Symmetry of Kelvin-wave dynamics and the Kelvin-wave cascade in the T=0 superfluid turbulence

The article considers implications of tilt symmetry (symmetry with respect to tilting of the coordinate axis with respect to which vortex motion is studied) in the non-linear dynamics of Kelvin waves. The conclusion is that although the spectrum of Kelvin wave is not tilt-invariant, this does not compromise tilt invariance of the Kelvin-wave cascade vividly argued now in the theory of superfluid turbulence. The article investigates the effect of strong kelvon interaction on the power-law exponent for the Kelvin-wave cascade and suggests a simple picture of the crossover from the classical Kolmogorov cascade to the quantum Kelvin-wave cascade, which does not encounter with mismatch of the energy distributions at the crossover and does not require a broad intermediate interval for realization of the crossover.