  We investigate the dynamics of a lattice soliton on a monatomic chain in the
presence of damping and external forces. We consider Stokes and hydrodynamical
damping. In the quasi-continuum limit the discrete system leads to a damped and
forced Boussinesq equation. By using a multiple-scale perturbation expansion up
to second order in the framework of the quasi-continuum approach we derive a
general expression for the first-order velocity correction which improves
previous results. We compare the soliton position and shape predicted by the
theory with simulations carried out on the level of the monatomic chain system
as well as on the level of the quasi-continuum limit system. For this purpose
we restrict ourselves to specific examples, namely potentials with cubic and
quartic anharmonicities as well as the truncated Morse potential, without
taking into account external forces. For both types of damping we find a good
agreement with the numerical simulations both for the soliton position and for
the tail which appears at the rear of the soliton. Moreover we clarify why the
quasi-continuum approximation is better in the hydrodynamical damping case than
in the Stokes damping case.
