A regularized continuum model for traveling waves and dispersive shocks of the granular chain

TL;DR

This paper develops a regularized continuum PDE model for granular crystals, analyzing traveling waves and dispersive shocks through analytical, numerical, and modulation techniques, and validating the model against discrete simulations.

Contribution

It introduces a novel regularized continuum PDE model that accurately approximates the discrete granular crystal system for analyzing wave phenomena.

Findings

The continuum model captures traveling wave solutions effectively.

Dispersive shock wave features are well-described by Whitham modulation theory.

Numerical simulations confirm the continuum model's accuracy in approximating the discrete system.

Abstract

In this paper we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations (DDEs). After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation (PDE). We then compute, both analytically and numerically, its traveling wave and periodic traveling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one.