  We study the statistical properties of the Kelvin waves propagating along
quantized superfluid vortices driven by the Gross-Pitaevskii equation. No
artificial forcing or dissipation is added. Vortex positions are accurately
tracked. This procedure directly allows us to obtain the Kevin-waves
occupation-number spectrum. Numerical data obtained from long time integration
and ensemble-average over initial conditions supports the spectrum proposed in
[L'vov and Nazarenko, JETP Lett 91, 428 (2010)]. Kelvin wave modes in the
inertial range are found to be Gaussian as expected by weak-turbulence
predictions. Finally the dissipative range of the Kelvin-wave spectrum is
studied. Strong non-Gaussian fluctuations are observed in this range.
