  We study bright solitons in a Bose-Einstein condensate with a spin-orbit
coupling that has been realized experimentally. Both stationary bright solitons
and moving bright solitons are found. The stationary bright solitons are the
ground states and possess well-defined spin-parity, a symmetry involving both
spatial and spin degrees of freedom; these solitons are real valued but not
positive definite, and the number of their nodes depends on the strength of
spin-orbit coupling. For the moving bright solitons, their shapes are found to
change with velocity due to the lack of Galilean invariance in the system.
