Periodic Travelling Waves in Dimer Granular Chains

TL;DR

This paper investigates the bifurcations and stability of periodic travelling waves in granular dimer chains, revealing conditions for spectral stability and continuity of solutions as the mass ratio varies, with numerical analysis connecting to monomer chains.

Contribution

It introduces a novel analysis of bifurcations from the anti-continuum limit and demonstrates stability criteria for waves with larger wavelengths in granular dimer chains.

Findings

Periodic waves are uniquely continued with respect to mass ratio.

Waves with larger than a critical wavelength are spectrally stable.

Numerical methods connect solutions from dimer to monomer chains.

Abstract

We study bifurcations of periodic travelling waves in granular dimer chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter and the periodic waves with the wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic travelling waves of granular monomer chains exist.