Considering that classical electrodynamics is essentially a covariant theory, we propose a simple heuristic approach to nonlinear Maxwell equations where certain scalar fields are coupled to the electromagnetic (EM) field. This leads us to introduce a new (nonlinear) version of the Poynting theorem. This model seems to have solutions in which coupled scalar fields and EM waves propagate. This work could provide students with a first introduction to the basics of nonlinear electrodynamics within the framework of a new research problem. The content of this article is suitable for advanced graduate students in theoretical particle physics.