An attractive analytic solution of the Maxwell's equation
Xiaorong Zou
TL;DR
This work presents an analytic, spectrally decomposed solution to Maxwell's equations in an isotropic, homogeneous, lossless medium by embedding the electric and magnetic fields into a complex form $F$ and performing a Fourier-mode based spectral analysis of the curl operator. By constructing eigenvector bases on subspaces $V^w$ for two cases ($r_w=0$ and $r_w\neq 0$), the authors derive closed-form time evolution for each spectral component and assemble the full solution via Fourier synthesis, preserving divergence constraints. The Main Theorem expresses the exact solution as a weighted sum over Fourier modes with exponential time factors $e^{-t\lambda_{d,w}/\sqrt{\mu\varepsilon}}$, providing a computationally efficient analytic analogue to Fourier-based representations. Two demonstrative examples illustrate the method: one recovering a classical cosine-based solution in the $r_w=0$ case, and another giving an explicit, fully analytic solution for a nondegenerate wavevector in the $r_w\neq 0$ case. The approach offers a rigorous, analyzable framework that parallels Fourier expansions while highlighting invariants and providing benchmarks for numerical solvers.
Abstract
In this paper, we provide an attractive analytic solution for Maxwell's equation for a given set of smooth periodic functions as initial condition with demonstrative examples. The complexity of the solution is comparable to the Fourier expansions of the initial functions.