Kelvin-Wave Multiplication by Sound Emission

A subsonic Kelvin quantum can emit a phonon and split into two softer quanta, increasing Kelvin population while losing energy. In the soft-phonon limit, rotational invariance relates the leading acoustic shear amplitude, odd under reversal of phonon axial momentum, to the energy released. Microscopic calculations support this dispersion dependence. In a reflecting cavity with coherent sound and Kelvin seeds, Gross--Pitaevskii evolution shows Kelvin gain, larger mean-square displacement and smaller mean-square slope; a $π$ shift of the initial sound phase reverses these changes.