  The soliton resolution conjecture for evolution PDEs of dispersive type
states (vaguely) that generic initial data of finite energy give rise
asymptotically to a set of receding solitons and a decaying background
radiation. In this letter, we investigate a possible extension of this
conjecture to discrete lattices of the Fermi-Pasta-Ulam-Tsingou type (rather
than PDEs) in two cases: the case of finite energy initial data and a more
general case where the initial data are a short range perturbation of a
periodic function. In the second case, inspired by rigorous results on the Toda
lattice, we suggest that the soliton resolution phenomenon is replaced by
something somewhat more complicated: a short range perturbation of a periodic
function actually gives rise to different phenomena in different regions. Apart
from regions of (asymptotically) pure periodicity and regions of solitons in a
periodic background, we also observe "modulated" regions of fast oscillations
with slowly varying parameters like amplitude and phase. We have conducted some
numerical calculations to investigate if this trichotomy (pure periodicity +
solitons + modulated oscillations) persists for any discrete lattices of the
Fermi-Pasta-Ulam-Tsingou type. For small perturbations of integrable lattices
like the linear harmonic lattice, the Langmuir chain and the Toda lattice, this
is true. But in general even chaotic phenomena can occur.
