We use two-dimensional plane strain finite difference calculations to study dynamic rupture on a material discontinuity interface between a compliant elastic layer and a stiffer elastic medium. Previous works established that rupture along a material interface governed by Coulomb friction propagates as a unidirectional narrow pulse associated with self-sharpening and divergent behavior. These effects are generated by coupling between spatial variations of slip along a material interface and local changes of normal stress. The simulations here employ a regularized friction law with a fading memory dependence of frictional strength on normal stress rather than the instantaneous Coulomb-type response. We find that the self-sharpening and divergent behavior found earlier with Coulomb friction exists also with regularized friction for large enough propagation distance. The parameters of regularized friction have to be fine tuned to produce apparent stability for a given propagation distance. However, eventually, the pulse always either diverges or dies. For cases with a single material interface the pulse strength increases with increasing contrast of shear wave velocity up to a maximum at ~30% contrast, beyond which the generalized Rayleigh wave does not exist. This is similar to earlier results with Coulomb friction. For models having layer width comparable to or somewhat larger than the imposed source size, there is strong dependence of the pulse strength and shape on the layer width and velocity. The pulse amplitude is modulated by regular oscillations with period proportional to the layer width and is amplified for a range of layer widths. The results suggest that rupture along an interface between a compliant layer and a stiffer surrounding medium, initiated by a failure of an asperity with size not larger than the layer width, can become a self-sustaining wrinkle-like pulse.