  The dynamics of domain walls in ultrathin ferromagnetic strips with
perpendicular magnetic anisotropy is studied from both numerical and analytical
micromagnetics. The influence of the interfacial Dzyaloshinskii-Moriya
interaction associated to a bi-layer strip arrangement has been considered,
giving rise to the formation of Dzyaloshinskii domain walls. Such walls possess
under equilibrium conditions an inner magnetization structure defined by a
certain orientation angle that make them to be considered as intermediate
configurations between Bloch and N\'eel walls. Two different dynamics are
considered, a field-driven and a current-driven dynamics, in particular, the
one promoted by the spin torque due to the Spin-Hall effect. Results show an
inherent asymmetry associated with the rotation of the domain wall
magnetization orientation before reaching the stationary regime, characterized
by a constant terminal speed. For a certain initial DW magnetization
orientation at rest, the rotation determines whether the reorientation of the
DW magnetization prior to reach stationary motion is smooth or abrupt. This
asymmetry affects the DW motion that can even reverse for a short period of
time. Additionally, it is found that the terminal speed in the case of the
current-driven dynamics may depend on either the initial DW magnetization
orientation at rest or the sign of the longitudinally injected current.
