Evolutionary constraints: Gluing in a toy model

TL;DR

This paper demonstrates that the evolutionary approach combined with gluing techniques simplifies generating initial data for electromagnetic pulses, contrasting with the complex elliptic methods traditionally used.

Contribution

It introduces a novel application of the evolutionary method with gluing to solve the Gauss constraint in electrodynamics, providing explicit initial data construction.

Findings

Evolutionary method simplifies solving the Gauss constraint.

Explicit initial data for electromagnetic pulses was constructed.

Elliptic methods require complex functional analysis for similar problems.

Abstract

It is possible to solve the Einstein constraint equations as an evolutionary rather than an elliptic system. Here we consider the Gauss constraint in electrodynamics as a toy model for thist. We use a combination of the evolutionary method with the gluing construction to produce initial data for an electromagnetic pulse surrounded by vacuum. It turns out that solving the evolutionary form of the constraint is straightforward, and explicitly yields the desired type of initial data. In contrast, proving the existence of a solution to the same problem within the elliptic setting requires sophisticated arguments based on functional analysis.