  Understanding the interplay between different wave excitations, such as
phonons and localized solitons, is crucial for developing coarse-grained
descriptions of many-body, near-integrable systems. We treat the
Fermi-Pasta-Ulam-Tsingou (FPUT) non-linear chain and show numerically that at
short timescales, relevant to the largest Lyapunov exponent, it can be modeled
as a random perturbation of its integrable approximation -- the Toda chain. At
low energies, the separation between two trajectories that start at close
proximity is dictated by the interaction between few soliton modes and an
intrinsic, apparent bath representing a background of many radiative modes. It
is sufficient to consider only one randomly perturbed Toda soliton-like mode to
explain the power-law profiles reported in previous works, describing how the
Lyapunov exponent of large FPUT chains decreases with the energy density of the
system.
