  We study theoretically the dynamics of composite domain walls (DW) in
multiferroic material GdFeO${}_3$ driven by magnetic field $H$. Two
antiferromagnetic orders of Fe and Gd spins interact Gd-ion displacement in
this system with coupling $c$ at low temperatures, and we have numerically
simulated the corresponding time-dependent Ginzburg-Landau equations in which
magnetic field $H$ couples to Fe-spin order parameter. We vary $H$ and $c$
systematically and calculate velocity $v$ and inner structure in a stationary
state for magnetic DW and magneto-electric DW. DW mobility $v/H$ increases with
$H$ and decreases with $c$ for both DWs, but their characteristics differ
between the two. We have also studied analytically the smooth characteristics
of magneto-electric DW by perturbation theory. Another finding is a splitting
instability at large H. A magneto-electric DW splits into a pair of magnetic DW
and electric DW when $c$ is large, while a magnetic DW splits when $c$ is
small. Internal structure of composite DW deforms with increasing $c$ and $H$,
and modulations in different order parameters separate in space. Their relative
distances show a noticeable enhancement with approaching the splitting
instability.
