The superradiant instability provides a promising avenue for detecting ultra-light axions through the formation of axion clouds around black holes. The way to observe the axion cloud includes observation of the spin of the black hole and the gravitational waves.To effectively observe these axions, it is crucial to have a precise understanding of the cloud’s evolution and its observable signatures. In this talk, I will discuss the impact of axion self-interaction on both the evolution of the cloud and its observational signatures. In particular, I will show that the self-interaction can produce signals in the different frequency bands.