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

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.