  In this paper, we study the problem of the nonlinear interaction of impulsive
gravitational waves for the Einstein vacuum equations. The problem is studied
in the context of a characteristic initial value problem with data given on two
null hypersurfaces and containing curvature delta singularities. We establish
an existence and uniqueness result for the spacetime arising from such data and
show that the resulting spacetime represents the interaction of two impulsive
gravitational waves germinating from the initial singularities. In the
spacetime, the curvature delta singularities propagate along 3-dimensional null
hypersurfaces intersecting to the future of the data. To the past of the
intersection, the spacetime can be thought of as containing two independent,
non-interacting impulsive gravitational waves and the intersection represents
the first instance of their nonlinear interaction. Our analysis extends to the
region past their first interaction and shows that the spacetime still remains
smooth away from the continuing propagating individual waves. The construction
of these spacetimes are motivated in part by the celebrated explicit solutions
of Khan-Penrose and Szekeres. The approach of this paper can be applied to an
even larger class of characteristic data and in particular implies an extension
of the theorem on formation of trapped surfaces by Christodoulou and
Klainerman-Rodnianski, allowing non-trivial data on the initial incoming
hypersurface.
