Breathers in oscillator chains with Hertzian interactions

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 -Schrödinger (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 sizable wave train lying at both sides of the localized excitation.