  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.
