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Our aim in the present work is to develop approximations for the collisionaldynamics of traveling waves in the context of granular chains in the presenceof precompression. To that effect, we aim to quantify approximations of therelevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV)equation and the Toda lattice. Using the availability in such settings of both1-soliton and 2-soliton solutions in explicit analytical form, we initializesuch coherent structures in the granular chain and observe the proximity of theresulting evolution to the underlying integrable (KdV or Toda) model. While theKdV offers the possibility to accurately capture collisions of solitary wavespropagating in the same direction, the Toda lattice enables capturing bothco-propagating and counter-propagating soliton collisions. The error in theapproximation is quantified numerically and connections to bounds establishedin the mathematical literature are also given.

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