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arXiv · 1304.5521

Vortex Filament Equation for a Regular Polygon

Francisco de la Hoz·Luis Vega
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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{\ss} 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.

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Francisco de la Hoz, Luis Vega. 2013-03-26. Vortex Filament Equation for a Regular Polygon. https://doi.org/10.1088/0951-7715%2F27%2F12%2F3031

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