Vortex filament on symmetric Lie algebras and generalized bi-Schrödinger flows

In this article, we display an evolving model on symmetric Lie algebras from a purely geometric way by using the Hamiltonian (or para-Hamiltonian) gradient flow of a fourth order functional called generalized bi-Schrodinger flows, which corresponds to the Fukumoto–Moffatt’s model in the theory of moving curves, or the vortex filament in physical words, in \(\mathbb {R}^3\). The theory of vortex filament in \(\mathbb {R}^3\) or \(\mathbb {R}^{2,1}\) up to the third-order approximation is shown to be generalized to symmetric Lie algebras in a unified way.