  We study the dynamics of skyrmions under spin-transfer torque in
Dzyaloshinskii-Moriya materials with easy-axis anisotropy. In particular, we
study the motion of a topological skyrmion with skyrmion number $Q=1$ and a
non-topological skyrmionium with $Q=0$ using their linear momentum, virial
relations, and numerical simulations. The non-topological $Q=0$ skyrmionium is
accelerated in the direction of the current flow and it either reaches a steady
state with constant velocity, or it is elongated to infinity. The steady-state
velocity is given by a balance between current and dissipation and has an upper
limit. In contrast, the topological $Q=1$ skyrmion converges to a steady-state
with constant velocity at an angle to the current flow. When the spin current
stops the $Q=1$ skyrmion is spontaneously pinned whereas the $Q=0$ skyrmionium
continues propagation. Exact solutions for the propagating skyrmionium are
identified as solutions of equations given numerically in a previous work.
Further exact results for propagating skyrmions are given in the case of the
pure exchange model. The traveling solutions provide arguments that a
spin-polarized current will cause rigid motion of a skyrmion or a skyrmionium.
