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

摘要

In this paper, we give evidence that the evolution of the Vortex FilamentEquation for a regular $M$-corner polygon as initial datum can be explained atinfinitesimal times as the superposition of $M$ one-corner initial data.Therefore, and due to periodicity, the evolution at later times can beunderstood as the nonlinear interaction of infinitely many filaments, one foreach corner. This interaction turns out to be some kind of nonlinear Talboteffect. We also give very strong numerical evidence of the transfer of energyand linear momentum for the $M$-corner case.