Vortex filament equation for a regular polygon

Abstract

In this paper, we study the evolution of the localized induction approximation (LIA), also known as vortex filament equation, Xt = Xs ∧Xss, for X(s, 0) 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 Gauß sum. We also study the fractal behavior of X(0, t), relating it with the so-called Riemann’s non-differentiable function, that, as proved by S. Jaffard, fits with the multifractal model of U. Frisch and G. Parisi, for fully developed turbulence. Keywords: vortex filament equation, Schrödinger map, generalized quadratic Gauß sums, pseudo-spectral methods, fractals