Breathers and surface modes in oscillator chains with Hertzian interactions

We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton’s cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and collisions acting on similar time scales, a situation corresponding e.g. to a modified Newton’s cradle with beads mounted on stiff cantilevers. Such systems support static and traveling breathers with unusual properties, including double exponential spatial decay, almost vanishing Peierls-Nabarro barrier and spontaneous directionreversing motion. We prove analytically the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions. Granular media are known to display a rich dynamical behavior originating from their complex spatial structure and different sources of nonlinearity (Hertzian contact interactions between grains, friction, plasticity). In the case of granular crystals (i.e. for grains organized on a lattice), nonlinear contact interactions lead to different types of localized wave phenomena that could be potentially used for the design of smart materials such as acoustic diodes [1]. Among the most studied types of excitations, solitary waves can be easily generated by an impact at one end of a chain of touching beads (see [7, 8] and references therein). In the absence of an original compression in the chain (the so-called precompression), these solitary waves differ from the classical KdV-type solitary waves, since they are highly-localized (with super-exponential decay) and their width remains unchanged with amplitude (see e.g. [12]). These properties originate from the fully nonlinear character of the Hertzian interaction potential V(r) = 2 5 γ (−r) 5/2 + (with γ > 0 and (a)+ = max(a, 0)), which yields a vanishing sound velocity in the absence of precompression. Discrete breathers (i.e. intrinsic localized modes) form another interesting class of excitations consisting of timeperiodic and spatially localized waveforms [6, 3]. These waves exist in diatomic granular chains under precompression [2, 13] (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 (for frequencies close to the bottom of the optic band). The situation is sensibly different for granular systems without precompression. In that case, localized oscillations can be generated on short transients in the form of transitory defect modes induced by a mass impurity (see [11] and references therein) but never occur time-periodically as proved in [5]. Indeed, uncompressed granular chains are described by the Fermi-Pasta-Ulam lattice with Hertzian interactions ÿn = V (yn+1 − yn) − V (yn − yn−1) (1) (or spatially inhomogeneous variants thereof), where yn(t) denotes the nth bead displacement from its reference position. For all T -periodic solutions of (1), the average inter-