Magnetic axion vortex and very light QCD axions

We show that a stable vortex soliton, carrying a constant magnetic flux, exists in the homogeneous medium of axions with a constant time-derivative. Axions can be bound in the vortex, having energy less than the axion mass. If the observed magnetic fields in galaxies are those of the vortex, the axion-photon coupling has to be smaller than ${10}^{-17}\mathrm{G}\mathrm{e}{\mathrm{V}}^{-1}$. Otherwise, axions decay too quickly to constitute dark matter in galaxies.