Superradiant instability of ultralight bosons can produce clouds around rotating black holes, whose continuous gravitational wave (GW) emission is a promising observational target. Precise predictions of the signal frequency and its evolution are essential for detecting such continuous GWs. For axions, self-interactions can populate multiple superradiant modes via nonlinear couplings, and GW emission can occur through various channels. To calculate the frequency shifts of GWs emitted through these channels, we employ relativistic perturbation theory based on a bilinear form. We apply this framework to self-interaction effects for the first time, and also revisit the treatment of the self-gravity contribution. Our results provide a simple and unified framework for calculating frequency shifts, including cases in which multiple modes are excited, and are relevant for next-generation GW observations.

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• Classical black holes
• Gravitational wave sources
• Astronomical black holes
• Axion-like particles
• Axions