Vortex Filament Equation for a Regular Polygon

TL;DR

This paper investigates the evolution of a vortex filament starting from a regular polygon, revealing that it remains polygonal at rational times and exhibits fractal behavior linked to multifractal functions.

Contribution

It demonstrates that vortex filament evolution preserves polygonal structure at rational times and characterizes it using generalized quadratic Gauss sums, supported by numerical simulations.

Findings

Vortex filament remains polygonal at rational times.

Evolution can be described by quadratic Gauss sums.

Fractal behavior relates to Riemann's non-differentiable function.

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.