The hyperbolic Ernst equation in a triangular domain

Abstract

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. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann-Hilbert problem. The formulation of the Riemann-Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle's corners-this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann-Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.

The hyperbolic Ernst–Maxwell equations in a triangular domain

“…In this section, we prove the three main results Theorems 1-3. We note that the proofs are conceptually similar to those in [18]. In particular, some parts of the proofs rather rely on the analyticity structure of the matrices U and V than on their specific expression.…”

“…In this section, we present the Lax pair and a Riemann-Hilbert problem corresponding to the hyperbolic Ernst-Maxwell equations (1.1). In this context, we also introduce the notation used throughout the paper, which is adopted from [18].…”

“…We seek (sufficiently regular) functions E(x, y) and H(x, y) such that problem for the hyperbolic Ernst equation was analyzed in the recent paper [18] by using the integrable structure of the equation.…”

“…In this paper, we generalize the approach adopted in [18] to the case of colliding electromagnetic plane waves. The Ernst-Maxwell equations (1.1) admit a Lax pair involving 3 × 3-matrices, whereas the Lax pair for Ernst's equation is given in terms of 2 × 2-matrices.…”

“…[1,4,14]) focuses on the generation of new exact solutions instead of a general approach for solving the Goursat problem for (1.1) as it is established here. However, besides [18], inverse scattering approaches have been used in order to solve a boundary problem for colliding gravitational waves [11] and more recently to analyze boundary value problems for the elliptic version of Ernst's equation [15,17,19]. All the papers [11,15,[17][18][19] are partially based on a general framework for solving boundary value problems known as the unified transform or Fokas method [9] (see also [2,6]).…”