Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Aiki, Masashi, Higaki, Mitsuo
Institutions: Tokyo University of Science, Kobe University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223199300