Quasi-continuum descriptions of rarefaction and dispersive shock waves in Fermi-Pasta-Ulam lattices with Hertzian potentials

TL;DR

This paper reviews quasi-continuum models of Fermi-Pasta-Ulam lattices with Hertzian potentials, deriving Whitham modulation equations to analyze dispersive shock waves and rarefaction waves, and validating the models against numerical simulations.

Contribution

It introduces analytical derivations of Whitham modulation equations for these models and demonstrates their effectiveness in approximating discrete wave phenomena.

Findings

Good agreement between analytical and numerical wave solutions

Edge speeds of DSWs can be effectively predicted

Quasi-continuum models accurately capture wave dynamics

Abstract

In the present work, we review two well-established quasi-continuum models of a Fermi-Pasta-Ulam lattice with Hertzian type potentials, and utilize these two models to approximate the discrete dispersive shock waves (DDSWs) which are numerically observed in the simulation of the lattice. To perform analysis on the various characteristics of the DDSW, we analytically derive the Whitham modulation equations of the two quasi-continuum models, which govern the slowly varying spatial and temporal dynamics of distinct parameters of the periodic solutions. We then perform a very useful reduction of the Whitham modulation system to gain a system of initial-value problems whose solutions can provide important insights on edge features of the DSWs such as their edge speeds. In addition, we also study the numerical rarefaction waves (RWs) of the lattice based on the two quasi-continuum models. In particular, we analytically compute and compare their self-similar solutions with the numerical discrete RW of the lattice. These comparisons made for both DSWs and RWs reveal to be reasonably good, which suggest the impressive performance of both quasi-continuum models.