Solving the characteristic initial value problem for colliding plane gravitational and electromagnetic waves

TL;DR

This paper introduces a generalized method for solving the nonlinear initial value problem for colliding plane gravitational and electromagnetic waves, transforming complex equations into linear integral equations based on evolving monodromy data.

Contribution

It extends the monodromy transform approach to handle nonanalytic behaviors on wavefronts in colliding wave scenarios.

Findings

The method successfully reformulates nonlinear equations as linear integral equations.

It enables the determination of wave interactions from initial data.

The approach handles complex wavefront behaviors in gravitational and electromagnetic collisions.

Abstract

A method is presented for solving the characteristic initial value problem for the collision and subsequent nonlinear interaction of plane gravitational or gravitational and electromagnetic waves in a Minkowski background. This method generalizes the monodromy transform approach to fields with nonanalytic behaviour on the characteristics inherent to waves with distinct wave fronts. The crux of the method is in a reformulation of the main nonlinear symmetry reduced field equations as linear integral equations whose solutions are determined by generalized (``dynamical'') monodromy data which evolve from data specified on the initial characteristics (the wavefronts).