Nonlinear waves in a strongly nonlinear resonant granular chain

Lifeng Liu (1) , Guillaume James (2) , Panayotis Kevrekidis (3, 4) , Anna Vainchtein (1)

1 Department of Mathematics
2 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
3 Theoretical Division and Center for Nonlinear Studies
4 Department of Mathematics and Statistics [University of Massachusetts]

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) dark-breather solutions, i.e., exponentially localized, time periodic states mounted on top of a non-vanishing background. 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. 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.

Domaines: Formation de Structures et Solitons [nlin.PS]; Systèmes dynamiques [math.DS]

Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein. Nonlinear waves in a strongly nonlinear resonant granular chain. 2015. ⟨hal-01161719⟩