  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.
