  We derive a closed-form expression for the phase shift experienced by 1+1
dimensional kinks colliding at ultra-relativistic velocities (gamma v >> 1),
valid for arbitrary periodic potentials. Our closed-form expression is the
leading order result of a more general scattering theory of solitary waves
described in a companion paper [Amin, Lim and Yang, arXiv:1308.0606]. This
theory relies on a small kinematic parameter 1/(gamma v) << 1 rather than a
small parameter in the Lagrangian. Our analytic results can be directly
extracted from the Lagrangian without solving the equation of motion. Based on
our closed-form expression, we prove that kink-kink and kink-antikink
collisions have identical phase shifts at leading order.
