Asymptotics to all orders of the Euler--Darboux equation in a triangle

TL;DR

This paper provides a detailed asymptotic analysis of the Euler--Darboux equation in a triangular domain, relevant for modeling gravitational wave interactions in relativity, especially near singularities.

Contribution

It introduces a full asymptotic expansion for solutions with singular boundary data, advancing understanding of wave behavior in gravitational contexts.

Findings

Derived asymptotic expansion near the triangle's diagonal.

Handled boundary data with singular derivatives at corners.

Linked asymptotics to curvature singularity formation.

Abstract

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.