Vortex Filament Equation for a Regular Polygon

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 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.

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