| Researchers have observed complex spatiotemporal phenomena in numerical investigations of models of the semiconductor Fabry-Perot interferometer. We consider a nondimensionalized version of a model that includes the coupling between the photoexcited carrier density and the temperature inside the cavity. This is an inhomogeneous system of reaction-diffusion partial differential equations depending nonlinearly on the refractive index, which in turn depends linearly on carrier density and temperature. For both the homogeneous and inhomogeneous versions of this system, we establish the existence of a locally unique, symmetric standing pulse solution. Further, we explain two mechanisms by which periodic solutions exhibiting edge oscillations may bifurcate out of the standing pulse in the inhomogeneous case.;The existence results follow from a geometric singular perturbation approach, in which we employ Fenichel's Invariant Manifold Theorems and the Exchange Lemma. By adapting a topological stability index to count eigenvalues and by computing the derivative of the Evans function at the origin, we prove that the standing pulse is linearly stable, and thus asymptotically stable, in a parameter regime far from the optics setting in the homogeneous case. Moreover, we explain two bifurcation mechanisms which cause a loss of stability, one of which arises as the physically relevant diffusion strength is approached and the other of which results as a reaction term is adjusted. The latter of these bifurcations cannot occur in the nondimensionalized optics model that we consider initially because of the omission of certain parameters; however, by restoring an additional parameter to control the form of this reaction term, we obtain a system in which both bifurcations take place.;We compute conditions for each of these bifurcation types to persist when an appropriate inhomogeneity is introduced as a regular perturbation of the homogeneous system. In the inhomogeneous case, eigenvalues cross the imaginary axis, away from the origin, in both bifurcations. We conjecture that in this setting, one of the two mechanisms yields in-phase edge oscillations, while the other produces out-of-phase edge oscillations. |
