Spectral boundary conditions and solitonic solutions in a classical Sellmeier dielectric

Electromagnetic field interactions in a dielectric medium represent a longstanding field of investigation, both at the classical level and at the quantum one. We propose a 1+1 dimensional toy-model which consists of an half-line filling dielectric medium, with the aim to set up a simplified situation where technicalities related to gauge invariance and, as a consequence, physics of constrained systems are avoided, and still interesting features appear. In particular, we simulate the electromagnetic field and the polarization field by means of two coupled scalar fields $phi$,$psi$ respectively, in a Hopfield-like model. We find that, in order to obtain a physically meaningful behaviour for the model, one has to introduce spectral boundary conditions depending on the particle spectrum one is dealing with. This is the first interesting achievement of our analysis. The second relevant achievement is that, by introducing a nonlinear contribution in the polarization field $psi$, with the aim of mimicking a third order nonlinearity in a nonlinear dielectric, we obtain solitonic solutions in the Hopfield model framework, whose classical behaviour is analyzed too.