On the Relationship Between the One-Corner Problem and the M-Corner Problem for the Vortex Filament Equation

Abstract

In this paper, we give evidence that the evolution of the Vortex Filament Equation for a regular M -corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. Therefore, and due to periodicity, the evolution at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner. This interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M -corner case.

Singularity formation for the 1-D cubic NLS and the Schrödinger map on <inline-formula><tex-math id="M1">\begin{document}$\mathbb S^2$\end{document}</tex-math></inline-formula>

“…This means that close to a corner there is a neat transfer of linear momentum. This fact has been confirmed in the numerical experiments done in [8] for regular polygons. Also, and due to the fact that the number of corners depends on the rationality of the time, this local transfer of momentum has a characteristic intermittent behavior.…”

“…Some more recent numerical simulations [8] show that the dynamics at time 0 + of any of the corners of the initial regular polygon is the one of the self-similar solution of (14) that is determined by the angle and location of the corner. As a consequence, the dynamics at 0 + can be understood as the non-linear interaction of infinitely many filaments (as q goes to infinity), one for each corner, that for infinitesimal times each resembles the one of the self-similar solution of (14) studied in [15].…”

Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

“…Bearing this in mind, we propose a numerical scheme (which will be explained in the following lines) and show a good agreement between the results thus obtained and the ones from the theoretical arguments. Then, as in de la Hoz and Vega ( 2018 ), we answer up to what extent the l -polygon problem and the one-corner problem are related. Consequently, not only can we compute the speed of the center of mass of the planar l -polygon, but the relationship also helps in comparing the trajectory of any of the corners of a regular planar polygon in both the Euclidean and hyperbolic cases.…”

On the energy of critical solutions of the binormal flow

“…, so that we obtain (15), and implicitly (16). Now we fix t > 0 and our purpose it to compute Ξ(t) and to obtain (14). Let 0 < ǫ < 1.…”