Periodic Travelling Waves and Compactons in Granular Chains
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We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.
KeywordsGranular chain Hertzian contact Hamiltonian lattice Periodic travelling wave Compacton Fully nonlinear dispersion
Mathematics Subject Classification37K60 70F45 70K50 70K75 74J30
The author is grateful to anonymous referees for several suggestions which essentially improved the paper, in particular for pointing out the question of the existence of compactons. Helpful discussions with J. Malick, B. Brogliato, A.R. Champneys and P.G. Kevrekidis are also acknowledged.
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