On the Relationship between the One-Corner Problem and the $M$-Corner Problem for the Vortex Filament Equation
This work provides a theoretical and numerical framework for understanding the nonlinear interaction of filaments in vortex dynamics, relevant to fluid dynamics and nonlinear wave phenomena.
The paper shows that the evolution of the Vortex Filament Equation for an M-corner polygon can be decomposed into M one-corner problems at infinitesimal times, leading to a nonlinear Talbot effect at later times. Strong numerical evidence is provided for energy and linear momentum transfer in the M-corner case.
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.