  We systematically investigate the weakly trapped spin-1 Bose-Einstein
condensates with spin-orbit coupling in an external Zeeman field. We find that
the mean-field ground state favors either a magnetized standing wave phase or
plane wave phase when the strength of Zeeman field is below a critical value
related to the strength of spin-orbit coupling. Zeeman field can induce the
phase transition between standing wave and plane wave phases, and we determine
the phase boundary analytically and numerically. The magnetization of these two
phases responds to the external magnetic field in a very unique manner, the
linear Zeeman effect magnetizes the standing wave phase along the direction of
the magnetic field, but the quadratic one demagnetizes the plane wave phase.
When the strength of Zeeman field surpasses the critical value, the system is
completely polarized to a ferromagnetic state or polar state with zero
momentum.
