Head-On Collision of a Pair of Coaxial Circular Vortex Filament

We consider the head-on collision of two coaxial vortex rings described as the motion of two circular vortex filaments under the localized induction approximation. We prove the existence of solutions to a system of nonlinear partial differential equations proposed by the author which exhibit head--on collision. We also give a necessary and sufficient condition for the initial configuration and parameters of the filaments for head-on collision to occur. Our results suggest that there exists a critical value $\gamma_{\ast}>1$ for the ratio $\gamma$ of the absolute value of the circulations such that when $\gamma\in [1,\gamma_{\ast}]$, two approaching rings will collide, and when $\gamma \in (\gamma_{\ast},\infty)$, the ring with the larger circulation passes through the other and then separate indefinitely.