The physical situation of the collision and subsequent interaction of plane
gravitational waves in a Minkowski background gives rise to a well-posed
characteristic initial value problem in which initial data are specified on the
two null characteristics that define the wavefronts. In this paper, we analyse
how the Abel transform method can be used in practice to solve this problem for
the linear case in which the polarization of the two gravitational waves is
constant and aligned. We show how the method works for some known solutions,
where problems arise in other cases, and how the problem can always be solved
in terms of an infinite series if the spectral functions for the initial data
can be evaluated explicitly.Comment: 14 pages. To appear in Class. Quantum Gra