  We derive exact static as well as moving solitonic solutions to the
one-dimensional spin-orbit-coupled F=1 Bose-Einstein condensates. The static
polar soliton is shown to be the ground state by the imaginary-time evolution
method. It shows a helical modulation of the order parameter due to the
spin-orbit coupling. In particular, the moving soliton exhibits a periodic
oscillation among the particle numbers of the hyperfine states. We further
explore the temporal evolution of the static polar soliton and find that the
spin-polarization exhibits dynamical oscillations. This disappearance and
re-emergence of the ferromagnetic state indicates the mixing of the
ferromagnetic and the antiferromagnetic manifolds.
