Integrable dynamics of knotted vortex filaments. (English) Zbl 1066.37049

Calini, Annalisa

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.
For the entire collection see [Zbl 1048.53002].

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

37N10	Dynamical systems in fluid mechanics, oceanography and meteorology
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35Q53	KdV equations (Korteweg-de Vries equations)
76B47	Vortex flows for incompressible inviscid fluids
53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

integrability; Floquet spectrum; vortex flament equation; \(N\)-phase solutions