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arXiv · 2610.29353

An Arnold-Type Stability Analysis of Vortex Filaments

Masashi Aiki·Mitsuo Higaki
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Abstract

We establish nonlinear orbital stability of circular arcs with opening angle $θ\le π$ and closed circles under the Localized Induction Equation (LIE), with prescribed endpoint constraints for arcs. Our Arnold-type variational approach combines two conserved quantities, the bending energy and a component of the fluid impulse, into a functional that we prove is comparable to the squared $H^2$-distance from the corresponding orbit for all admissible curves. For circular arcs with angle $θ>π$, we prove nonlinear orbital instability using the inverse generalized Hasimoto transformation.

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Masashi Aiki, Mitsuo Higaki. 2026-10-21. An Arnold-Type Stability Analysis of Vortex Filaments. https://arxiv.org/abs/2610.29353

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