  In Einstein's theory of relativity, the interaction of two collinearly
polarized plane gravitational waves can be described by a Goursat problem for
the Euler--Darboux equation in a triangular domain. In this paper, using a
representation of the solution in terms of Abel integrals, we give a full
asymptotic expansion of the solution near the diagonal of the triangle. The
expansion is related to the formation of a curvature singularity of the
spacetime. In particular, our framework allows for boundary data with
derivatives which are singular at the corners. This level of generality is
crucial for the application to gravitational waves.
