Strongly nonlinear waves in locally resonant granular chains
Résumé
We explore a recently proposed locally resonant granular system bearing
harmonic internal resonators in a chain of beads interacting via Hertzian
elastic contacts. In this system, we propose the existence of two types of
configurations: (a) small-amplitude periodic traveling waves and (b) darkbreather
solutions, i.e. exponentially localized, time-periodic states mounted
on top of a non-vanishing background. A remarkable feature distinguishing
our results from other settings where dark breathers are observed is the
complete absence of precompression in the system, i.e. the absence of a linear
spectral band. We also identify conditions under which the system admits
long-lived bright breather solutions. Our results are obtained by means
of an asymptotic reduction to a suitably modified version of the so-called
discrete p-Schrödinger (DpS) equation, which is established as controllably
approximating the solutions of the original system for large but finite times
(under suitable assumptions on the solution amplitude and the resonator
mass). The findings are also corroborated by detailed numerical computations.
Long-lived bright breathers are proved to exist over long but finite times, after
which numerical simulations indicate that the breathers disintegrate. In line
with these results, we prove that the only exact time-periodic bright breathers
consist of trivial linear oscillations, without contact interactions between
discrete elements.