Collisions of Vortex Filament Pairs

We consider the problem of collisions of vortex filaments for a model introduced by Klein et al. (J Fluid Mech 288:201–248, 1995) and Zakharov (Sov Phys Usp 31(7):672–674, 1988, Lect. Notes Phys 536:369–385, 1999) to describe the interaction of almost parallel vortex filaments in three-dimensional fluids. Since the results of Crow (AIAA J 8:2172–2179, 1970) examples of collisions are searched as perturbations of antiparallel translating pairs of filaments, with initial perturbations related to the unstable mode of the linearized problem; most results are numerical calculations. In this article, we first consider a related model for the evolution of pairs of filaments, and we display another type of initial perturbation leading to collision in finite time. Moreover, we give numerical evidence that it also leads to collision through the initial model. We finally study the self-similar solutions of the model.