Vortex Filament Equation for a regular polygon in the hyperbolic plane

TL;DR

This paper investigates the evolution of the vortex filament equation for regular polygons in hyperbolic space, demonstrating numerical methods' effectiveness and revealing superposition properties similar to Euclidean cases.

Contribution

It provides the first numerical analysis of VFE for polygons in hyperbolic space and shows the superposition principle applies in this non-Euclidean setting.

Findings

Numerical solutions agree with algebraic predictions.

Polygon ends grow exponentially in hyperbolic space.

Superposition of one-corner evolutions at infinitesimal times.

Abstract

The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.