https://hal.science/hal-01417957Liu, LifengLifengLiuDepartment of Mathematics - PITT - University of Pittsburgh - Pennsylvania Commonwealth System of Higher Education (PCSHE)James, GuillaumeGuillaumeJamesBIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems - Inria Grenoble - Rhône-Alpes - Inria - Institut National de Recherche en Informatique et en Automatique - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - LJK - Laboratoire Jean Kuntzmann - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - Inria - Institut National de Recherche en Informatique et en Automatique - CNRS - Centre National de la Recherche Scientifique - UGA [2016-2019] - Université Grenoble Alpes [2016-2019]Kevrekidis, PanayotisPanayotisKevrekidisDepartment of Mathematics and Statistics [University of Massachusetts] - UMASS - University of Massachusetts SystemTheoretical Division and Center for Nonlinear Studies - LANL - Los Alamos National LaboratoryVainchtein, AnnaAnnaVainchteinDepartment of Mathematics - PITT - University of Pittsburgh - Pennsylvania Commonwealth System of Higher Education (PCSHE)Strongly nonlinear waves in locally resonant granular chainsHAL CCSD2016[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]James, Guillaume2016-12-16 11:17:452023-03-15 08:53:382016-12-16 11:17:45enJournal articles10.1088/0951-7715/29/11/34961We explore a recently proposed locally resonant granular system bearingharmonic internal resonators in a chain of beads interacting via Hertzianelastic contacts. In this system, we propose the existence of two types ofconfigurations: (a) small-amplitude periodic traveling waves and (b) darkbreathersolutions, i.e. exponentially localized, time-periodic states mountedon top of a non-vanishing background. A remarkable feature distinguishingour results from other settings where dark breathers are observed is thecomplete absence of precompression in the system, i.e. the absence of a linearspectral band. We also identify conditions under which the system admitslong-lived bright breather solutions. Our results are obtained by meansof an asymptotic reduction to a suitably modified version of the so-calleddiscrete p-Schrödinger (DpS) equation, which is established as controllablyapproximating the solutions of the original system for large but finite times(under suitable assumptions on the solution amplitude and the resonatormass). The findings are also corroborated by detailed numerical computations.Long-lived bright breathers are proved to exist over long but finite times, afterwhich numerical simulations indicate that the breathers disintegrate. In linewith these results, we prove that the only exact time-periodic bright breathersconsist of trivial linear oscillations, without contact interactions betweendiscrete elements.