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Integrable dynamics of knotted vortex filaments. (English) Zbl 1066.37049
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 5th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 5–12, 2003. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-8-7/pbk). 11-50 (2004).
This series of lectures adresses relationships between integrability (the Floquet spectrum of a given solution), geometry and topology of closed curve solutions of the vortex flament equation. Among others, the author discusses the explicit algebro-geometric construction of N-phase solutions of the vortex flament equation, following Krichever’s use of the Baker-Akhiezer function.
MSC:
37N10Dynamical systems in fluid mechanics, oceanography and meteorology
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K25Relations of infinite-dimensional systems with differential geometry
35Q53KdV-like (Korteweg-de Vries) equations
76B47Vortex flows
53C44Geometric evolution equations (mean curvature flow, Ricci flow, etc.)