  Recently, it has been found that a $Q$-ball can amplify waves incident upon
it, due to rotation in the internal space and the interaction of the two modes
in the complex scalar field. While the spherically symmetric 3D case has been
investigated previously, here we explore the 3D axi-symmetric case, which is
numerically much more challenging. The difficulty comes because a partial wave
expansion is needed, and the different partial waves can not be separated, for
either the background spinning Q-ball solution or the perturbative scattering
on top of it. A relaxation method and a high dimensional shooting method are
applied to compute the Q-ball solutions and the amplification factors
respectively. We also classify the behavior of the amplification factors and we
discuss their bounds and the superradiance criteria.
