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.
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