Magnetic fields in inhomogeneous axion stars

We study the time evolution of magnetic fields in various configurations of spatially inhomogeneous pseudoscalar fields, which are the coherent superposition of axions. The new induction equation for the magnetic field, which accounts for this inhomogeneity, is derived for such systems. Based on this equation, we study, first, the evolution of two Chern-Simons (CS) waves interacting with a linearly decreasing pseudoscalar field. The nonzero gradient of the pseudoscalar field results in the mixing between these CS waves. Then, we consider the problem in a compact domain, when an initial CS wave is mirror symmetric. In this situation, the inhomogeneity of a pseudoscalar field acts as the effective modification of the $\alpha$-dynamo parameter. Thus, we conclude that the influence of a spatially inhomogeneous pseudoscalar field on the magnetic field evolution strongly depends on the geometry of the system.