## 4 An Extended Formulation for Electrodynamics  
In order to fix the notation, we start by writing the set of Maxwell equations in SI unit convention:  
\[\
abla\cdot\mathbf{D}\,=\,\rho_{s} \tag{1}\]  
\[\
abla\times\mathbf{H}-\frac{\partial\mathbf{D}}{\partial t}\,=\,\mathbf{J}_{s} \tag{2}\]  
\[\
abla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}\,=\,0 \tag{3}\]  
\[\
abla\cdot\mathbf{B}\,=\,0 \tag{4}\]  
where, as usual, \(\mathbf{D}\) is the displacement field, \(\rho_{s}\) represents the charge sources in the medium, \(\mathbf{H}\) is the magnetic strength field, \(\mathbf{J}_{s}\) is the current source in the medium, \(\mathbf{E}\) is the electric field and \(\mathbf{B}\) is the magnetic induction field. In a vacuum and in the absence of field sources, the right hand sides of Equations (1) and (2) are set to zero. In many relevant cases, these equations are coupled with some constitutive relations--for instance: \(\mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}\) (\(\mathbf{P}\) being the polarization density) and \(\mathbf{H}=\mathbf{B}/\mu_{0}-\mathbf{M}\) (\(\mathbf{M}\) being the magnetization). As is customary, \(\varepsilon_{0}\) and \(\mu_{0}\) denote the dielectric constant and the magnetic permeability in a vacuum, respectively. To complete the set, it is customary to add the equation expressing the [PERSON]'s force on a given charge \(q\) traveling in an electromagnetic field with velocity \(\mathbf{v}\); i.e.,  
\[\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}) \tag{5}\]  
We now introduce the extensions of the equations as proposed in [14; 15], wherein the possibility to have nonzero divergence (\(\
abla\cdot\mathbf{E}\
eq 0\)) for the electric field in a vacuum, even in the absence of sources, was postulated. This assumption produces an extra current term in the [PERSON]'s law. As a consequence of this revision, two new variables and two new unknowns were added to the standard system. The main motivation for this extension was the possibility to expand the set of possible solutions. The reviewed model has the advantage of incorporating solitary signal-packets with compact support (solitons) within the framework of a differential theory. The term \"soliton\" commonly refers to solutions of certain nonlinear equations that are initially localized in space and evolve while maintaining their shape (see, e.g., [16]). In our case, these will be smooth electromagnetic waves with compact support traveling (in the simplest situation) along a straight trajectory at the speed of light. Note that these entities are not available in the solution space of the classical Maxwell equations. In suitable circumstances, these solitons may be assimilated to \"photons\", as those emanated or absorbed by matter upon proper solicitations. With this interpretation, the photon is not the \"carrier\" of the electromagnetic field anymore, but becomes a pure wave, carrying along electric and magnetic components. In the present paper, we do not want to stress the reader with these aspects. More insights on these issues can be found in [15]. For the time being, let us suppose we are working in a vacuum. According to [14], by letting the divergence of the electric field be different from zero, we arrive at the following formulation:  
\[\frac{\partial\mathbf{E}}{\partial t}\,=\,c^{2}\,\
abla\times\mathbf{B}\,-\,\rho\,\mathbf{V} \tag{6}\]  
\[\frac{\partial\mathbf{B}}{\partial t}\,=\,-\,\
abla\times\mathbf{E} \tag{7}\]\[\
abla\cdot\mathbf{B}\,=\,0 \tag{8}\] \[\frac{\partial p}{\partial t}\,=\,\hat{\mu}\,\rho\,\mathbf{E}\cdot\mathbf{V}\] (9) \[\rho\left[\,\frac{DV}{Dt}\,+\,\hat{\mu}\left(\mathbf{E}\,+\,\mathbf{V}\, \times\,\mathbf{B}\right)\right]\,=\,-\
abla p \tag{10}\]  
where, by definition, \(\rho=\
abla\cdot\mathbf{E}\) is the divergence of the electric field.