Generalized Bi-Schr\"odinger Flows and Vortex Filament on Symmetric Lie Algebras

TL;DR

This paper develops a geometric framework for third-order vortex filament models on symmetric Lie algebras, extending previous integrable approaches and broadening the theoretical understanding of vortex dynamics in higher dimensions.

Contribution

It introduces a purely geometric method using generalized bi-Schrödinger flows to model vortex filaments on symmetric Lie algebras, surpassing integrability restrictions.

Findings

Established third-order vortex filament models on symmetric Lie algebras

Overcame limitations of integrable methods in higher dimensions

Connected geometric flows with vortex filament theory

Abstract

The theory of the vortex filament in three-dimensional fluid dynamics, consisting mainly of the models up to the third-order approximation, is an attractive subject in both physics and mathematics. Many efforts have been devoted to the extension of the theory to higher-dimensional symmetric Lie algebras. However, such a generalization known in literature is strongly restricted by the integrable method. In this article, we endeavor to establish the third-order models of the vortex filament on symmetric Lie algebras in a purely geometric way by generalized bi-Schr\"odinger flows. Our generalization overcomes the limitation of integrability and creates successfully the desired models on Hermitian or para-Hermitian symmetric Lie algebras. Combining the result in this article with what have been known in literature for the leading-order and the second-order models, we actually exhibit the basic models and the related theory of the vortex filament on symmetric Lie algebras up to the third-order approximation.