  In this paper we consider two models of soliton dynamics (the sine Gordon and
the \phi^4 equations) on a 1-dimensional lattice. We are interested in
particular in the behavior of their kink-like solutions inside the Peierls-
Nabarro barrier and its variation as a function of the discreteness parameter.
We find explicitly the asymptotic states of the system for any value of the
discreteness parameter and the rates of decay of the initial data to these
asymptotic states. We show that genuinely periodic solutions are possible and
we identify the regimes of the discreteness parameter for which they are
expected to persist. We also prove that quasiperiodic solutions cannot exist.
Our results are verified by numerical simulations.
