  Taking into account the coupling between the position of the wall and an
internal degree of freedom, namely its phase $\phi$, we examine, in the rigid
wall approximation, the dynamics of a magnetic domain wall subject to a weak
pinning potential. We determine the corresponding force-velocity
characteristics, which display several unusual features when compared to
standard depinning laws. At zero temperature, there exists a bistable regime
for low forces, with a logarithmic behavior close to the transition. For weak
pinning, there occurs a succession of bistable transitions corresponding to
different topological modes of the phase evolution. At finite temperature, the
force-velocity characteristics become non-monotonous. We compare our results to
recent experiments on permalloy nanowires.
