  In this article we use an electromagnetic Lagrangian constructed so as to
include dispersive effects in the description of an electromagnetic wave
propagating in the Quantum Electrodynamic Vacuum. This Lagrangian is Lorentz
invariant, includes contributions up to six powers in the electromagnetic
fields and involves both fields and their first derivatives. Conceptual
limitations inherent to the use of this higher derivative Lagrangian approach
are discussed. We consider the one-dimensional spatial limit and obtain an
exact solution of the nonlinear wave equation recovering the Korteveg-de Vries
type periodic waves and solitons given in S. V. Bulanov et al., Phys. Rev. D,
101, 016016 (2020).
