The global existence, uniqueness and C^1-regularity of geodesics in   nonexpanding impulsive gravitational waves
  We study geodesics in the complete family of nonexpanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we prove existence and uniqueness of continuously differentiable geodesics (in the sense of Filippov) and use a C^1-matching procedure to explicitly derive their form. 
Jiri Podolsky, Clemens S\"amann, Roland Steinbauer, Robert Svarc
gr-qc hep-th math-ph math.MP