Breathers in a locally resonant granular chain with precompression
Résumé
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 Schrödinger (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.