  We derive the equations of motion of a Domain Wall in a thin-film magnet
coupled to the surface states of a Topological Insulator in the presence of of
both an electric field along the Domain Wall and a magnetic field perpendicular
to the junction. We show how the electric field acts as a chirality stabilizer
holding off the appearance of Walker breakdown and enhancing the terminal
velocity. We also propose a mechanism to reverse the Domain Wall chirality in a
controllable manner by tuning the chiral current flowing through the Wall. An
input from a weak perpendicular magnetic field is required in order to break
the reflection symmetry that protects the degeneracy of the chirality vacuum.
