Nanoptera in weakly nonlinear woodpile and diatomic granular chains
arXiv:2102.07322
Abstract
We study ``nanoptera'', which are non-localized solitary waves with exponentially small but non-decaying oscillations, in two singularly-perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from Stokes phenomena and appear as special curves, called Stokes curves, are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis demonstrates that traveling waves in a singularly perturbed woodpile chain have a single Stokes curve, across which oscillations appear. Comparing these asymptotic predictions with numerical simulations reveals that this accurately describes the non-decaying oscillatory behavior in a woodpile chain. We perform a similar analysis of a diatomic Hertzian chain, that the nanpteron solution has two distinct exponentially small oscillatory contributions. We demonstrate that there exists a set of mass ratios for which these two contributions cancel to produce localized solitary waves. This result builds on prior experimental and numerical observations that there exist mass ratios that support localized solitary waves in diatomic Hertzian chains without precompression. Comparing asymptotic and numerical results in a diatomic Hertzian chain with precompression reveals that our exponential asymptotic approach accurately predicts the oscillation amplitude for a wide range of system parameters, but it fails to identify several values of the mass ratio that correspond to localized solitary-wave solutions.
29 pages, 9 figures
References in corpus (10)
- How Hertzian solitary waves interact with boundaries in a 1-D granular medium
- Highly Nonlinear Solitary Waves in Periodic Dimer Granular Chains
- Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures
- Intrinsic Energy Localization through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals
- Wave propagation in single column woodpile phononic crystals: Formation of tunable band gaps
- Traveling Waves and their Tails in Locally Resonant Granular Systems
- Delayed Scattering of Solitary Waves from Interfaces in a Granular Container
- Characterizing Traveling Wave Collisions in Granular Chains Starting from Integrable Limits: the case of the KdV and the Toda Lattice
- Granular chains with soft boundaries: Slowing the transition to quasi-equilibrium
- The equilibrium phase in heterogeneous Hertzian chains