TITLE:
Lie Symmetries and Exact Solutions of 2D Proper-Time Maxwell’s Equations
AUTHORS:
Joshua Owolabi Adeleke
KEYWORDS:
Electrical Engineering, Theoretical Physics, Lie Symmetry Analysis, Proper-Time Formulation, Maxwell’s Equations, Conservation Laws, Exact Solutions, 2D Wave Equation
JOURNAL NAME:
Open Access Library Journal,
Vol.12 No.10,
October
20,
2025
ABSTRACT: 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
μ
u
μ
=−
c
2
, where
u
μ
is the 4-velocity, using the Minkowski metric
η
μν
=diag(
−1,1,1
)
. Here,
b=
c
2
+
u
x
2
+
u
y
2
is the propagation speed,
γ=b/c
is the Lorentz factor, and
ζ=r/τ
is the similarity variable.