Multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in (3+1) Minkowski spacetime
Tony Salvi
TL;DR
This work constructs multi-phase high-frequency solutions to the Klein-Gordon-Maxwell equations in Lorenz gauge on 3+1 Minkowski space by combining a low-order geometric optics approximation with a carefully structured error analysis. The authors prove the existence of a 1-parameter MPHF family $(A_\lambda,\Phi_\lambda)$ on a time interval independent of $\lambda$, with the background $(A_0,\Phi_0)$ solving a Klein-Gordon-Maxwell null-transport system, revealing a backreaction effect. A detailed bootstrap argument, Strichartz and energy estimates, and a carefully prepared initial data framework ensure uniform-in-$\lambda$ control and gauge propagation, yielding convergence to the background and small, controlled error in $H^{1/2}$. The results demonstrate how coherent phase interactions and null-structure-like features enable stable high-frequency limits in a nonlinear gauge theory, with implications for backreaction phenomena in related hyperbolic systems.
Abstract
We study a 1-parameter family (Aλ, Φλ)λ 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 λ small enough the family of solutions exists on an interval uniform in λ 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, Φ0) is not solution to Klein-Gordon-Maxwell equations but to a Klein-Gordon-Maxwell null-transport type system.