Julian Mauersberger: The hyperbolic Ernst equation in a triangular domain

The collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. In this talk we first recall how the vacuum Einstein field equations reduce to the single hyperbolic Ernst equation in the situation of colliding plane waves. Then we use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann--Hilbert problem. At the end of the talk we consider the special case of collinear polarization, in which the asymptotic behavior of the solution near the possible curvature singularity can be determined, as well as a generalization for colliding electromagnetic plane waves in Einstein--Maxwell theory.