  In recent experiments with ultracold gases a Raman coupling scheme is used to
produce both spin-orbit (SO) and Zeeman-type couplings [Y.-J. Lin et al.,
Nature 471, 83 (2011)]. Their competition drives a phase transition to a
magnetized state with broken $Z_2$ symmetry. Using a hydrodynamic approach we
study a confined binary condensate subject to both SO and Zeeman-type
couplings. We find that in the limit of small healing length and in the phase
with unbroken symmetry, the boundary magnetization profile has an analytical
solution in the form of a sine-Gordon soliton. The soliton is bound to the edge
of the system by the nontrivial boundary condition resulting from the combined
effect of the SO coupling and the drop in the particle density. The same
boundary condition is important in the magnetized phase as well, where we
characterize numerically the boundary spin structure. We further discuss how
the nontrivial magnetization structure affects the density profile near the
boundary, yet another prediction that can be tested in current experiments of
spin-orbit coupled condensates.
