Lie Symmetry Analysis of 1D Proper-Time Maxwell’s Equations: Exact Solutions and Conservation Laws

We present a complete Lie symmetry analysis of the one-dimensional proper-time formulation of Maxwell’s equations with velocity-dependent propagation speed b=√c²+u². Through systematic application of symmetry methods, we: 1) classify all Lie point symmetries of the system, 2) derive exact invariant solutions via symmetry reduction, and 3) construct conserved quantities using Noether’s theorem. The solutions exhibit characteristic propagation at speed b , maintaining relativistic causality through the constraint u^μu_μ=-c². Numerical verification confirms solution stability under appropriate discretization. This work establishes a rigorous mathematical foundation for proper-time electrodynamic systems, with applications to particle acceleration and high-energy astrophysical phenomena. The novelty lies in the application of these methods to the proper-time formulation, revealing new solution structures tied to the relativistic propagation speed b .