Vortex Filament Equation for a Regular Polygon

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 Gauß 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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