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

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.

arXiv comment18 pages, 9 figures

References in corpus (16)

• Energy trapping and shock disintegration in a composite granular medium
• Anomalous wave reflection from the interface of two strongly nonlinear granular media
• Discrete Breathers in One-Dimensional Diatomic Granular Crystals
• How Hertzian solitary waves interact with boundaries in a 1-D granular medium
• Highly Nonlinear Solitary Waves in Periodic Dimer Granular Chains
• Localized Breathing Modes in Granular Crystals with Defects
• Wave localization in strongly nonlinear Hertzian chains with mass defect
• Compactons and Chaos in Strongly Nonlinear Lattices
• Experimental evidence of shock mitigation in a Hertzian tapered chain
• Pulse Velocity in a Granular Chain
• Strongly nonlinear wave dynamics in a chain of polymer coated beads
• Damped-Driven Granular Crystals: An Ideal Playground for Dark Breathers and Multibreathers
• Experimental evidence of solitary wave interaction in Hertzian chains
• Solitary and shock waves in discrete double power-law materials
• On the existence of solitary traveling waves for generalized Hertzian chains
• Counterpropagating Two-Soliton Solutions in the FPU Lattice

Cited by in corpus (7)

• Nonlinear Coherent Structures in Granular Crystals
• Nonlinear waves in flexible mechanical metamaterials
• Shock and Rarefaction Waves in Generalized Hertzian Contact Models
• On the Whitham modulation equations for the Toda lattice and the quantitative characterization of its dispersive shocks
• Integrable Approximations of Dispersive Shock Waves of the Granular Chain
• Nanoptera in weakly nonlinear woodpile and diatomic granular chains
• On the generation and propagation of solitary waves in integrable and non-integrable nonlinear lattices