Analytical Solutions of 3D Maxwell's Equations via Infinite-Order Curl-Operator Expansions
Microsoft（微软）
David Wei Ge
备注：
54 pages, 72 figures. Major revision: added convergence conditions in the solution theorems; separated the uniqueness theorem into two theorems with independent proofs; added sections on deformed trigonometric functions and comparison with existing analytical methods; revised numerical results with a new FDTD initial-value finding and reorganized presentation
Solving Maxwell's equations to obtain explicit analytical representations of electromagnetic fields in open-space settings with general initial conditions and source terms remains a fundamental challenge. We address this problem through an operator-based construction, developing an infinite-order curl-operator expansion that yields analytical solutions as function-to-function mappings from initial data and source terms to electromagnetic fields. The framework is applied to representative cases, including Gaussian initial conditions, Gaussian sources, harmonic sources, and Ricker wavelet sources. In these applications, a family of deformed sine and cosine functions emerges naturally, enabling explicit analytical representations of the solutions. The analysis further shows that these deformed trigonometric functions are intrinsic to the general solution formulas, with their structure determined by the initial conditions and source terms. The results are compared with numerical simulations based on the FDTD method and further applied to a systematic study of mesh refinement effects. This study shows that, for the problems considered, the error measures approach limiting values as the mesh is refined, indicating that further step size reduction alone may produce little additional improvement once the mesh is sufficiently fine. The results demonstrate that the analytical solutions provide theoretical insight into electromagnetic field behavior while serving as practical benchmark solutions for computational electromagnetics.
Symmetrization of the Maxwell--Neumann--Poincar'e operator, spectral decomposition in $\mathbf{H}(\mathrm{curl},D)$ traces, and boundary localisation of SPRs
Northeast Normal University（东北师范大学）；City University of Hong Kong（香港城市大学）
Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients
考虑了$\mathbb{R}^3$中的非线性卷曲-卷曲问题$\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$，该问题与具有Kerr-type非线性物质定律的非线性Maxwell方程组有关.我们证明了一个有界的，圆柱对称系数$V$和亚临界圆柱对称非线性$f$的对称基态解的存在性。新的存在性结果扩展了一类问题的基态类型的解决方案是已知的。它是基于紧性的对称函数由于狮子，新的重排型不等式布洛克和最近的扩展Nehari流形技术由Szulkin和Weth。
Stability of an incompressible plasma-vacuum interface with displacement current in vacuum
研究了理想不可压缩磁流体力学中等离子体-真空界面的自由边界问题。与真空磁场服从前麦克斯韦动力学的div-curl系统的经典说法不同，为了更好地理解真空中电场的影响，我们不忽略真空区域中的位移电流，并考虑电场和磁场的麦克斯韦方程组。在[Trakhinin Y. arXiv：1812.08675]，我们证明了线性化常系数问题的能量先验估计.这个估计的推导过程是基于各种方法，包括二次对称化的真空麦克斯韦方程组，推导出的双曲演化方程的接口功能和建设的退化Kreiss型对称化椭圆双曲问题的总压力。
