  We consider a thin ferromagnetic layer to which an external field or a
current are applied along an in plane easy axis. The perpendicular hard axis
anisotropy constant is large so that the out of plane magnetization component
is smaller than the in plane components. A perturbation approach is used to
obtain the profile and velocity of the moving domain wall. The dynamics of the
in plane components of the magnetization is governed by a reaction diffusion
equation which determines the speed of the profile. We find a simple analytic
expression for the out of plane magnetization showing a symmetric distortion
due to the motion in addition to the asymmetric component due to the
Dzyaloshinskii--Moriya interaction. The results obtained complement previous
studies in which either the Dzyalozhinskii vector or the out of plane hard axis
anisotropy were assumed small. In the regime studied the Walker breakdown is
not observed but the reaction diffusion dynamics predicts a slowing down of the
domain wall for sufficiently large magnetic field. The transition point depends
on the applied field, saturation magnetization and easy axis anisotropy.
