Energy spectrum of the Kelvin-wave cascade on quantized vortices
In plain words
When superfluid vortex lines wiggle, the wiggles (Kelvin waves) are expected to pass energy to ever shorter wavelengths until it is radiated as sound. Two competing theories predict different spectra, and no measurement in a superfluid has decided between them.
Precise statement
For weakly nonlinear helical waves on a quantized vortex line with circulation $\kappa = h/m_4$ (about $10^{-3}\ \mathrm{cm}^2/\mathrm{s}$ in 4He) at $T \to 0$, two theories predict $E(k) \sim k^{-7/5}$ (Kozik-Svistunov, local six-wave interactions) and $E(k) \sim k^{-5/3}$ (L'vov-Nazarenko, effective nonlocal four-wave interactions). Determine the realized exponent and prefactor in superfluid 4He, or in Gross-Pitaevskii and vortex-filament models with resolved wave ranges.
What would settle it
A measurement of Kelvin-wave amplitude spectra on quantized vortices, or simulations resolving two decades in k with exponent errors below 0.05.
Status in the literature
Unverified note
Gross-Pitaevskii and vortex-filament simulations favour $-5/3$; a 2026 experiment on a classical vortex (Barckicke, Gissinger, Falcon, arXiv:2607.07535) observed a six-wave cascade but could not separate the two exponents; no superfluid measurement exists.