The Vortex Filament Equation: the Talbot effect and the transfer of energy and momentum

I will present some recent results obtained in collaboration with V. Banica and F. de la Hoz on the evolution of vortex filaments according to the so called Localized Induction Approximation  (LIA). This approximation is given by a non-linear geometric partial differential equation, that is known under the name of the Vortex Filament Equation (VFE). The aim of the talk is threefold. First, I will recall the Talbot effect of linear optics.  Secondly, I will give some explicit solutions of VFE where this Talbot effect is also present. Finally, I will consider some questions concerning the transfer of energy and momentum for these explicit solutions.