Ultra-light particles, such as axions, form a macroscopic condensate around a
highly spinning black hole by the superradiant instability. Due to its
macroscopic nature, the condensate opens the possibility of detecting the axion
through gravitational wave observations. However, the precise evolution of the
condensate must be known for the actual detection. For future observation, we
numerically study the influence of the self-interaction, especially interaction
between different modes, on the evolution of the condensate in detail. First,
we focus on the case when condensate starts with the smallest possible angular
quantum number. For this case, we perform the non-linear calculation and show
that the dissipation induced by the mode interaction is strong enough to
saturate the superradiant instability, even if the secondary cloud starts with
quantum fluctuations. Our result indicates that explosive phenomena such as
bosenova do not occur in this case. We also show that the condensate settles to
a quasi-stationary state mainly composed of two modes, one with the smallest
angular quantum number for which the superradiant instability occurs and the
other with the adjacent higher angular quantum number. We also study the case
when the condensate starts with the dominance of the higher angular quantum
number. We show that the dissipation process induced by the mode coupling does
not occur for small gravitational coupling. Therefore, bosenova might occur in
this case.
