Existence of Standing Pulse Solutions to an Inhomogeneous Reaction–Diffusion System

We prove the existence of locally unique, symmetric standing pulse solutions to homogeneous and inhomogeneous versions of a certain reaction–diffusion system. This system models the evolution of photoexcited carrier density and temperature inside the cavity of a semiconductor Fabry–Pérot interferometer. Such pulses represent the fundamental nontrivial mode of pattern formation in this device. Our results follow from a geometric singular perturbation approach, based largely on Fenichel's theorems and the Exchange Lemma.

[1]  A. V. Grigor'yants,et al.  Spontaneous oscillations, transverse diffusion instability and spatial dissipative structures in optical bistability and multistability , 1987 .

[2]  Neil Fenichel Geometric singular perturbation theory for ordinary differential equations , 1979 .

[3]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[4]  J. Dockery Existence of standing pulse solutions for an excitable activator-inhibitory system , 1992 .

[5]  J. Rubin,et al.  Bifurcations and edge oscillations in the semiconductor Fabry-Pérot interferometer , 1997 .

[6]  A. M. Sergent,et al.  Loss measurements in GaAs and AlxGa1−xAs dielectric waveguides between 1.1 eV and the energy gap , 1976 .

[7]  Neil Fenichel Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[8]  H. Gibbs Optical Bistability Controlling Light With Light , 1985 .

[9]  G. Hernandez,et al.  Fabry-Perot Interferometers , 1986 .

[10]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[11]  Christopher K. R. T. Jones,et al.  Tracking invariant manifolds with di erential forms in singularly per-turbed systems , 1994 .

[12]  N. Kopell,et al.  Construction of the Fitzhugh-Nagumo Pulse Using Differential Forms , 1991 .

[13]  Richardson,et al.  Spatiotemporal oscillations in a semiconductor étalon. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[14]  A. V. Grigor'yants,et al.  Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors , 1988 .

[15]  Christopher K. R. T. Jones,et al.  Invariant manifolds and singularly perturbed boundary value problems , 1994 .