Breathers in oscillator chains with Hertzian interactions

Abstract

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schrödinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors.

Breather Solutions of the Discrete p-Schrödinger Equation

“…The DpS equation was recently introduced to describe small amplitude oscillations in a class of mechanical systems consisting of a chain of touching beads confined in smooth local potentials [12,13,26], the most well known example of such systems being Newton's cradle [10]. In this context, the p-Laplacian involved in (1) accounts for the fully-nonlinear character of Hertzian interactions between beads (with p = 5/2 in the case of contacting spheres).…”

“…pinned to some lattice sites and time-periodic) or traveling along the lattice. Such solutions can be generated from localized initial conditions or may arise from modulational instabilities of periodic waves [12,13,26]. Although such properties are classical in the context of anharmonic Hamiltonian lattices [9], energy localization is particularly strong in the DpS equation.…”

“…The second method is based on a continuum limit obtained when p is close to 2, where one recovers a logarithmic stationary nonlinear Schrödinger equation with a Gaussian homoclinic solution. This limiting procedure is a first step towards the rigorous justification of previous formal continuum approximations [17,26,13]. Moreover, it is interesting to note that the case p ≈ 2 is physically sound in the context of granular chain models [16,27,19].…”

“…The case i = 1 yields the so-called even-parity modes (or site-centered solutions), and the case i = 2 corresponds to odd-parity modes (bond-centered solutions). These solutions were numerically computed in [12,26,13] for p = 5/2 (see also [9,23] and references therein in the case p = 4 with an additional on-site cubic nonlinearity).…”

“…The first one is reminiscent of the method of successive approximations [20,25], but presents some limitations since a convergence analysis is not available. The second one relies on formal continuum approximations which provide quantitatively correct results at least for p = 5/2 and p = 4 [17,26,13]. However, in these works the continuum approximation is performed with a finite mesh size, so that the continuum problem is not properly justified.…”