Breathers in a locally resonant granular chain with precompression

TL;DR

This paper investigates the existence and stability of discrete breathers in a precompressed granular chain with internal resonators, deriving an effective NLS equation and identifying two types of localized oscillatory states.

Contribution

It introduces a novel analytical and numerical study of breathers in a resonant granular chain, including the derivation of an effective NLS equation and analysis of breather bifurcations.

Findings

Existence of traveling bright and dark breathers in the system.

Analytical evidence supported by numerical simulations for breather types.

Stability analysis of stationary dark breathers.

Abstract

We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schr\"odinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, time-periodic states mounted on top of a non-vanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.