Geometry-controlled zeros of resonant Kelvin-wave conversion in four spatial dimensions

Abstract

Kelvin waves carry energy along vortices, but their resonant interactions contain information that is absent from the linear spectrum. We identify a geometry that exposes this information on a vortex surface in four spatial dimensions. Perpendicular incoming waves cancel the complete local-area four-wave coupling, leaving the finite nonlocal interaction mediated by the surrounding fluid. This interaction restores conversion at generic wave-number ratios, yet changes sign and vanishes at a selected ratio while momentum and frequency remain exactly matched. Matching to Gross-Pitaevskii (GP) theory, including the microscopic core response, establishes a zero of the full coupling at every sufficiently small nonzero wave number, with ratio tending to $λ_*\simeq1.469$. Finite-grid GP coefficients exhibit a sign reversal for the same resonant quartets. Separate coherent-mixing calculations and weak-probe spectra connect the coupling sign to the generated-wave phase and the ordering of reflection-even and reflection-odd modes. The resulting suppression removes a leading channel for changing propagation directions without changing the paired wavelengths. These results isolate a nonlinear effect of the surrounding flow that local tension misses, providing a microscopic benchmark for vortex-wave theories and a starting point for studying angular redistribution beyond the Kelvin spectrum.