  We study current induced magnetization dynamics in a long thin ferromagnetic
wire with Dzyaloshinskii-Moriya interaction (DMI). We find a spiral domain wall
configuration of the magnetization and obtain an analytical expression for the
width of the domain wall as a function of the interaction strengths. Our
findings show that above a certain value of DMI a domain wall configuration
cannot exist in the wire. Below this value we determine the domain wall
dynamics for small currents, and calculate the drift velocity of the domain
wall along the wire. We show that the DMI suppresses the minimum value of
current required to move the domain wall. Depending on its sign, the DMI
increases or decreases the domain wall drift velocity.
