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Vortex filament on symmetric Lie algebras and generalized bi-Schrödinger flows. (English) Zbl 1410.53064
In this paper, the authors present 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-Schrödinger flow, 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 paper is interesting and presents a subject of great interest not only in physics but also in Riemannian and pseudo-Riemannian geometry, namely the theory of moving curves. In the final part of the paper, the authors apply the technique of Sym and Pohlmeyer to produce geometric realizations of the equation of generalized bi-Schrödinger flows. Then, the authors use the geometric concept of PDEs with given curvature representation proposed in the category of Yang-Mills theory. This concept is applied with great success in the theory of the matrix nonlinear Schrödinger-like equation.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A04 Curves in Euclidean and related spaces
53C35 Differential geometry of symmetric spaces
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