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

Abstract

In the present work, we review two well-established quasicontinuum models of a Fermi-Pasta-Ulam lattice with Hertzian potentials, and we utilize these two models to approximate the discrete dispersive shock waves which are numerically observed in the simulation of the lattice. To perform analysis on various characteristics of the discrete dispersive shock wave, we analytically derive the Whitham modulation equations of the two quasicontinuum 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 dispersive shock waves such as their edge speeds. In addition, we also study the numerical rarefaction waves of the lattice based on the two quasicontinuum models. In particular, we analytically compute and compare their self-similar solutions with the numerical discrete rarefaction wave of the lattice. These comparisons made for both dispersive shock waves and rarefaction waves reveal to be reasonably good, which suggest the impressive performance of both quasicontinuum models in approximating the nonlinear dispersive wave patterns of the granular crystal lattice.