Nanoptera in weakly nonlinear woodpile and diatomic granular chains

TL;DR

This paper investigates nanoptera, non-localized solitary waves with tiny oscillations, in Hertzian chains, using exponential asymptotics to predict oscillation amplitudes and conditions for localized waves, validated by numerical simulations.

Contribution

It applies exponential asymptotics to analyze nanoptera in woodpile and diatomic Hertzian chains, revealing conditions for localized solitary waves and matching predictions with simulations.

Findings

Nanoptera arise from Stokes phenomena crossing in the complex plane.

Asymptotic predictions accurately match numerical oscillation amplitudes.

Certain mass ratios in diatomic chains lead to localized solitary waves.

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.