On the vortex filament in 3-spaces and its generalizations

Abstract

In this article, we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations. We shall present some effective geometric tools applied in the study, such as the Schrodinger flow, the geometric Korteweg-de Vries (KdV) flow and the generalized bi-Schrodinger flow, as well as the complex and para-complex structures. It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating, since it relates to almost complex structures and the G2 structure on $${\mathbb{S}^6}$$ . As a new result in this survey, we describe the equation of generalized bi-Schrodinger flows from ℝ1 into a Riemannian surface.