Lie Symmetries and Exact Solutions of 2D Proper-Time Maxwell’s Equations

We present a Lie symmetry classification of 2D proper-time Maxwell’s equations, deriving an 8-dimensional Lie algebra and constructing rotationally invariant Bessel function solutions and self-similar solutions with Hadamard regularization. A central-difference/Verlet scheme is stable under the standard Courant condition for numerical solutions. The results model transverse plasma waves in relativistic jets and particle beams in plasma wakefield accelerators [1], preserving causality via $u^{\mu }{u}_{\mu }=-c^2$ , where $u^{\mu }$ is the 4-velocity, using the Minkowski metric ${\eta }_{\mu \nu }=\text{diag}\left(-1,1,1\right)$ . Here, $b=\sqrt{c^2{u}_x^2{u}_y^2}$ is the propagation speed, $\gamma =b/c$ is the Lorentz factor, and $\zeta =r/\tau$ is the similarity variable.Subject AreasElectric Engineering, Theoretical Physics