# Long wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes

## TL;DR

This paper analyzes the properties of long wavelength solitary waves in Hertzian chains across different nonlinearity regimes, deriving explicit formulas and relations that connect wave speed, amplitude, and nonlinearity, and comparing them to known solutions.

## Contribution

It provides a comprehensive analytical framework linking solitary wave characteristics to nonlinearity levels in Hertzian chains, including explicit formulas and asymptotic comparisons.

## Key findings

- Asymptotic amplitude relates to continuum sound speed.
- Ratio of asymptotic to peak amplitude indicates nonlinearity degree.
- Explicit formulas for wave height, width, impulse, and energy.

## Abstract

Properties of solitary waves in pre-compressed Hertzian chains of particles are studied in the long wavelength limit using a well-known continuum model. Several main results are obtained by parameterizing the solitary waves in terms of their wave speed and their asymptotic amplitude. First, the asymptotic amplitude is shown to be directly related to the continuum sound speed, and the ratio of asymptotic amplitude to peak amplitude is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. Second, an algebraic relation is derived that determines the dynamical nonlinearity ratio in terms of the ratio of the solitary wave speed to the sound speed. In particular, highly supersonic solitary waves correspond to highly nonlinear propagating pulses in weakly compressed systems, and slightly supersonic solitary waves correspond to weakly nonlinear propagating pulses in strongly compressed systems. Third, explicit formulas for the physical height, width, impulse and energy of the solitary waves are obtained in both the strongly nonlinear regime and the weakly nonlinear regime. Asymptotic expansions are used to show that in the strongly nonlinear regime, solitary waves are well-approximated by Nesterenko's compacton (having the same wave speed), while in the weakly nonlinear regime, solitary waves coincide with solitons of the Korteweg-de Vries (KdV) equation, with the same wave speed. All of these results are illustrated by means of exact solitary wave solutions, including the physically important case that models a chain of spherical particles.