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 direction-reversing motion. We prove analytically the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions.