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On the vortex filament in 3-spaces and its generalizations

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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 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 G2 structure on \({\mathbb{S}^6}\). As a new result in this survey, we describe the equation of generalized bi-Schrödinger flows from ℝ1 into a Riemannian surface.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 11531012, 11926307 and 12071080).

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Qing Ding

  2. School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, 341000, China

    Shiping Zhong

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  1. Qing Ding
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  2. Shiping Zhong
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Correspondence to Qing Ding.

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In Memory of Professor Zhengguo Bai (1916–2015)

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Ding, Q., Zhong, S. On the vortex filament in 3-spaces and its generalizations. Sci. China Math. 64, 1331–1348 (2021). https://doi.org/10.1007/s11425-020-1839-5

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