Breathers in oscillator chains with Hertzian interactions

TL;DR

This paper investigates the existence, properties, and dynamics of breathers in Hertzian oscillator chains, revealing their high localization, unique mobility, and interactions, with implications for shock wave control.

Contribution

It proves nonexistence of breathers in uncompressed Hertzian chains and demonstrates their existence and behavior when an on-site potential is added, using numerical and asymptotic models.

Findings

Breathers are highly localized in Hertzian chains.

Traveling breathers can be excited and exhibit spontaneous direction reversal.

High defect sizes cause almost total reflection of breathers.

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. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors.