A steady solution to the hydrodynamic equation and incommensurate magnetization in a U(2) invariant superfluid

At the zero temperature limit, a one-dimensional steady solution to the hydrodynamic equation of a U(2) invariant superfluid is obtained. This solution reveals that the magnitude of magnetization is always directly proportional to the particle number density. Furthermore, the problem can be interpreted as a particle's motion in a central force field. It is demonstrated that the particle's orbits are elliptical in shape, with a precession angle determined by a non-zero mass current. This suggests that the spatial periods of the three component magnetizations are not commensurate. These findings indicate that the coupling of mass superflow and magnetization distortions usually results in an incommensurate magnetization.