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Title: Strongly nonlinear waves in locally resonant granular chains

Abstract

In this paper, 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. 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. Finally, in line with these results, we prove that the only exact time-periodic bright breathers consist ofmore » trivial linear oscillations, without contact interactions between discrete elements.« less

Authors:
 [1];  [2];  [3];  [1]
  1. Univ. of Pittsburgh, PA (United States). Dept. of Mathematics
  2. INRIA Grenoble-Rhone-Alpes, Saint Ismier (France)
  3. Univ. of Massachusetts, Amherst, MA (United States). Dept. of Mathematics and Statistics; Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Univ. of Pittsburgh, PA (United States); Univ. of Massachusetts, Amherst, MA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States); INRIA Grenoble-Rhone-Alpes, Saint Ismier (France)
Sponsoring Org.:
USDOE; National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO); Rhone-Alpes Complex Systems Inst. (IXXI) (France)
OSTI Identifier:
1360694
Alternate Identifier(s):
OSTI ID: 1326243
Report Number(s):
LA-UR-15-25011
Journal ID: ISSN 0951-7715
Grant/Contract Number:  
AC52-06NA25396; DMS-1007908; DMS-1506904; FA9550-12-1-0332; W911NF-15-1-0604
Resource Type:
Accepted Manuscript
Journal Name:
Nonlinearity (Print)
Additional Journal Information:
Journal Name: Nonlinearity (Print); Journal Volume: 29; Journal Issue: 11; Journal ID: ISSN 0951-7715
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; locally resonant granular chain; periodic traveling wave; discrete breather

Citation Formats

Liu, Lifeng, James, Guillaume, Kevrekidis, Panayotis, and Vainchtein, Anna. Strongly nonlinear waves in locally resonant granular chains. United States: N. p., 2016. Web. doi:10.1088/0951-7715/29/11/3496.
Liu, Lifeng, James, Guillaume, Kevrekidis, Panayotis, & Vainchtein, Anna. Strongly nonlinear waves in locally resonant granular chains. United States. doi:10.1088/0951-7715/29/11/3496.
Liu, Lifeng, James, Guillaume, Kevrekidis, Panayotis, and Vainchtein, Anna. Fri . "Strongly nonlinear waves in locally resonant granular chains". United States. doi:10.1088/0951-7715/29/11/3496. https://www.osti.gov/servlets/purl/1360694.
@article{osti_1360694,
title = {Strongly nonlinear waves in locally resonant granular chains},
author = {Liu, Lifeng and James, Guillaume and Kevrekidis, Panayotis and Vainchtein, Anna},
abstractNote = {In this paper, 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. 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. Finally, 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.},
doi = {10.1088/0951-7715/29/11/3496},
journal = {Nonlinearity (Print)},
number = 11,
volume = 29,
place = {United States},
year = {2016},
month = {9}
}

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    Works referencing / citing this record:

    Waves in strongly nonlinear discrete systems
    journal, July 2018

    • Nesterenko, Vitali F.
    • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Issue 2127
    • DOI: 10.1098/rsta.2017.0130

    Waves in strongly nonlinear discrete systems
    journal, July 2018

    • Nesterenko, Vitali F.
    • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Issue 2127
    • DOI: 10.1098/rsta.2017.0130