Breathers and surface modes in oscillator chains with Hertzian interactions

Abstract

We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton's cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and collisions acting on similar time scales, a situation corresponding e.g. to a modified Newton's cradle with beads mounted on stiff cantilevers. Such systems support static and traveling breathers with unusual properties, including double exponential spatial decay, almost vanishing Peierls-Nabarro barrier and spontaneous direction-reversing motion. We prove analytically the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions.

2012 International Symposium on Nonlinear Theory and its Applications (NOLTA 2012), Spain (2012)

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• Traveling Waves and their Tails in Locally Resonant Granular Systems
• Breathers in a locally resonant granular chain with precompression
• Direct measurement of superdiffusive and subdiffusive energy transport in disordered granular chains
• Periodic Travelling Waves in Dimer Granular Chains
• Quasiperiodic granular chains and Hofstadter butterflies
• Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions
• Traveling waves for the Mass in Mass model of granular chains
• Nonlinear Resonances and Antiresonances of a Forced Sonic Vacuum
• Granular chains with soft boundaries: Slowing the transition to quasi-equilibrium
• Solitary Waves in Mass-in-Mass Lattices
• Solitary Waves in a Discrete Nonlinear Dirac equation
• Low-frequency discrete breathers in long-range systems without on-site potential
• Impulse-induced generation of stationary and moving discrete breathers in nonlinear oscillator networks
• Breather solutions of the discrete p-Schrödinger equation
• Discrete breathers in ${\varphi }^4$ and related models
• Nonlinear waves in a strongly nonlinear resonant granular chain
• Linear and nonlinear dynamics of isospectral granular chains
• Breathers in lattices with alternating strain-hardening and strain-softening interactions
• Long time stability of small amplitude Breathers in a mixed FPU-KG model
• From Newton's cradle to the discrete p-Schrödinger equation
• Breather arrest in a chain of damped oscillators with Hertzian contact