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

TL;DR

This work analyzes dispersive and rarefaction wave dynamics in a Hertzian FPU lattice using two quasi-continuum models, log-KdV and generalized-KdV, to approximate the discrete system. It derives Whitham modulation equations for both models via averaging, reduces them to simple-wave ODEs to extract DSW edge features, and obtains self-similar RW solutions from dispersionless limits. DSW fitting provides analytic predictions for edge speeds, which are then validated against numerical simulations of both the lattice and the quasi-continuum models; results show good agreement for small initial jumps and α near 1. The study demonstrates that these quasi-continuum descriptions, including their Lagrangian structure and modulation theory, effectively capture key wave phenomena in granular chains and offer a framework for further data-driven and bi-directional extensions.

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.

Figures (9)

• Figure 1: The discrete dispersive shock wave of the lattice \ref{['eq: granular crystals']}. The left panel $(a)$ shows the space-time dynamics of the DDSW, where the two dashed black lines represent the theoretical predictions on the linear and solitonic edges of the DDSW based on the DSW-fitting results (See Section \ref{['sec: DSW fitting']}), while the right panel $(b)$ depicts the spatial profile of the lattice DDSW at $t = 500$. Notice that $\alpha = 1.1$.
• Figure 2: The discrete rarefaction wave. The panel $(a)$ and $(b)$ refer to the density plot of the magnitude of the field $u_n$ and the spatial profile of the discrete RW of the lattice \ref{['eq: granular crystals']} at $t = 500$, respectively. Also, note that $\alpha = 1.1$.
• Figure 3: The Gaussian "dark" solitary wave in Eq. \ref{['eq: solitart-wave for log-kdv']}. The left panel $(a)$ depicts the spatial profile of the dark solitary wave at $\tau = 0$, while the right panel $(b)$ displays the space-time evolution dynamics of such solitary wave with a speed of propagation $c = 0.5$.
• Figure 4: The potential curves. Panel $(a)$ depicts the potential curve \ref{['eq: Potential curves of log-kdv']} of the log-KdV equation \ref{['eq: log-kdv approximation']}, while $(b)$ shows that \ref{['eq: potential curve of g-kdv']} of the generalized-KdV equation \ref{['eq: generalized KdV']}.
• Figure 5: The comparisons of the rarefaction waves: Panels $(a)$ and $(b)$ depicts the comparison of the RWs between the log-KdV, generalized-KdV self-similar solutions in Eqs. \ref{['eq: self-similar solution']}, \ref{['eq: self-similar solution of gkdv']} and the associated lattice RWs of the discrete model \ref{['eq: granular crystals']} at $t = 200$, where we utilize the parameter value $\alpha = 1.1$.