Characterizing Traveling Wave Collisions in Granular Chains Starting from Integrable Limits: the case of the KdV and the Toda Lattice

TL;DR

This paper develops approximations for traveling wave collisions in granular chains using integrable models like KdV and Toda, quantifying their accuracy and applicability to different collision scenarios.

Contribution

It introduces a method to approximate granular chain soliton interactions with integrable equations, analyzing the accuracy for various collision types.

Findings

KdV accurately models same-direction soliton collisions.

Toda lattice captures both co- and counter-propagating collisions.

Numerical errors are quantified and linked to mathematical bounds.

Abstract

Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both 1-soliton and 2-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both co-propagating and counter-propagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.