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

  • Christopher K. R. T. Jones
  • Jonathan E. Rubin


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.

Standing waves inhomogeneous reaction–diffusion system geometric singular perturbation semiconductor Fabry–Pérot interferometer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Balkarei, Yu. I., Grigor'yants, A., and Rzhanov, Yu. A. (1987). Spontaneous oscillations, transverse diffusion instability and spatial dissipative structures in optical bistability and multistability, Sov. J. Quant. Electron. 17, 72–75.Google Scholar
  2. Balkarei, Yu. I., Grigor'yants, A., Rzhanov, Yu. A., and Elinson, M. I. (1988). Regenerative oscillations, spatio-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors, Opt. Commun. 66, 161–166.Google Scholar
  3. Dockery, J. D. (1992). Existence of standing pulse solutions for an excitable activator-inhibitory system, J. Dynam. Diff. Eq. 4, 231–257.Google Scholar
  4. Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21, 193–226.Google Scholar
  5. Fenichel, N. (1974). Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23, 1109–1137.Google Scholar
  6. Fenichel, N. (1977). Asymptotic stability with rate conditions, II, Indiana Univ. Math. J. 26, 81–93.Google Scholar
  7. Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31, 53–98.Google Scholar
  8. Fife, P. C. (1979). Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin.Google Scholar
  9. Gardner, R. (1995). Review of Traveling Wave Solutions of Parabolic Systems, by Volpert, A. I., Volpert, V. A., and Volpert, V. A. Bull. (New Ser.) Am. Math. Soc. 32, 446–452.Google Scholar
  10. Gibbs, H. M. (1985). Optical Bistability: Controlling Light with Light, Academic Press, Orlando, FL.Google Scholar
  11. Grindrod, P. (1991). Patterns and Waves, Clarendon Press, Oxford.Google Scholar
  12. Hernandez, G. (1986). Fabry-Perot Interferometers, Cambridge University Press, Cambridge.Google Scholar
  13. Jones, C. K. R. T. (1995). Geometric singular perturbation theory. In Johnson, R. (ed.), Dynamical Systems, Lecture Notes in Mathematics 1609, Springer-Verlag, Berlin, pp. 44–118.Google Scholar
  14. Jones, C. K. R. T., and Kopell, N. (1994). Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Diff. Eq. 108, 64–88.Google Scholar
  15. Jones, C., Kopell, N., and Langer, R. (1991). Construction of the FitzHugh-Nagumo pulse using differential forms. In Swinney, H., Aris, G., and Aronson, D. (eds.), Patterns and Dynamics in Reactive Media, IMA Volumes in Mathematics and Its Applications 37, Springer-Verlag, New York, pp. 101–116.Google Scholar
  16. Merz, J. L., Logan, R. A., and Sergent, M. A. (1976). Loss measurements in GaAs and AlxGa1-xAs dielectric waveguides between 1.1eV and the energy gap, J. Appl. Phys. 47, 1436–1450.Google Scholar
  17. Newell, A. C., and Moloney, J. V. (1992). Nonlinear Optics, Addison-Wesley, Redwood City, CA.Google Scholar
  18. Rubin, J. E. (1996a). The Generation of Edge Oscillations in an Inhomogeneous Reaction-Diffusion System, Ph.D. thesis, Brown University, Providence, RI.Google Scholar
  19. Rubin, J. E. (1996b). Stability and bifurcations for standing pulse solutions to an inhomogeneous reaction-diffusion system (preprint).Google Scholar
  20. Rubin, J. E., and Jones, C. K. R. T. (1996). Bifurcations and edge oscillations in the semiconductor Fabry-Pérot interferometer (preprint).Google Scholar
  21. Rzhanov, Yu. A., Richardson, H., Hagberg, A. A., and Moloney, J. V. (1993). Spatiotemporal oscillations in a semiconductor étalon, Phys. Rev. A 47, 1480–1491.Google Scholar
  22. Tin, S.-K., Kopell, N., and Jones, C. K. R. T. (1994). Invariant manifolds and singularly perturbed boundary value problems, SIAM J. Numer. Anal. 31, 1558–1576.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • Christopher K. R. T. Jones
    • 1
  • Jonathan E. Rubin
    • 2
  1. 1.Division of Applied MathematicsBrown University, ProvidenceRhode Island
  2. 2.Department of MathematicsThe Ohio State UniversityColumbus

Personalised recommendations