  We report on reversible electric-field-driven magnetic domain wall motion in
a Cu/Ni multilayer on a ferroelectric BaTiO$_3$ substrate. In our
heterostructure, strain-coupling to ferroelastic domains with in-plane and
perpendicular polarization in the BaTiO$_3$ substrate causes the formation of
domains with perpendicular and in-plane magnetic anisotropy, respectively, in
the Cu/Ni multilayer. Walls that separate magnetic domains are elastically
pinned onto ferroelectric domain walls. Using magneto-optical Kerr effect
microscopy, we demonstrate that out-of-plane electric field pulses across the
BaTiO$_3$ substrate move the magnetic and ferroelectric domain walls in unison.
Our experiments indicate an exponential increase of domain wall velocity with
electric field strength and opposite domain wall motion for positive and
negative field pulses. Magnetic fields do not affect the velocity of magnetic
domain walls, but independently tailor their internal spin structure, causing a
change in domain wall dynamics at high velocities.
