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Traveling Waves for the Mass in Mass Model of Granular Chains

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Abstract

In the present work, we consider the mass in mass (or mass with mass) system of granular chains, namely, a granular chain involving additionally an internal (or, respectively, external) resonator. For these chains, we rigorously establish that under suitable “anti-resonance” conditions connecting the mass of the resonator and the speed of the wave, bell-shaped traveling-wave solutions continue to exist in the system, in a way reminiscent of the results proven for the standard granular chain of elastic Hertzian contacts. We also numerically touch upon settings, where the conditions do not hold, illustrating, in line also with recent experimental work, that non-monotonic waves bearing non-vanishing tails may exist in the latter case.

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Authors and Affiliations

  1. Lederle Graduate Research Tower, Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Panayotis G. Kevrekidis & Haitao Xu

  2. Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87544, USA

    Panayotis G. Kevrekidis

  3. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS, 66045–7523, USA

    Atanas G. Stefanov

Authors
  1. Panayotis G. Kevrekidis
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  2. Atanas G. Stefanov
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  3. Haitao Xu
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Correspondence to Atanas G. Stefanov.

Additional information

Stefanov’s research is supported in part by NSF-DMS 1313107. Kevrekidis acknowledges support from the National Science Foundation under grant DMS-1312856, from ERC and FP7-People under grant 605096, and from the ARO (under grant W911NF-15-1-0604). P.G.K.’s work at Los Alamos is supported in part by the U.S. Department of Energy.

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Kevrekidis, P.G., Stefanov, A.G. & Xu, H. Traveling Waves for the Mass in Mass Model of Granular Chains. Lett Math Phys 106, 1067–1088 (2016). https://doi.org/10.1007/s11005-016-0854-6

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  • DOI: https://doi.org/10.1007/s11005-016-0854-6

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