Differential approximation for Kelvin-wave turbulence

I present a nonlinear differential equation model (DAM) for the spectrum of Kelvin waves on a thin vortex filament. This model preserves the original scaling of the six-wave kinetic equation, its direct and inverse cascade solutions, as well as the thermodynamic equilibrium spectra. Further, I extend DAM to include the effect of sound radiation by Kelvin waves. I show that, because of the phonon radiation, the turbulence spectrum ends at a maximum frequency $\omega^* \sim (\epsilon^3 c_s^{20} / \kappa^{16})^{1/13}$ where $\epsilon$ is the total energy injection rate, $c_s$ is the speed of sound and $\kappa$ is the quantum of circulation.