PreprintCC BY 4.0arXiv
An Arnold-Type Stability Analysis of Vortex Filaments
Abstract
We establish nonlinear orbital stability of circular arcs with opening angle θ≤ 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 H2-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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BibTeX
@misc{aiki2026arnoldtype,
title = {{An Arnold-Type Stability Analysis of Vortex Filaments}},
author = {Masashi Aiki and Mitsuo Higaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29353},
eprint = {2610.29353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29353},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Arnold-Type Stability Analysis of Vortex Filaments
AU - Masashi Aiki
AU - Mitsuo Higaki
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29353
UR - https://arxiv.org/abs/2610.29353
AB - We establish nonlinear orbital stability of circular arcs with opening angle θ≤ 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.
ER -