Vortex filament equation for a regular polygon

Abstract

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