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Vortex filament equation for a regular polygon

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 14 Nov 2014Embargo end date: 01 Jan 2013 Spain Publisher:IOP PublishingJournal:Nonlinearity, volume 27, pages 3,031-3,057 (issn: 0951-7715, eissn: 1361-6544,

Authors: Luis Vega; Luis Vega; Francisco de la Hoz;

doi: 10.1088/0951-7715/27/12/3031 , 10.48550/arxiv.1304.5521

arXiv: http://arxiv.org/abs/1304.5521

handle: 20.500.11824/48

Vortex filament equation for a regular polygon

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Abstract

In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf X(s, t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau�� sum. We also study the fractal behavior of $\mathbf X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal.

31 pages, 15 figures (27 pages in the final version of Nonlinearity)

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Spain

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Keywords

Mathematics - Analysis of PDEs, 35Q55, 11L05, 65M20, 28A80, FOS: Mathematics, Mathematics - Numerical Analysis, Dynamical Systems (math.DS), Numerical Analysis (math.NA), Mathematics - Dynamical Systems, Analysis of PDEs (math.AP)

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