  The motion of a N\'eel-like ${180}^{\circ}$ domain wall induced by a
time-dependent staggered spin-orbit field in the layered collinear
antiferromagnet Mn$_2$Au is explored. Through an effective version of the two
sublattice nonlinear $\sigma$-model which does not take into account the
antiferromagnetic exchange interaction directed along the tetragonal c-axis, it
is possible to replicate accurately the relativistic and inertial traces
intrinsic to the magnetic texture dynamics obtained through atomistic spin
dynamics simulations for quasistatic processes. In the case in which the
steady-state magnetic soliton motion is extinguished due to the abrupt shutdown
of the external stimulus, its stored relativistic exchange energy is
transformed into a complex translational mobility, being the rigid domain wall
profile approximation no longer suitable. Although it is not feasible to carry
out a detailed follow-up of its temporal evolution in this case, it is possible
to predict the inertial-based distance travelled by the domain wall in relation
to its steady-state relativistic mass. This exhaustive dynamical
characterization for different time-dependent regimes of the driving force is
of potential interest in antiferromagnetic domain wall-based device
applications.
