Spectrum of Kelvin-wave turbulence in superfluids

We derive a type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature, that describes step-by-step (local) energy cascade over scales caused by 4-wave interactions. Resulting new energy spectrum $E\text{Sb}LN(k)\propto k^{-5\mathrm{/}3}$ must replace in future theory (e.g. in finding the quantum turbulence decay rate) the previously used spectrum $E\text{Sb}KS(k)\propto k^{-7\mathrm{/}5}$, which was recently shown to be inconsistent due to nonlocality of the 6-wave energy cascade.