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We investigate the dynamics of a lattice soliton on a monatomic chain in thepresence of damping and external forces. We consider Stokes and hydrodynamicaldamping. In the quasi-continuum limit the discrete system leads to a damped andforced Boussinesq equation. By using a multiple-scale perturbation expansion upto second order in the framework of the quasi-continuum approach we derive ageneral expression for the first-order velocity correction which improvesprevious results. We compare the soliton position and shape predicted by thetheory with simulations carried out on the level of the monatomic chain systemas well as on the level of the quasi-continuum limit system. For this purposewe restrict ourselves to specific examples, namely potentials with cubic andquartic anharmonicities as well as the truncated Morse potential, withouttaking into account external forces. For both types of damping we find a goodagreement with the numerical simulations both for the soliton position and forthe tail which appears at the rear of the soliton. Moreover we clarify why thequasi-continuum approximation is better in the hydrodynamical damping case thanin the Stokes damping case

Soliton dynamics in damped and forced Boussinesq equations