  We theoretically study the current-induced dynamics of a transverse magnetic
domain wall in bi-layer nanowires consisting of a ferromagnet on top of a
nonmagnet having strong spin-orbit coupling. Domain wall dynamics is
characterized by two threshold current densities, $J_{th}^{WB}$ and
$J_{th}^{REV}$, where $J_{th}^{WB}$ is a threshold for the chirality switching
of the domain wall and $J_{th}^{REV}$ is another threshold for the reversed
domain wall motion caused by spin Hall effect. Domain walls with a certain
chirality may move opposite to the electron-flow direction with high speed in
the current range $J_{th}^{REV} < J < J_{th}^{WB}$ for the system designed to
satisfy the conditions $J_{th}^{WB} > J_{th}^{REV}$ and \alpha > \beta, where
\alpha is the Gilbert damping constant and \beta is the nonadiabaticity of spin
torque. Micromagnetic simulations confirm the validity of analytical results.
