  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.
