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.

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