On the Evolution of the Vortex Filament Equation for regular $M$-polygons with nonzero torsion

TL;DR

This paper investigates the evolution of regular polygons with nonzero torsion under the Vortex Filament Equation, revealing non-periodic, multifractal, helical trajectories and connecting these solutions to the instability of smooth vortex solutions.

Contribution

It extends the understanding of VFE solutions to polygons with torsion, showing their complex, non-periodic evolution and multifractal behavior, supported by algebraic and numerical methods.

Findings

Solutions are polygons at rational times.

Evolution is non-periodic and trajectories are helical for large times.

Links between vortex filament solutions and instability of smooth curves.

Abstract

In this paper, we consider the evolution of the Vortex Filament equation (VFE): \begin{equation*} \mathbf X_t = \mathbf Xs \wedge \mathbf Xss, \end{equation*} taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point $\mathbf X(0,t)$ is not planar, and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann's non-differentiable function that are as close to smooth curves as desired, when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the selfsimilar solutions of VFE have finite renormalized energy.