Regular Polygonal Vortex Filament Evolution and Exponential Sums

Abstract

When taking a regular planar polygon of \(M\) sides and length \(2\pi \) as the initial datum of the vortex filament equation, \(\mathbf{X}_{t}= \mathbf{X}_{s}\wedge \mathbf{X}_{ss}\), the solution becomes polygonal at times of the form \(t_{pq} = (p/q)(2\pi /M^{2})\), with \(\gcd (p,q)=1\), and the corresponding polygon has \(Mq\) sides, if \(q\) is odd, and \(Mq/2\) sides, if \(q\) is even. Moreover, that polygon is skew (except when \(q = 1\) or \(q = 2\), where the initial shape is recovered), and the angle \(\rho \) between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time \(t_{pq}\), \(\cos ^{q}(\rho /2) = \cos (\pi /M)\), if \(q\) is odd, and \(\cos ^{q}(\rho /2) = \cos ^{2}(\pi /M)\), if \(q\) is even. Since the transition of one side of the polygon to the next one is given by a rotation in \(\mathbb{R}^{3}\) determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle \(2\pi /M\), which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.