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 ${\omega }^{\ast }\sim ({ϵ}^3{c}_s^{20}\mathrm{/}{\kappa }^{16}{)}^{1\mathrm{/}13}$ where $ϵ$ is the total energy injection rate, $c_s$ is the speed of sound and $\kappa$ is the quantum of circulation.