Mutual-friction self-focusing of a Hasimoto soliton and its torsion-driven arrest: a conserved amplitude–torsion pairing on a superfluid vortex filament

Abstract

Abstract A quantized vortex in superfluid helium-4 moves under its self-induced velocity and, at finite temperature, under mutual friction with the normal fluid. In the local-induction approximation the frictionless filament is integrable and its localized excitation is the Hasimoto soliton, a coherent Kelvin-wave packet with curvature amplitude ν and torsion τ. We show that undriven mutual friction of strength α can focus such a packet before damping it. Differentiating against the evolving line element gives exact rate laws for the curvature norm ∫κ² ds and momentum ∫κ²τ ds of any smooth decaying or periodic Frenet-regular profile. On the rigid-sech collective-coordinate manifold these laws close as ν̇ = 2αν(ν² − τ²) and τ̇ = 4αν²τ, and conserve τ/(ν² + τ²). Within this reduction the amplitude grows for ν > τ, reaches νₘₐₓ = (ν₀² + τ₀²)/(2τ₀) exactly when ν = τ, and then decays, while the renormalized line length, hence the local-induction energy, decreases monotonically. The peak time and late-time decay rate follow in closed form. Direct three-dimensional curve evolution verifies both rate laws and the rise to the peak; the post-peak branch remains a reduced-system prediction. For a packet of width 100 μm the predicted peak occurs after 0.13 s at 1.7 K and 0.03 s at 2.1 K, within the scale of particle-tracking experiments.