  We consider a binary Bose-Einstein condensate (BEC) with nonlinear repulsive
interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the
presence of the trapping harmonic-oscillator (HO) potential, we report the
existence of even, odd, and asymmetric spatial modes. They feature alternating
domains with opposite directions of the pseudo-spin, i.e., anti-ferromagnetic
structures, which is explained by the interplay of the linear couplings, HO
confinement, and repulsive self-interaction. The number of the domains is
determined by the strength of the SO coupling. The modes are constructed
analytically in the weakly nonlinear system. The dynamical stability of the
modes is investigated by means of the Bogoliubov-de Gennes equations and direct
simulations. A notable result is that the multi-domain-wall (DW) structures are
stable, alternating between odd and even shapes, while the simplest single-DW
structure is unstable. Thus, the system features a transition to the complex
ground states under the action of the SO coupling. The addition of the Zeeman
splitting transforms the odd modes into asymmetric ones via spontaneous
symmetry breaking. The results suggest possibilities for switching the binary
system between states with opposite (pseudo) magnetization by external fields,
and realization of similar stable states and dynamical effects in solid-state
and nonlinear-optical settings emulated by the SO-coupled BECs.
