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In this article, we devote to a mathematical survey on the theory of the vortex filament in 3dimensional spaces and its generalizations. We shall present some effective geometric tools applied in the study, such as the Schrödinger flow, the geometric Korteweg-de Vries (KdV) flow and the generalized bi-Schrödinger 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 G 2 structure on S 6 . As a new result in this survey, we describe the equation of generalized bi-Schrödinger flows from R 1 into a Riemannian surface.

In this article, the coupled matrix nonlinear Schrödinger (NLS) type equations are gauge equivalent to the equation of Schrödinger flow from R 1 {{\mathbb{R}}}^{1} to complex Grassmannian manifold G ˜ n , k = GL ( n , C ) ∕ GL ( k , C ) × GL ( n − k , C ) , {\widetilde{G}}_{n,k}={\rm{GL}}\left(n,{\mathbb{C}})/{\rm{GL}}\left(k,{\mathbb{C}})\times {\rm{GL}}\left(n-k,{\mathbb{C}}), which generalizes the correspondence between Schrödinger flow to the complex 2-sphere C S 2 ( 1 ) ↪ C 3 {\mathbb{C}}{{\mathbb{S}}}^{2}\left(1)\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\mathbb{C}}}^{3} and the coupled Landau-Lifshitz (CLL) equation. This gives a geometric interpretation of the matrix generalization of the coupled NLS equation (i.e., CLL equation) via Schrödinger flow to the complex Grassmannian manifold G ˜ n , k {\widetilde{G}}_{n,k} . Finally, we explicit soliton solutions of the Schrödinger flow to the complex Grassmannian manifold G ˜ 2 , 1 {\widetilde{G}}_{2,1} .