In 1877 John Kerr described an experiment that demonstrated a quadratic change in refractive index in a plate glass placed in a strong external electric field. This results in a nonlinear relationship between the average electric polarization within the materials and the intensity of the applied electric field. This opened the door for a new area of electromagnetic material science by incorporating nonlinearity into the basic Maxwell system, which in general describes a linear relationship between the electric and magnetic fields. Since then multiple other nonlinear effects have been found in materials that need to be incorporated into Maxwell's equations to accurately model the dynamical evolution of the polarization driven by the electric field.
In this thesis, we explore a model of one linear and two nonlinear effects that are incorporated into the Maxwell's Equations via the macroscopic polarization. This will include a single linear Lorentz dispersion, the nonlinear instantaneous electronic Kerr response as well as the non-instantaneous Raman vibrational response. We will consider one spatial dimension and investigate electromagnetic (EM) wave propagation in these nonlinear materials. To do so, we will include these effects in our constitutive equations for the relationship between the electric field and displacement and reduce our system of partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs) by assuming traveling wave solutions. Using linear stability analysis from dynamical systems theory allows us to predict behavioral changes in the electric and magnetic fields for an EM traveling wave passing through a material. We will consider the stability of steady states through an eigenvalue analysis of the linearized ODE system and consider the character of arising bifurcations. We have proved that varying the response time parameter of the Lorentz and Raman effects produces a degenerate Hopf bifurcation, and the varying the velocity of our traveling wave solution results in a pitchfork bifurcation. We will also look for changes in behavior arising from a Leapfrog time discretization of the ODE system of the Maxwell Lorentz-Kerr model relative to those in the continuum case, with the predicted stability being preserved in the discrete case under certain conditions.