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

TL;DR

This paper investigates how the evolution of the Vortex Filament Equation for polygonal initial data with multiple corners can be understood as a superposition of single-corner evolutions, revealing complex nonlinear interactions akin to a Talbot effect.

Contribution

It provides a novel explanation of the multi-corner vortex filament evolution as a superposition of single-corner solutions and demonstrates nonlinear interactions through numerical evidence.

Findings

Superposition of single-corner evolutions explains multi-corner dynamics.

Identification of a nonlinear Talbot effect in filament interactions.

Numerical evidence of energy and momentum transfer in multi-corner cases.

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.