Vortex filament equation for a regular polygon

Abstract

Abstract.In this paper, we study the evolution of the vortex filament equation (VFE),with X(s, 0) being 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 was proved by Jaffard to be a multifractal.

The Evolution of the Local Induction Approximation for a Regular Polygon

“…Moreover, it is absolutely in agreement with our numerical simulations (see [8] for more details). Therefore, we think that there is concluding evidence that this formula is valid for any q.…”

“…Bearing in mind (7), it is not too complicate to show [8] that this new ψ also satisfies (6). A very important property of the NLS equation is the fact that it is invariant by the Galilean transformations: if ψ is a solution of (6), so isψ…”

“…However, in a recently submitted paper [8], we have considered for the first time the evolution of (3)- (4), taking a planar regular polygon of M sides as the initial datum.…”

On the energy of critical solutions of the binormal flow

“…In this section we give some evidence that supports the conjecture made in [13] about the evolution of a regular planar polygon according to the binormal flow.…”

“…Our final result is an observation that uses Theorem 1.1 to reinforce the conjecture done in [13] about the evolution of a regular planar polygon according to the binormal flow (see also [17], [21], [15]). In that paper, and after some theoretical arguments, it is conjectured that the evolution of a regular polygon is periodic in time, and that at rational multiples of the time period the curve is a skew polygon with the same angle between consecutive sides.…”