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Section: New Results

Nonlinear waves in granular chains

Participants : Guillaume James, Bernard Brogliato.

Granular chains made of aligned beads interacting by contact (e.g. Newton's cradle) are widely studied in the context of impact dynamics and acoustic metamaterials. When a large number of beads are present, their dynamics can be described by infinite-dimensional differential equations, which possess a limited smoothness when unilateral Hertzian contact interactions are considered. In this context, we have developed and analyzed new reduced-order models describing nonlinear wave propagation in such systems. In the work [25] (collaboration with D.Pelinovsky, McMaster Univ.), we analyze small amplitude slowly modulated compression waves in the limit when the exponent of the Hertz force is close to unity. From a multiple scale analysis, we derive a Korteweg-de Vries equation with logarithmic nonlinearity allowing to approximate wave profiles, in particular solitary wave solutions. In the work [50] (collaboration with Y. Starosvetsky, Technion IIT), we prove existence of spatially localized nonlinear modes (breathers) in the DpS equation, an amplitude equation describing small oscillations in Newton's cradle over long time scales. For Hertz force exponents close to unity, we show that breather envelopes are well approximated by a Gaussian solution of the logarithmic nonlinear Schrödinger equation. This result is generalized to traveling localized oscillations (traveling breathers) generated by an impact in Newton's cradle (G. James, article in preparation). The existence of breathers is also analyzed in granular metamaterials consisting of hollow beads with internal masses (G. James) in collaboration with L. Liu, A. Vainchtein (Pittsburgh Univ.) and P. Kevrekidis (UMass Amherst) - article in preparation. In addition the LZB model introduced in [15] has been extensively used to numerically investigate wave phenomena in chains of aligned balls (tapered, monodisperse, anti-tapered, stepped chains). Thorough comparisons with experimental results reported in the Granular Matter literature have been made. The results are reported in the monograph [16] .