  We study the statistical and dynamical behavior of turbulent Kelvin waves
propagating on quantized vortices in superfluids, and address the controversy
concerning the energy spectrum that is associated with these excitations.
Finding the correct energy spectrum is important because Kelvin waves play a
major role in the dissipation of energy in superfluid turbulence at near-zero
temperatures. In this paper, we show analytically that the solution proposed in
Ref. \cite{10LN} enjoys existence, uniqueness and regularity of the pre-factor.
Furthermore, we present numerical results of the dynamical equation that
describes to leading order the non-local regime of the Kelvin wave dynamics. We
compare our findings with the analytical results from the proposed local and
non-local theories for Kelvin wave dynamics and show an agreement with the
non-local predictions. Accordingly, the spectrum proposed in Ref. \cite{10LN}
should be used in future theories of quantum turbulence. Finally, for weaker
wave forcing we observe an intermittent behavior of the wave spectrum with a
fluctuating dissipative scale, which we interpreted as a finite-size effect
characteristic to mesoscopic wave turbulence.
