanced when a precompression is applied to the chain. Depending on parameters, this results either in the absence of traveling breather excitation on long time scales, or in the formation of a “nanopteron” characterized by a sizeable wave train lying at both sides of the localized excitation. Key words: Hamiltonian lattice, Hertzian contact, discrete breather, direction-reversing wave, surface mode, discrete p-Schr¨odinger equation. Preprint submitted to Physica D 20 January 2013 1 Introduction The study of nonlinear waves in granular crystals is the object of intensive research, both from a theoretical perspective and for practical purposes, e.g. for the design of shock absorbers [55,20], acoustic lenses [58] or diodes [4]. Due to the nonlinear interactions between grains, several interesting types of localized waves can be generated in chains of beads in contact. Solitary waves are the most studied type of excitations and can be easily generated by an impact at one end of a chain [47,17,26,41,51,55]. These solitary waves, in the absence of an original compression in the chain (the so-called precompression), differ from classical ones (i.e. KdV-type solitary waves [48]) due to the fully nonlinear character of Hertzian contact interactions. Indeed, their decay is super-exponential and their width remains unchanged with amplitude [16,60]. Another interesting class of excitations consists of time-periodic and spatially localized oscillations. Such waves may correspond to Anderson modes [27] in the presence of spatial disorder, or to defect modes localized at an impurity in a granular chain under precompression [61]. A different class of spatially localized oscillations that occur in the absence of defects consists of discrete breathers, which originate from the combined effects of nonlinearity and spatial discreteness (see the review [19]). These waves exist in diatomic granular chains under precompression [3,62,30], with their frequency lying between the acoustic and optic phonon bands and can be generated e.g. through modulational instabilities. However, because precompression suppresses the fully nonlinear character of Hertzian interactions, these excitations inherit the usual properties of discrete breathers, i.e. their spatial decay is exponential and their width diverges at vanishing amplitude, i.e. for frequencies close to the bottom of the optic band [30]. For granular systems without precompression, the above discussion raises the question of existence of spatially localized oscillations. Defect modes induced by a mass impurity have been numerically observed in unloaded granular chains [24,34], but these excitations were found to occur only on short transients. The existence of long-lived localized oscillations as been only reported for granular chains including on-site potentials in addition to the usual Hertzians interaction [32,59]. Models in this class describe e.g. the small amplitude waves in a classical Newton’s cradle [28], which consists of a chain of beads attached to pendula (see figure 1, left). In [32], static and moving breathers were numerically observed as a result of modulational instabilities of periodic traveling waves, and extremely stable static breathers were generated from specific initial conditions. In addition, a reduced model, the so-called discrete p-Schr¨odinger (DpS) equation was derived as an asymptotic model 1Corresponding author. E-mail: [email protected] 2 for small amplitude oscillations in the Newton’s cradle, and successfully reproduced the above localization phenomena. The discrete breathers possess special properties both in the original cradle model and the simplified DpS system, i.e. their spatial decay is super-exponential and their width remains nearly constant at small amplitude. Fig. 1. Left : prototypical Newton’s cradle. Right : stiff Newton’s cradle, consisting of an array of clamped cantilevers decorated by spherical beads (displacements are amplified for clarity). In this paper we extend the above results in two directions. Firstly, we prove in section 2 the nonexistence of time-periodic spatially localized oscillations in uncompressed granular chains in the absence of local potentials. This result seems surprising at a first glance, because Hertzian models of granular chains fall within the class of Fermi-Pasta-Ulam (FPU) lattices, which sustain discrete breathers under some general assumptions on the interaction potentials and particle masses (see [50] and references therein). However these conditions do not hold for uncompressed granular chains. Using a simple averaging argument, we show that the non-attracting character of Hertzian interactions between grains (repulsive under contact, and vanishing in the absence of contact) precludes the existence of time-periodic localized oscillations, both for spatially homogeneous or inhomogeneous chains. Our second contribution concerns the case of (generalized) Newton’s cradles, i.e. spatially homogeneous granular chains incorporating a local harmonic or anharmonic potential (taking the form of an even quartic polynomial). We analyze the existence and qualitative properties of time-periodic and traveling breathers, and the possible excitation of traveling breathers from an impact at the end of a chain. We consider two different situations corresponding respectively to the absence or existence of a precompression of the chain, and yielding quite different dynamical behaviors. The case without precompression is examined in section 3, were we obtain the following results. (1) In section 3.2, we use a Newton method to compute branches of site- and bond-centered breathers parametrized by their frequency ω > ω0(ω0be- ing the linear frequency of the local oscillators). These branches bifurcate from the trivial equilibrium when ω→ω0(where analytical approximations of breather profiles are also obtained), and can be continued up to a strongly nonlinear regime. The spectral stability of these breather solutions is analyzed for some parameter values, and depends on the type 3 (and strength) of the local anharmonicity. In addition, we observe a neardegeneracy in the spectrum (associated with a so-called “pinning mode” [1]), resulting in a transition from static to traveling breathers under very small perturbations. Having obtained traveling breathers from small perturbations of static ones, we study in section 3.3 if they can arise from much simpler initial conditions, and attempt to excite the first site of a Newton’s cradle. (2) In section 3.3.1, we identify four different dynamical regimes depending on the parameter values (and time scales) considered. The first one corresponds to small initial excitations, and long (generally finite) time intervals on which the dynamics of Newton’s cradle and the DpS equation are similar (this is in agreement with recent theoretical results of [2]). In this regime, the main stress wave takes the form of a traveling breather propagating almost freely along the chain. The second regime is obtained for harmonic on-site potentials in the limit of large amplitude perturbations, were soliton-like excitations are observed, a situation reminiscent of [47,26]. In the third regime, which occurs for soft on-site potentials and large to moderate initial excitations, a significant part of the energy does not propagate and remains trapped near the boundary (forming a so-called surface mode [46]), while a small amplitude traveling breather is generated. The last regime corresponds to hard on-site potentials and large to moderate initial excitations (or sufficiently long time scales). In that case, the initial perturbation produces a “boomeron” (direction-reversing traveling breather) reminiscent of excitations previously obtained in particular integrable models (see [13] and references therein). (3) In section 3.3.2, we examine possible experimental realizations of these kinds of granular lattices and the related observation of moving breathers after an impact (i.e. an initial excitation of the first site of the chain). As it follows from the results of section 3.3.1, the DpS regime giving rise to (almost) freely-propagating breathers is realized for small enough impact velocities. Combining this result with suitable scaling arguments, we deduce that moving breathers would not be observable in practice in a classical Newton’s cradle acting under gravity. In addition, we argue that reasonably simple mechanical systems with stiff local oscillators could be tailored so that the DpS regime takes place. As a prototype for which this situation occurs, we consider the chain of identical clamped cantilevers represented in figure 1. Each cantilever is decorated by two spherical beads attached to its center, and the beads of two successive cantilevers are tangent at the ground state. Using a reduced oscillator chain model of this system (calibrated for realistic material parameter values), we check that an impact on the first cantilever generates indeed a moving breather. Lastly, the case of a Newton’s cradle under precompression is studied in sec- 4 tion 4. Since precompression adds effectively a linear component to Hertzian interactions, this system falls within a more standard class of models, the socalled mixed Klein-Gordon/Fermi-Pasta-Ulam lattices [35]. Traveling or static breathers close to envelope solitons of the (focusing) continuum nonlinear Schr¨odinger (NLS) equation exist in such systems, at least in the small amplitude limit and on long transients [21,22]. They can be easily generated from modulational instabilities, starting from “well-prepared” initial data leading to similar dynamics in the original lattice and the NLS equation. However, for a highly localized initial disturbance of the chain, such as the excitation of a single particle, no correspondence with the NLS equation has been mathematically established up to now. Whether traveling breathers may form or not after an impact is therefore a nontrivial theoretical problem. Another interesting question concerns the qualitative properties of the corresponding breathers (if they form), and in particular the differences compared to traveling breathers in uncompressed chains. In numerical simulations, we observe the formation of an important dispersive wave train and the generation or absence of a traveling breather depending on the local potential. Traveling breather propagation does not occur (at least on the timescale of the simulations) with our choice of harmonic and soft on-site potentials, a property that we (heuristically) relate to the growth rates of modes during modulational instability. A traveling breather is observed in the hard potential case, but is found much less localized than in the absence of precompression. In addition, the main pulse is sitting on a sizeable nondecaying oscillatory tail extending at both sides, the ensemble forming a so-called “nanopteron” [5]. According to our findings, the precompression introduces therefore a very different phenomenology (compared to the case of section 3), where dispersion becomes much more dominant and different effects of the local anharmonicity are observed. To conclude this paper, we state in section 5 some relevant theoretical problems left open in this study, and discuss our results from a more general perspective in connection with possible experiments. 2 Non-existence of breathers in FPU chains with repulsive interactions We consider an infinite chain of particles of masses mn>0, interacting with their nearest neighbors via anharmonic potentials Vn. This type of system (which can be thought of in general, i.e., for unequal masses mn, as a spatially inhomogeneous FPU lattice) corresponds to the Hamiltonian H=X n∈Z mn 2˙x2 n+Vn(xn+1 −xn) (1) 5 where xndenotes the particle displacements from the ground state. We consider interaction potentials Vnof the form Vn(x) = Wn[(−x)+], where (a)+= max(a,0), Wn∈C1(R+,R+), W′ n(0) = 0 and W′ n(x)>0 for all x > 0. The form of Vnimplies that particle interactions are repulsive under compression (i.e. for x < 0) and unilateral (interaction forces vanish under extension, i.e. for x > 0). Moreover we assume W′ n(x)≤f(x)∀x∈[0, r],∀n≥n0,(2) for some real constant r > 0, integer n0and a monotone increasing function f∈C0([0, r]) satisfying f(0) = 0. For example, these assumptions are satisfied with f(x) = supn≥n0W′ n(x) if the functions Wnare convex in [0, r] and belong to some finite set for n≥n0(this is the case in particular for spatially periodic systems). Another example is given by Hertzian interactions Wn(x) = 1 αn+ 1 γnxαn+1, where the coefficients γn, αn>0 depend on material properties and particle geometry. In that case one can choose f(x) = γ xα(and r= 1) provided γn≤γand αn≥α > 0 for all n≥n0. The Hamiltonian (1) leads to the equations of motion mn¨xn=V′ n(xn+1 −xn)−V′ n−1(xn−xn−1), n ∈Z.(3) In what follows we show that under the above assumptions, the only timeperiodic breather solutions of (3) are trivial equilibria. Due to the translational invariance of (1), breathers are defined as time-periodic solutions which converge (uniformly in time) towards translations xn=c±∈Ras n→ ±∞. This implies that relative particle displacements vanish at infinity, i.e. one has lim n→±∞ kxn−xn−1kL∞(0,T )= 0 (4) for a T-periodic breather. In what follows, we prove in fact a more general nonexistence result of nontrivial periodic solutions vanishing as n→+∞. Theorem 1 All time-periodic solutions of (3) satisfying lim n→+∞kxn−xn−1kL∞= 0 (5) are independent of tand increasing with respect to n. 6 Proof. Let us consider a T-periodic solution of (3) and integrate (3) over one period. This yields the equality ¯ Fn=¯ Fn+1, where ¯ Fn=1 TRT 0V′ n−1(xn(t)−xn−1(t)) dt is the average interaction force between masses n−1 and n. Consequently ¯ Fn=¯ Fis independent of n. Now let us check that ¯ Fvanishes thanks to the bound (2) uniform in n. We have for all n |¯ F|=1 TZT 0W′ n−1[ (xn−1(t)−xn(t))+]dt ≤kW′ n−1[ (xn−1−xn)+]kL∞. Taking into account (5) and (2), the above inequality yields for nlarge enough |¯ F| ≤ kf[ (xn−1−xn)+]kL∞=f[k(xn−1−xn)+kL∞] since fis increasing. It follows that |¯ F| ≤ f(kxn−1−xnkL∞)→0 as 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 Fn(t) = V′ n−1(xn(t)−xn−1(t)) are negative, continuous and satisfy RT 0Fn(t)dt = 0 as shown previously, we have consequently Fn(t) = 0 for all tand n. Using (3), this implies ¨xn= 0 and thus xnis an equilibrium solution (due to timeperiodicity). Moreover one has xn≥xn−1since Fn= 0. We note that the above arguments do not work if an on-site potential is added to (1), because the average interaction forces are no more independent of n. In the next section, we numerically show the existence of breathers for such type of nonlinear lattices. 3 Breathers in uncompressed granular chains with local potentials 3.1 Models and methods In this section we consider an extension of the Hertzian chain (1) incorporating local potentials. We analyze the existence and qualitative properties of time- 7 periodic and traveling breathers (section 3.2), and illustrate how to excite traveling breathers from simple initial conditions (section 3.3). Our approach is based both on numerical and asymptotic methods. In section 3.2.1, the modified Gauss-Newton method introduced in [8] is used to compute branches of breather solutions bifurcating from the ground state. Their spectral stability is investigated in section 3.2.2, in relation with their translational motion under perturbations. More precisely, their Floquet spectra display (in addition to the usual double eigenvalue +1) an extra pair of eigenvalues very close to unity. As an effect of this near-degeneracy, we show that small perturbations of the breathers along an associated pinning mode generate a translational motion with negligible radiation, according to the process analyzed in [1]. In addition, the concept of Peierls-Nabarro barrier [12,42] allows one to approximate the amount of energy required for the depinning of stable breathers. In addition to the direct approach described above, the main qualitative properties of small amplitude breathers are also captured from the asymptotic limit of the DpS equation. In particular, we derive quasi-continuum approximations of the breather profiles valid at small amplitude. These approximate breathers have a compact support, which provides a reasonable approximation to the super-exponential decay of the exact breathers (in analogy to what is known for the approximation of solitons in uncompressed granular chains [47,16]). Having observed the mobility of static breathers under small perturbations in section 3.2, we explore in section 3.3 the excitation of traveling breathers from a velocity perturbation at the end of a semi-infinite chain. In section 3.3.1, we show the relevance of the DpS equation for describing (over long finite times) the traveling breather propagation after a small amplitude initial perturbation. In addition, we show the failure of the DpS equation for capturing new types of waves (surface modes and boomerons) showing up for anharmonic local potentials and (mainly) for larger perturbations. In section 3.3.2, we argue that the traveling breather excitations obtained in section 3.3.1 are relevant in the context of impact mechanics, i.e. can be obtained in granular chains with local potentials for realistic parameter values. 3.1.1 Hertzian granular chains with on-site potentials We consider a nonlinear lattice with the Hamiltonian H=X n 1 2˙y2 n+W(yn) + V(yn+1 −yn),(6) where V(r) = 2 5(−r)5/2 +.(7) 8 The system (6) corresponds to a chain of identical particles in the local potential W, coupled by the classical Hertz potential Vdescribing contacts between smooth non-conforming surfaces. Unless explicitly stated, the on-site potential Wwill be chosen harmonic with W(y) = 1 2y2.(8) In that case, the dynamical equations read ¨yn+yn= (yn−1−yn)3/2 +−(yn−yn+1)3/2 +.(9) Figure 1 depicts two examples of such systems. In practical situations, the assumption of a local harmonic potential implies that the model will be valid for small amplitude waves and suitable time scales on which higher order terms can be neglected. In order to capture higher order effects, different parts of our analysis will be extended to symmetric anharmonic local potentials W(y) = 1 2y2+s 4y4,(10) where the parameter smeasures the degree of anharmonicity. In the work [32], long-lived static and traveling breather solutions of (9) have been numerically observed, starting from suitably chosen localized initial conditions, or from small perturbations of unstable periodic traveling waves. However, the classical result of MacKay and Aubry [40] proving the existence of static breathers near the anti-continuum limit does not apply in that case. Indeed, if Hertzian interactions forces are cancelled (or equivalently, if one considers breathers in the limit of vanishing amplitude), one obtains an infinite lattice of identical linear oscillators, and the nonresonance assumption of reference [40] is not satisfied. The anti-continuum limit can be only used for models incorporating anharmonic on-site potentials, and under the assumption of weak Hertzian interactions whose applicability is rather limited [33]. Moreover, other existence proofs based on spatial dynamics and the center manifold theorem [31] do not apply, due to the fully-nonlinear character of interaction forces (the same remark holds true in the case of traveling breathers [56]). Variational tools [50] may be suitable to obtain existence proofs in this context, but this question is outside the scope of the present paper, where we chiefly resort to numerical and asymptotic methods. In the following section, we recall the relation between (9) and the asymptotic model given by DpS equation [32]. 9 42 44 46 48 50 52 54 56 58 1e−16 1e−14 1e−12 1e−10 1e−8 1e−6 1e−4 1e−2 1 n |yn(0)| 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 ωb supn|yn| Fig. 4. Left plot: moduli of the initial breather positions computed by the Newton method, plotted in semi-logarithmic scale. Computations are performed for a chain of 99 particles and a site-centered breather with frequency ωb= 1.1. Right plot: maximal amplitude at t= 0 of breather solutions of (9) as a function of their frequency ωb. The continuous line corresponds to bond-centered breathers, and the dashed line to site-centered breathers (note that the two graphs are very close). increases with the breather amplitude but remains very small in our parameter range (e.g. EP N is close to 10−4for ωb= 1.5 and s=−1/6). The harmonic case yields even much smaller barriers (by 3 −4 orders of magnitude for ωb= 1.3). As shown by figure 5, the smaller relative energy difference between sitecentered and bond-centered breathers is also achieved in the harmonic case. In order to correctly interpret the results of figure 5, we should stress that our computation of EP N yields sometimes only a rough approximation, but captures nevertheless the correct orders of magnitude (which is our objective here, since the above definition of EP N provides itself only an approximation of the true Peierls-Nabarro barrier). This originates from the finite precision of the breather computation and the very small values of the energy ratio EP N /Ebc (or equivalently EP N /Esc) that we obtain. If one approximates the numerical error on breather positions by the incremental error ǫinc, then the relative error made on EP N is of order ǫinc Ebc/EP N . Since we get ǫinc ∼10−8, the relative error on EP N is of order unity in the worst case of figure 5 (s= 0, ωb= 1.5). The above results indicate that extremely small perturbations of the breathers are capable of putting them into motion (even more critically for harmonic on-site potentials), a phenomenon that will be illustrated in the next section. 3.2.2 Breather stability and mobility In this section we examine the stability properties of spatially antisymmetric and symmetric breather solutions of (11) and (6), and link these properties with the existence of traveling breather solutions. The linear (spectral) sta- 16 1 1.1 1.2 1.3 1.4 1.5 10−20 10−15 10−10 10−5 100 ωb Energies 1 1.1 1.2 1.3 1.4 1.5 10−15 10−10 10−5 100 ωb Energies 1 1.05 1.1 1.15 1.2 1.25 1.3 10−15 10−10 10−5 100 ωb Energies Fig. 5. Approximate Peierls-Nabarro barrier computed as a function of breather frequency, for different degrees of anharmonicity of the on-site potential (top left plot : s=−1/6, top right plot : s= 0, bottom plot : s= 1). The red curves give the energy Ebc of bond-centered breathers defined by (6). The blue curves correspond to the approximate Peierls-Nabarro barriers EP N (see text), and the black curve to the relative energy ratio EP N /Ebc between the energy barrier and the bond-centered breather energy. bility of breather solutions of (11) is investigated by means of the perturbation [36]: vn(t) = exp(iµt) [un+ (anexp(λt) + b⋆ nexp(λ⋆t))] (22) where unis a spatially symmetric or antisymmetric solution of (14) homoclinic to 0. The resulting linear problem for the eigenvalue λand the eigenvector (an, bn)T(where Tdenotes transpose) is solved by standard numerical linear algebra solvers and the results are depicted by means of the spectral plane (λr, λi) of the eigenvalues λ=λr+iλi. Note that in this Hamiltonian system, whenever λis an eigenvalue, so are λ⋆,−λand −λ⋆. In addition, the breather stability properties remain qualitatively unchanged for all values of µ. This follows from the scale invariance of (11) pointed out in section 3.2.1, which also implies the linear dependence of the eigenvalues λon µ. However, we note in passing that this simplification is obviously not valid for the model (9). 17 From the bottom panels of Fig. 6, we can infer that spatially antisymmetric solutions are spectrally stable (due to the absence of eigenvalues of non-vanishing real part) and therefore should be structurally robust, a result confirmed by our direct numerical simulations (data not shown here). −10 0 10 −0.05 0 0.05 un n−10 0 10 −5 0 5 un n −1 0 1 −2 0 2 λi λr−1 0 1 −10 0 10 λi λr Fig. 6. The profiles (top panels) and the linear stability (bottom panels) of the spatially 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 linearly stable. On the other hand, the stability and associated dynamical properties are more interesting in the case of the site-centered solution of Fig. 7. In this case, we can observe the presence of a real eigenvalue pair. As can be seen in the bottom panel of Fig. 7, the real part of the relevant eigenvalue pair (which corresponds to the instability growth rate) grows linearly with the eigenvalue parameter µ, inducing a progressively stronger instability for larger amplitude solutions. The dynamical manifestation of this instability is illustrated in Figure 8. Here we perturb the dynamically unstable solution of the right panel of Fig. 7 by a uniformly distributed random perturbation (of amplitude 0.01). The projection of this random field on the unstable eigenvector of the site centered mode excites the manifestation of the dynamical instability of this mode which is, in turn, illustrated in the space-time evolution (where the colorbar corresponds to the field |vn(t)|2) of Fig. 8. Clearly, the instability of the site-centered mode is associated with a “translational” eigenmode of the linearization problem, whose excitation induces the motion of the localized mode. Having determined the spectral stability of bond-centered and site-centered breather solutions in the DpS equation, we now consider the same problem for their analogues in the original lattice (6), including in our analysis the effect of a possible addition of a local anharmonic potential (10). We have computed the Floquet spectrum of (6)-(10) linearized at the bondcentered breather and the site-centered breather, for different values of the breather frequency ωb∈(1,2] and the anharmonicity parameter s∈[−1,1]. The Floquet spectrum includes a quadruplet of eigenvalues close to +1 and eigenvalues on the unit circle accumulating near e±i2π/ωb. The spectral proper- 18 −10 0 10 −0.06 −0.04 −0.02 0 0.02 un n−10 0 10 −6 −4 −2 0 2 4 un n −0.02 0 0.02 −1 0 1 λr λi −0.02 0 0.02 −10 0 10 λr λi 2 4 6 8 10 2 4 6 8 10 12 14 16x 10−3 Max(λr) µ Fig. 7. The top panel is directly analogous to the results of Fig. 6, but for the case of the site-centered solution. The presence of a (rather small) real eigenvalue pair of linearly growing magnitude as µincreases can be observed in the spectral plane and is more clearly highlighted in the figure of the bottom panel. n t 0 500 1000 1500 2000 2500 3000 −10 −5 0 5 10 5 10 15 20 Fig. 8. The figure shows the space-time contour plot of the square modulus of the field |vn(t)|2for equation (11). The initial condition is a site-centered localized mode (from the right panel of Fig. 7), perturbed by a uniformly distributed random perturbation of amplitude 0.01. The perturbation leads to the manifestation of the instability of the site-centered mode which, in turn, leads to its mobility. ties of these discrete breathers differ from usual ones [43] for several reasons. Firstly, no bands of continuous spectrum are present on the unit circle for the infinite chain. This is due to the fact that system (9) linearized at yn= 0 (the limit of a breather solution at infinity) consists of an infinite chain of uncoupled identical linear oscillators, and thus the phonon band reduces to a single frequency, equal to unity in the present case. Secondly, another nonstandard property originates from the quadruplet of eigenvalues close to +1. Due to the Hamiltonian character of (9), +1 is always at least a double eigenvalue of 19 the Floquet matrix. In addition, we always find an extra pair of eigenvalues in the immediate vicinity of +1 corresponding to a pinning mode (see below). This contrasts with the case of Klein-Gordon lattices, where this situation is a codimension-one phenomenon, occurring near critical values of the coupling constant and for particular classes of on-site potentials [1,7]. In what follows we describe the evolution of the quadruplet of eigenvalues close to +1 for ωb= 1.1 and s∈[−1,1]. The following figures display the moduli and arguments of these eigenvalues for the bond-centered breather (figure 9) and the site-centered breather (figure 10). For the bond-centered breather, a pair of Floquet multipliers λ, λ−1emerges from the unit circle after a collision at +1, for s > sb 0≈0.26. For the site-centered breather, a pair of multipliers λ, λ−1(with λ > 1) exists for s < ss 0≈0.05, and enters the unit circle for s > ss 0after a collision at +1. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −5 0 5 x 10−4 s arg(λ) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.9996 0.9998 1 1.0002 1.0004 s |λ| Fig. 9. Arguments (upper plot) and moduli (lower plot) of the quadruplet of Floquet eigenvalues λclose to +1, corresponding to system (9)-(10) linearized at the bond-centered breather. Computations are performed for ωb= 1.1, and eigenvalues are plotted as a function of the anharmonicity parameter s∈[−1,1]. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1x 10−3 s arg(λ) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.9996 0.9998 1 1.0002 1.0004 1.0006 s |λ| Fig. 10. Same plot as in figure 9, for the site-centered breather with ωb= 1.1. From the above spectral study, one can infer that for harmonic on-site poten- 20 tials (i.e. s= 0) and ω= 1.1, the site-centered breather is weakly unstable and the bond-centered breather is spectrally stable. These results agree with the above results obtained for the DpS equation. This provides a consistent picture, given that the DpS equation correctly approximates breather profiles of amplitudes ǫ=O((ωb−1)2) for ωb≈1 (section 3.2.1). The DpS admits weakly unstable site-centered and stable bond-centered breather solutions, and approximates the dynamics of (9) for O(ǫ) initial data on times of order O(ǫ−1/2) [2]. Hence, we expect a parallel to the instability of site-centered modes of the DpS dynamics in Eq. (9). Note that these instabilities are extremely small for ωbclose to 1, because the instability of the site-centered breather is already very weak at the renormalized (slow) time-scale of the DpS equation (see figure 7), and becomes O(ǫ1/2) 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-centered breathers display a change of stability at the two different critical values s= sb,s 0>0 (ss 0being quite small), after which their dynamical stability differs from the stability of the DpS breathers. It would be interesting to analyze the possible bifurcations of new types of time-periodic breathers near these critical values of s(perhaps subcritical pitchfork bifurcations at the points s=sb 0,s=ss 0), and this problem will be considered in a future work. In what follows we illustrate the effect of the additional Floquet eigenvalues close to +1 on the breather dynamics, considering the case ωb= 1.1 and s= 0. Figure 11 compares an eigenvector associated with one of these eigenvalues and the renormalized discrete gradient gn=yn+1(0) −yn−1(0) Pn|yn+1(0) −yn−1(0)|2, which reveals that the two profiles are very close. The associated mode will thus be referred to as a translation mode or pinning mode, and the effect of a perturbation along its direction is to shift the breather center [7]. This is precisely the type of mode associated with the instabilities reported in figures 9 and 10 for s∈[−1,1]. The existence of this mode has the effect of enhancing the breather mobility. To illustrate this, we perturb at t= 0 the velocity components of a stationary breather, adding the discrete gradient gnmultiplied by a velocity factor c. The kinetic energy imprinted to the lattice is then c2/2. We consider below the energy density at the n-th site, which is defined from (6): en=1 2˙y2 n+W(yn) + 2 5(yn−yn+1)5/2 +.(23) Fig. 12 shows the energy density plot in the system of Eqs. (6)-(8), for a bond-centered breather (with frequency ωb= 1.1) perturbed with c= 2 · 21 10−4. This perturbation results in a translational motion of the breather at an almost constant velocity with negligible dispersion. A nearly identical figure is obtained for the site-centered breather having the same frequency (result not shown). These results illustrate the strong mobility of discrete breathers in the present model. They are consistent with the approximation EP N of the Peierls- Nabarro barrier computed previously, since we found EP N ≈1.77 ·10−11 for s= 0 and ωb= 1.1 (see figure 5). The above momentum perturbation increases the kinetic energy of the bond-centered breather by c2/2 = 2 ·10−8, which is well-above EP N . To describe the effect of breather perturbations below the Peierls-Nabarro barrier, it is convenient to consider the breather energy center X=Pn′+m n=n′−mnen Pn′+m n=n′−men (24) with n′being the location of the maximum energy density of the breather and m > 0 an integer which accounts for the width of the breather (we have fixed m= 5). Figure 12 displays X(t) for c= 3 ·10−6, i.e. c2/2 = 4.5·10−12 lying below EP N . In that case, only the unstable site-centered breather is able to move 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 mobility is obtained for c > cc≈6.19 ·10−6. The value of the Peierls-Nabarro barrier resulting from dynamical simulations is thus c2 c/2≈1.92·10−11, which is quite close to the approximation EP N computed previously. The same features can be observed for anharmonic on-site potentials, depending on the stability or instability of the site- or bond-centered breather. More precisely, for s > sb 0, arbitrarily small perturbations along the pinning mode put the unstable bond-centered breather into motion, whereas there is a perturbation threshold for bond-centered breather mobility when s < sb 0. Similarly, for s < ss 0, arbitrarily small perturbations along the pinning mode give rise to site-centered breather motion, and a perturbation threshold for site-centered breather mobility is found when s > ss 0. In addition, in the absence of a mobility threshold, the escape time of the moving breather diverges when the size of perturbation goes to 0. 3.3 Waves resulting from a localized perturbation 3.3.1 Traveling breathers, boomerons and surface modes Having demonstrated the mobility of breather modes in the DpS equation, in direct analogy with the dynamics of the full oscillator model, we attempt the excitation of the first site of a Newton’s cradle and the associated DpS chain, 22 5 10 15 20 25 30 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 n gn 5 10 15 20 25 30 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 n gn Fig. 11. Pinning mode (full line) and discrete gradient (dashed line) corresponding to a bond-centered (left plot) and site-centered (right plot) stationary breather in system (9), for the breather frequency ωb= 1.1. The components of the pinning mode correspond to particle positions at t= 0 (initial particle velocities vanish). n t 30 40 50 60 70 80 90 0 1 2 3 4 5 6 7 8 9 10 x 104 0 1 2 3 4 5 6 x 10−5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 105 36 36.5 37 37.5 38 38.5 39 t X Fig. 12. Left plot: energy density plot of a moving breather in system (6)-(8), obtained by perturbing along the pinning mode a bond-centered stationary breather with frequency ωb= 1.1. The initial velocity perturbation has a magnitude c= 2 ·10−4, and the traveling breather propagates at velocity v≈5.364 ·10−4. The same type of perturbation applied to a site-centered breather results in a nearly identical figure with v≈5.446 ·10−4(result not shown). Right plot: time-evolution of the breather energy center of a bond-centered (full line) and site-centered (dashed line) breather, for a different velocity perturbation c= 3 ·10−6, which corresponds to an increase of kinetic energy below the Peierls-Nabarro barrier. and observe the ensuing space-time evolution. Consider the equation (12) on a semi-infinite lattice with n≥1 and a free end boundary condition at n= 1. We numerically compute the solution of (12) with the initial condition A1(0) = −i, An(0) = 0 for n≥2.(25) It can be clearly seen in Fig. 13 that the result is the formation of a localized excitation which is traveling robustly through the chain. This is the traveling 23 breather resulting from the mobility of the discrete breathers that we considered before. In addition to this strongly localized excitation, a small amplitude dispersive wave train having a rather complex structure is emitted from the boundary. Until relatively large values of τ, the modulus of the solution displays small variations within an extended region which is broadening with time (shaded area in the left panel of figure 13) and decays rather sharply when nis further increased (see the solution profile in the right panel of figure 13). We cannot consider this excitation to be a surface mode of the chain [46], as its profile is fairly extended and decays slowly with time. It is rather reminiscent of a self-similar spreading modulated periodic pattern, as shown by the space-time diagram of figure 14. At the sharp edge of this regular pattern, high frequency spatial oscillations are generated in a region which is initially small but become thicker as time increases. This behavior is reminiscent of dispersive shocks studied in the FPU model and other nonlinear dispersive Hamiltonian systems (see [25,15] and references therein). Additional small amplitude traveling breathers eventually escape from the dispersive wave train, as shown in the right panel of figure 13. In addition, between the large amplitude traveling breather and the highly oscillatory region one observes a slowly modulated periodic pattern of very small amplitude. Its structure is detailed in the right panels of figures 13 (inset) and 14. This pattern appears nearly stationary on the timescale of the simulation, as shown in figure 14 (left plot). According to the above numerical observations, the initial perturbation generates a rather complex dynamics which has yet to be explained. The modulated periodic wave following the traveling breather and the macroscopic evolution of the highly oscillatory region might be described in the framework of Whitham’s equation [15], using periodic traveling wave solutions of the DpS equation explicitly computed in [32]. For all ǫ > 0 small enough, the above solution of DpS corresponds to an approximate solution of (9) given by (13), satisfying yapp n(0) = 0, ˙yapp 1(0) = 2ǫ+O(ǫ3/2), ˙yapp 2(0) = O(ǫ3/2) and ˙yapp n(0) = 0 for n≥3. Figures 15 compares this approximate solution and the solution of (9) with initial condition yn(0) = 0,˙y1(0) = 2ǫ, ˙yn(0) = 0 for n≥2 (26) for a small value of ǫ. One can see that the DpS equation and the full oscillator model give rise to similar dynamics, i.e. the initial impulse splits into a traveling breather followed by a small oscillatory tail and a dispersive wave train. Note that the profiles of the exact and approximate solutions are quite close over a long transient, but the traveling breather velocity is slightly overestimated by the DpS approximation (13). Figure 16 (left panel) describes the energy density in Newton’s cradle after the initial perturbation (26). The propagation of a localized excitation at an almost constant velocity is clearly visible, as well as the broadening and decay of the dispersive wave train emit- 24 200 400 600 800 1000 1200 1400 1600 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 400 600 800 1000 1200 1400 0.000 0.001 0.002 0.003 0.004 0.005 0.006 n n |An(τ)| |An(τ)| Fig. 13. Evolution of |An(τ)|under the DpS equation (12), for the initial condition An(0) = −i δn,1(δi,j denotes the usual Kronecker symbol). The space-time diagram (left panel) displays the solution in grey levels. Notice the robust propagating localized mode (traveling breather), as well as the presence of a weak residual (fairly extended) excitation near the boundary. The spatial profile of the solution is shown at two different times (right panel), the blue curve corresponding to τ= 502.5 and the black one to τ= 4000. In the latter case, one can notice the broadening and decay of a dispersive wave train, and the emission of two additional small amplitude traveling breathers, following the main robust localized excitation. The inset provides a zoom on a small amplitude extended wave following the main localized excitation, for τ= 4000. 400 600 800 1000 1200 1400 −0.004 −0.002 0.000 0.002 0.004 0.006 n Re An(τ) Fig. 14. Real part of the solution An(τ) computed in figure 13. The space-time diagram (left panel) shows the propagating localized excitation followed by different modulated spatially periodic patterns. The right panel displays (for τ= 4000) the details of the small amplitude oscillations behind the large amplitude traveling breather. ted from the boundary. The modulated periodic patterns previously obtained with the DpS equation correspond (via the ansatz (13)) to modulated periodic traveling waves propagating in Newton’s cradle. Their structure is detailed in figure 16 (right panel). The different strips correspond (from top to bottom) to the traveling breather excitation followed by a small oscillatory tail, and the dispersive wave train including a highly oscillatory region followed by slowly modulated periodic traveling waves. As we previously conjectured, different parts of the wave train might be described (via Whitham’s equation) as mod- 25 010 20 30 40 50 60 70 80 90 100 −1.5 −1.0 −0.5 0.0 0.5 1.0 n ˙yn(t) 010 20 30 40 50 60 70 80 90 100 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 n ˙yn(t) Fig. 22. Particle velocities for s= 1 and the initial condition (26) with ˙y1(0) ≈1.87. The traveling breather profile is shown at two different times close to direction-re- versing (the left panel corresponds to t≈238, and the right panel to t≈241). Nonlinear waves confined between the traveling breather and the boundary are also visible. ics. This analysis is done in section 3.3.2.1, where we derive a dimensionless parameter λcharacterizing the DpS regime. From this study, we recover the fact that the usual Newton’s cradle acting under gravity lies far beyond the DpS regime (it supports soliton-like excitations instead of traveling breathers [26,47]). However, the analysis of section 3.3.2.1 can be used as a guide to define other mechanical models supporting traveling breather excitations. In section 3.3.2.2, we introduce such a system consisting of a chain of stiff cantilevers decorated by spherical beads, calibrated using realistic material parameter values. Traveling breathers described by the DpS equation could have interesting applications for the control of stress waves in granular systems, since they allow for a coherent and highly-localized energy transport. Moreover, the internal oscillation of traveling breathers may allow resonance phenomena to occur, opening e.g. the possibility of breather interactions with defect modes of the chain. As an application, we show in section 3.3.2.2 that the above cantilever chain can act effectively as a granular shock reflector. 3.3.2.1 Dimensional analysis of the DpS limit In sections 3.1.2 and 3.3.1, we have described the DpS limit and the associated dynamical regime for equation (9) written in a normalized form (or for its generalization with local anharmonicity (10)). In this section, we consider a chain of identical beads of mass msitting in local anharmonic potentials, described by the Hamiltonian H=X n m 2˙x2 n+k 2x2 n+˜s 4x4 n+2 5γ(xn−xn+1)5/2 +,(27) where γis the nonlinear stiffness of Hertzian interactions, kthe linear stiffness of local potentials and ˜smeasures the strength of local anharmonicity. Our 32 aim is to analyze the evolution of the initial condition xn(0) = 0,˙xn(0) = V δn,1.(28) In what follows, we use suitable scalings to rewrite this problem in normalized form. This allows us to analyze in which parameter regime the DpS equation drives the dynamics, which induces the propagation of a traveling breather as described in section 3.3.1. As we shall see, this case occurs when the initial velocity is sufficiently small compared to a reference value depending on parameters. Let us first consider two interacting beads, one being initially at rest and the other having an initial velocity V, and temporarily neglect the local restoring force of the on-site potentials. After collision, their contact time is approximately equal to 2.43 τhwith τh= [m2/(γ2V)]1/5, and their maximal compression distance is close to 0.76 δ, where δ= (mV 2/γ)2/5[39,17]. Moreover, the stiffness constant of Hertzian interactions linearized at precompression δis of the order of κh=γ√δ. Including back the local restoring forces, the displacement ξat which Hertzian and local forces equilibrate satisfies γ ξ3/2=k ξ and is given by ξ= (k/γ)2. In addition, the period of local oscillations is 2πτcwith τc= (m/k)1/2. Now we are ready to perform a suitable rescaling of (27). Setting xn(t) = ξ yn(t/τc), the Hamiltonian (27) is mapped to the normalized form (6)-(10) with s= ˜sξ2/k. Moreover, the initial condition (28) reads in dimensionless form yn(0) = 0,˙yn(0) = λ5/2δn,1,(29) where λ=κh k(30) measures the relative strengths of the Hertzian interaction at initial velocity Vand the local potential. Since κh=m/τ2 h, we have equivalently λ=m kτ2 h ,(31) i.e. λ1/2measures (up to a multiplicative constant) the relative duration of local oscillations and binary collisions of free beads. From (29) and the results of section 3.3.1, we deduce that the DpS regime giving rise to (almost) freely-propagating breathers takes place when λ5/2is small enough. According to definition (30), the linear stiffness of local oscillators must dominate the effective Hertzian stiffness κh, which depends on the initial velocity. Equivalently (from equation (31)), local linearized oscillations must be sufficiently fast compared to the collision of two free beads occurring at the given initial velocity. In addition, one can observe that λ5/2=V/Vc 33 with Vc=m−1/2k5/2γ−2. Consequently, the DpS equation drives the dynamics when the initial velocity is small compared to the reference velocity Vc determined by the model parameters. To illustrate the above computations, let us consider the excitation of the first bead of a classical Newton’s cradle (as represented in figure 1), in which the local potential is due to gravity. In this system, impact propagation is usually analyzed under the assumption of small bead displacements [28], leading to the case ˜s= 0 of (27). The period of local oscillations (generally of the order of a second) being much larger than the collision time between two beads (typically of the order of 0.1 ms [39]), we are then extremely far from the DpS regime. For example, for a classical Newton’s cradle with strings of length 50 cm and binary collision time 2.43 τh= 0.077 ms (value taken from [39] for an impact velocity of 1.1 m.s−1), one obtains λ5/2≈1.75 ·1019. In that case, the propagating wave takes the form of a soliton-like excitation which is reminiscent of the primary compression pulse represented in figure 17. In section 3.3.2.2 below, we introduce a mechanical system for which local oscillators are much stiffer and the DpS dynamics becomes relevant. 3.3.2.2 Application to a chain of cantilevers decorated by spherical beads Several types of mechanical models have been devised to analyze the properties of discrete breathers experimentally, see e.g. [11,54,37,23]. In this section we introduce a simplified model of the cantilever system sketched in figure 1 (right picture). We consider the form (27) analyzed previously (restricting ourselves to harmonic on-site potentials) and examine the more general situation when the lattice is spatially inhomogeneous. With this model, we shall observe that a moving breather generated by an impact on the first cantilever can be almost totally reflected by a localized impurity corresponding to a moderate increase of the bead radii on a single cantilever. We begin by introducing a simplified model of the cantilever system of (the right panel of) figure 1, where cantilever compression is neglected and bead deformations are treated quasi-statically. More precisely, each bead is seen as an elastic medium at equilibrium, clamped at a cantilever at one side, and either free or in contact with one bead of a neighboring cantilever at the opposite side. So any bead deformation is fully determined by two cantilever positions, and can be approximated by Hertz’s contact law. In addition, each cantilever decorated by two spherical beads is described by a point-mass model which approximates the dynamics of the slower bending mode, following a classical approach in the context of atomic force microscope cantilevers [52]. Under these approximations, our model incorporates a single degree of freedom per cantilever, namely its maximal deflection. 34 The point-mass model is obtained as follows. Using a rod model and under the assumption of small deflection, a cantilever clamped at both ends and bent by a force applied to its mid-point can be represented by an equivalent linear spring of stiffness k= 192 E Iℓ−3, where Eis the cantilever’s Young modulus, ℓits length and I=w h3/12 its area moment of inertia, w, h denoting the cantilever width and thickness respectively (see e.g. [38], pp. 77 and 81). For a cantilever without attached beads, the first bending mode frequency satisfies ωmin ≈ 22.4 [EI/(ρA)]1/2ℓ−2([38], p.102) where ρdenotes the cantilever density and A=w h its cross section. A single cantilever is then represented by an effective mass m∗=k/ω2 min ≈0.38 mc, where mc=ρAℓ is the exact cantilever mass. The effective mass of a cantilever decorated by two beads of masses mbis then m=m∗+ 2mb. For beads of radius Rand density dwe fix consequently m= 0.38 mc+ (8/3)πdR3. 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 Eand Poisson coefficient ν. We denote by Rn=R˜ Rnthe radius of the two beads of the nth cantilever (Rbeing a reference value and ˜ Rnan adimensional number), xn(t) the maximal cantilever deflections and mn= 0.38 mc+ (8/3)πdR3 ntheir effective masses. The array of decorated cantilevers is then described by the Hamiltonian H=X n mn 2˙x2 n+k 2x2 n+2 5γn(xn−xn+1)5/2 +,(32) where γn=γ ηnis the nonlinear stiffness constant of Hertzian interactions between two beads on different cantilevers nand n+ 1, defined by γ=E√2R 3(1−ν2) and ηn= [2 ˜ Rn˜ Rn+1/(˜ Rn+˜ Rn+1)]1/2(see e.g. [38]). Note that mnand γn are constant in the particular case when Rnis constant, which leads to the homogeneous system (27) (or its normalized form (6)) previously analyzed. Setting xn(t) = ξ yn(t/τc) as in section 3.3.2.1, the Hamiltonian (32) yields the following equations of motion in dimensionless form µn¨yn+yn=ηn−1(yn−1−yn)3/2 +−ηn(yn−yn+1)3/2 +,(33) where µn=mn/m. In particular, if all beads have radius R(i.e. ˜ Rn= 1) then ηn=µn= 1 and one recovers system (9). Our main purpose is to analyze an impact problem in a chain of Ncantilevers with free end boundary conditions, where the first cantilever is hit by a striker at t= 0. For this purpose we consider a simpler initial condition where all cantilevers with index n≥2 are initially at rest and the first cantilever has initial velocity Vand zero deflection. This corresponds to fixing the initial condition (28), which yields (29) in rescaled form. 35 Numerical simulations are performed for a chain of N= 200 stainless steel cantilevers with ρ= 8 ·103kg.m−3,E= 193 GPa, ℓ= 25 mm, w= 5 mm, h= 1 mm, decorated by teflon beads with d= 2.2·103kg.m−3,E= 1.46 GPa, ν= 0.46 [51]. All beads have radius R= 2.38 mm, except at the middle of the chain where ˜ R100 can be tuned. We fix the impact velocity V= 1 m.s−1, which yields τh≈0.047 ms. Since τc≈0.025 ms, we have λ≈0.29 and λ5/2is small. Consequently, according to the results of section 3.3.2.1, the DpS approximation is valid in the spatially homogeneous case, or in sufficiently long homogeneous segments of a chain including defects. As a result, the initial impact generates a traveling breather and a fairly extended wave train emitted from the boundary, as previously described in section 3.3. From our simulations, the traveling breather velocity is close to 2030 sites per second. Evaluating the traveling breather characteristics at n= 80, we find a maximal bead velocity close to 0.5 m.s−1(i.e. half the impact velocity), a maximal cantilever deflection close to 11 µm and a maximal interaction force close to 2.8 N. The pulse duration is close to 3.8 ms and the period of internal oscillations close to 0.14 ms ≈T0/(1.1), T0= 2πτc being the period of linear local oscillations. When the breather reaches the defect site, it appears to be almost totally reflected for a large enough inhomogeneity, whereas it remains significantly transmitted for a sufficiently small inhomogeneity. This phenomenon is illustrated by figure 23, which compares the cases ˜ R100 = 1.6 (almost total reflection) and ˜ R100 = 1.1 (partial reflection), showing the high sensitivity of the reflection to the strength of the inhomogeneity. After the breather reflection by the defect for ˜ R100 = 1.6, a small part of the vibrational energy remains loosely trapped near the defect site (an excitation not visible in figure 23 due to smallness of interaction forces). Such phenomena resulting from breatherdefect interactions have been already numerically observed in different types of Klein-Gordon lattices (see [63,9] and references therein). In the present model, almost total reflection occurs for physically realistic parameter values, which suggests potential applications of such systems as shock wave reflectors. To close this section, let us note that the other dynamical regimes identified in section 3.3 may be also of practical interest. Another shock-redirection mechanism exists for hard anharmonic on-site potentials, which yield directionreversing traveling breathers for large enough impact velocities (see figure 21). This suggests that both reflection mechanisms (based either on defects or hard local anharmonicity) could be combined to devise an efficient shock wave reflector working for a large range of impact velocities. In addition, the surface modes observed for a soft on-site anharmonicity may have potential applications for energy scavenging, or for the design of acoustic diodes [4]. 36 Fig. 23. Space-time diagrams showing the interaction forces fn=−γn(xn−xn+1)3/2 + in system (32) for the impact problem described in the text (forces are expressed in N). Forces are represented in grey levels, white corresponding to vanishing interactions (i.e. beads not in contact) and black to a minimal negative value of the contact force. Left plot : ˜ R100 = 1.6. Right plot : ˜ R100 = 1.1. 4 Traveling breathers under precompression In section 3 we have analyzed the properties of discrete breathers in chains of oscillators coupled by fully nonlinear Hertzian interactions. We have obtained highly-localized static breathers, which display a super-exponential spatial decay and have an almost constant width in the small amplitude limit. Moreover, small perturbations of the static breathers along a pinning mode generate traveling breathers propagating at an almost constant velocity with very small dispersion. These properties are largely due to the fully-nonlinear coupling between oscillators, which reduces the phonon band to a single frequency. Intuitively, the absence of linear coupling terms enhances localization, because linear dispersion tends to disperse localized wave packets. Though this phenomenon can be compensated by nonlinearity, breathers in nonlinear lattices with phonon bands generally have a slow exponential spatial decay in the limit of vanishing amplitude (see e.g. [30,31]). Moreover, due to resonance with phonons, exact traveling breathers are generally superposed on nondecaying oscillatory tails, a phenomenon mathematically analyzed in a number of works (see [56,29,49] and references therein). Only under special choices of the speed (or the system parameters) can it then be the case that the amplitude of these oscillatory tails exactly vanishes [44,45]. Due to these noticeably different breather properties in the presence or absence of phonon band, it is interesting to consider physical systems possessing a tunable phonon band, allowing to pass from one situation to the other. This is the case in particular for granular crystals under tunable precompression, since the latter results in a perturbation of the interaction potential inducing an additional harmonic component. 37 In this section, we incorporate the precompression effect to model (6), formally analyze the existence of discrete breathers through the phenomenon of modulational instability, and numerically demonstrate that the existence of a phonon band can drastically modify the outcome of an initial impact. 4.1 Granular chain under precompression, and correspondence to the NLS equation We consider the system (6)-(10) with the modified interaction potential V(r) = 2 5(d−r)5/2 ++d3/2r−2 5d5/2,(34) where d > 0 is a parameter. We have thus for r≈0 V(r) = v1 r2 2+v2 r3 3+v3 r4 4+O(|r|5), with v1=3 2d1/2, v2=−3 8d−1/2, v3=−1 16 d−3/2.This modified potential possesses a harmonic component of size d1/2in the neighborhood of the origin, and it becomes linear for r≥d. The first term of (34) corresponds to the classical Hertzian potential including a precompression effect. For example, this type of interaction can be achieved in the cantilever system of figure 1 by applying a force at both ends when the cantilevers are unclamped, which results in a uniform compression of all the beads by a distance d, and by clamping the cantilevers at this new equilibrium state. The second and third terms of (34) do not modify the equations of motion, and just aim at putting the modified Hertz potential in a standard form with V(0) = 0, V′(0) = 0. System (6)-(10)-(34) consists of a mixed Klein-Gordon - FPU lattice. The dynamical equations linearized at yn= 0 admit solutions consisting of sinusoidal waves (or phonons) yn(t) = A ei(qn−ωt)+ c.c., which obey the dispersion relation ω2(q) = 1 + 2v1(1 −cos q),(35) where q∈[0, π] denotes the wavenumber and ωthe phonon frequency. Due to precompression, phonon frequencies belong now to a band of finite width (of size O(d1/2) when d≈0). For this class of systems combining anharmonic local and interaction potentials, the modulational instability (MI) of small amplitude periodic and standing waves has been studied in a number of references (see e.g. [18,35]). This phenomenon can be analyzed through the continuum nonlinear Schr¨odinger equation, which describes the slow spatio-temporal modulation of small amplitude phonons under the effects of nonlinearity and dispersion. Indeed, from 38 the general results of [21,22], system (6)-(10)-(34) admits solutions of the form yn(t) = ǫ A[ǫ2t, ǫ(n−c t)] ei(qn−ωt)+c.c. +O(ǫ3/2) (36) on time intervals of length O(ǫ−2) (ǫbeing a small parameter), where A(τ, ξ) satisfies the NLS equation i ∂τA=−1 2ω′′(q)∂2 ξA+h|A|2A. (37) In the above expressions, ωis given by (35), c=ω′(q) is the group velocity, h=β/ω,ω′′ =v1ω2/ω and we have (see [21], equation (2.12) p. 557) β=16v2 2(sin q)2(1 −cos q)2 4v1(1 −cos q)2+ 3 +3 2[ 4v3(1 −cos q)2+s], ω2= cos q−v1 ω2(sin q)2. The so-called focusing case of the NLS equation [48] occurs for ω′′(q)h < 0, i.e. under the condition Φdef =−β ω2>0.(38) In that case the spatially homogeneous solutions of (37) given by A(τ) = r0e−i h r2 0τare exponentially unstable, with maximum growth rate σmax = |h|r2 0. To analyze the occurrence of MI, one can observe that ω(q) admits a unique inflection point in the interval (0, π), at the wavenumber q=qc∈(0, π/2) satisfying cos qc=v1(1 −cos qc)2. In the generic case when β(qc)6= 0, it follows that Φ changes sign at q=qc(since w2changes sign). Consequently, MI generically occurs for wavenumbers in some interval lying at one side of qc. This interval may extend or not up to one edge of the phonon band, depending on parameter values. For q= 0 (in-phase mode) the condition Φ >0 reduces to s < 0, and for q=π(out-of-phase mode) it reduces to 16v3+s > 0 (these conditions have been also obtained in [18] through a Hill’s type analysis). Another important property of the focusing case of the NLS equation (37) is the existence of sech-shaped soliton solutions [48]. By the ansatz (36), these solutions yield small amplitude solutions of (6)-(10)-(34) taking the form yn(t) = ǫ M ei[qn−(ω−ǫ2ω′′ /2)t+ϕ] cosh [ǫ(n−c t)] +c.c. +O(ǫ3/2) (39) on time intervals of length O(ǫ−2), with M= (−ω′′/h)1/2. Such solutions therefore correspond to “long-lived” traveling breather excitations of the chain. The existence of exact (permanent) traveling breather excitations close to the form (39) is a delicate question. This result has been proved in special 39 cases for Klein-Gordon lattices (i.e. chains of linearly coupled nonlinear oscillators) and FPU lattices, see [56,29] and references therein. These traveling breather solutions have been obtained as small amplitude homoclinic orbits of an advance-delay differential equation lying on a finite-dimensional center manifold. The homoclinic orbits converge at infinity towards a periodic orbit, whose size can be made exponentially small with respect to the amplitude of the homoclinics, but does not generically vanish. As a result, the traveling breathers can be seen as coherent structures connecting two identical wave trains, whose amplitude can be made exponentially small in ǫ(such waves are called “nanopterons”, following the denomination of [5]) . Of course, these results do not directly apply to the study of traveling breathers in system (6)-(10)-(34) including both an anharmonic interaction potential and an onsite potential. However, in principle only minor adaptations of the analysis of [29] should be required to prove the same type of results (due to the evenness of W, which preserves the reversibility symmetry of the FPU system used in [29]). From the above theory, one can drawn several differences with respect to the case without precompression studied in section 3, linked with different features of the NLS and DpS asymptotic limits. Firstly, the anharmonicity parameter senters the cubic coefficient hof the NLS equation, thereby influencing the focusing or defocusing dynamics of small modulated periodic waves. On the contrary, the DpS equation (12) does not depend on s, as well as the evolution (over long finite times) of small initial data in (6)-(7)-(10). In addition, the profile of (39) becomes loosely localized in the small amplitude limit ǫ→0. This is in contrast with the traveling breathers numerically obtained in section 3.3 in the DpS regime, which are well described by the approximation (12)- (13)-(25), resulting in a constant width in the small amplitude limit. Such differences suggest that the outcome of an impact may be considerably affected by the precompression, and strongly differ from the phenomenology described in section 3.3. This problem will be numerically examined in the next section. 4.2 Excitation of traveling breathers In the numerical simulations performed in this section, we fix d= 1/2 so that v1≈1.06, v2≈ −0.53 and v3≈ −0.17, qc≈1.17, and we consider different values of the anharmonicity parameter s= 1, s= 0 and s=−1/6. Before analyzing the excitation of the first site of the precompressed chain (and comparing the results to the study of section 3.3), we illustrate below some features of the MI that will help to interpret the results. For all the above parameter values, there exists a band of unstable phonon modes characterized by Φ(q)>0 within the NLS approximation. This band and the graph of Φ are 40 shown in figure 24. In addition, the graph of |h(q)|indicates the strength of the MI (since the maximum exponential growth rate is proportional to |h(q)| at fixed wave amplitude). 0.0 −1.0 −0.5 0.0 0.5 1.0 0.0 −0.04 −0.02 0.00 0.02 0.04 0.06 0.0 −1.0 −0.5 0.0 0.5 π 2 π 2 π 2 π π π q q q Φ,|h|Φ,|h|Φ,|h| Fig. 24. Graphs of the MI coefficients for different values of s, corresponding to s= 1 (top), s= 0 (middle) and s=−1/6 (bottom). In each panel, the black curve shows the MI coefficient Φ(q) defined by (38) as a function of wavenumber q. Modulational instability occurs in the bands where Φ >0, delimited by red lines. The blue curve displays the absolute value |h(q)|of the cubic coefficient of the NLS equation (37), which is proportional (up to a factor depending solely on wave amplitude) to the maximum instability growth rate of the unstable mode with wavenumber q. To illustrate the MI phenomenon, we integrate (6)-(10)-(34) numerically for initial conditions xn(0) = asin (qn) (1 + bcos (2nπ/N)),(40) ˙xn(0) = −a ω cos (qn) (1 + bcos (2nπ/N)) corresponding to slowly modulated phonons, with a= 0.15, b= 0.01, a wavenumber qin the band of unstable modes (see fig. 24), and ωdetermined by (35). We consider a chain of Nparticles with periodic boundary conditions. Figure 25 displays the results for s=−1/6, q=π/4 and N= 200. The initial perturbation generates a traveling breather over a long transient (at the end of which a splitting of the pulse occurs). 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