Phonon radiation is shown numerically to damp the motion of a relativistica
lly driven phi (4) soliton, even in the extremely weak discreteness limit.
At higher discreteness, the soliton mobility is characterized by a disconti
nuous dependence on the driving force; jumps are related to phonon and brea
ther-radiation thresholds. Moreover, the speed of a frictionless soliton ca
nnot be lowered below a certain threshold value, or else it might get trapp
ed between two adjacent chain sites.