Multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in (3+1) Minkowski spacetime

TL;DR

This paper constructs a family of high-frequency solutions to Klein-Gordon-Maxwell equations in Minkowski spacetime, demonstrating their existence under certain conditions and analyzing their limiting behavior as solutions to a null-transport system.

Contribution

It introduces a new approach to high-frequency solutions in Klein-Gordon-Maxwell equations, based on an initial ansatz and geometric optics approximation, without requiring smallness assumptions.

Findings

Solutions exist uniformly in the parameter for small enough values.

The solutions approximate a geometric optics-based ansatz.

The limit solutions satisfy a null-transport type system, not the original equations.

Abstract

We study a 1-parameter family (A{\lambda}, {\Phi}{\lambda}){\lambda} of multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in the (3+1)-dimensional Minkowski spacetime. This family is based on an initial ansatz. We prove that for {\lambda} small enough the family of solutions exists on an interval uniform in {\lambda} only function of the initial ansatz. These solutions are close to an approximate solution constructed by geometric optics. The initial ansatz needs to be regular enough, to satisfy a polarization condition and to satisfy the constraints for Maxwell null-transport in Lorenz gauge, but there is no need for smallness of any kind. The phases need to interact in a coherent way. We also observe that the limit (A0, {\Phi}0) is not solution to Klein-Gordon-Maxwell equations but to a Klein-Gordon-Maxwell null-transport type system.