Exploring nonlinear vacuum electrodynamics beyond Maxwell's Equations

In this article, we provide a review of the current state of the research on the nonlinear electrodynamics of vacuum in the presence of strong electromagnetic fields. We discuss the parametrized post-Maxwellian formalism that can be facilitate calculations in the weak electromagnetic field approximation. The post-Maxwellian parameters are a set of parameters used in non-linear electrodynamics to describe the behavior of electromagnetic fields in the presence of strong fields. We also discuss the equations for the electromagnetic field in the Born-Infeld electrodynamics and in non-linear Heisenberg-Euler electrodynamics, as well as the exact solution in the form of a plane elliptically polarized wave. We represent the Lagrangian of nonlinear electrodynamics in vacuum in detail. We demonstrate that the Lagrangian can be expressed as a series of integer powers of invariants J(1) and J(2), which allows for the identification of nonlinear electrodynamics that are consistent with experiments carried out in a weak electromagnetic field. Finally, for methodological purposes, the article emphasizes the importance of the systematic implementation of the calculation of nonlinear electrodynamics effects, regardless of any nonlinear theory. Such an approach can facilitate a deeper understanding of the underlying physics and contribute to the development of novel experimental techniques.