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

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Abstract

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.

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REFERENCES

  • 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 

  • 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 

  • Dockery, J. D. (1992). Existence of standing pulse solutions for an excitable activator-inhibitory system, J. Dynam. Diff. Eq. 4, 231–257.

    Google Scholar 

  • Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21, 193–226.

    Google Scholar 

  • Fenichel, N. (1974). Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23, 1109–1137.

    Google Scholar 

  • Fenichel, N. (1977). Asymptotic stability with rate conditions, II, Indiana Univ. Math. J. 26, 81–93.

    Google Scholar 

  • Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31, 53–98.

    Google Scholar 

  • Fife, P. C. (1979). Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin.

    Google Scholar 

  • 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.

  • Gibbs, H. M. (1985). Optical Bistability: Controlling Light with Light, Academic Press, Orlando, FL.

    Google Scholar 

  • Grindrod, P. (1991). Patterns and Waves, Clarendon Press, Oxford.

    Google Scholar 

  • Hernandez, G. (1986). Fabry-Perot Interferometers, Cambridge University Press, Cambridge.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Newell, A. C., and Moloney, J. V. (1992). Nonlinear Optics, Addison-Wesley, Redwood City, CA.

    Google Scholar 

  • Rubin, J. E. (1996a). The Generation of Edge Oscillations in an Inhomogeneous Reaction-Diffusion System, Ph.D. thesis, Brown University, Providence, RI.

    Google Scholar 

  • Rubin, J. E. (1996b). Stability and bifurcations for standing pulse solutions to an inhomogeneous reaction-diffusion system (preprint).

  • Rubin, J. E., and Jones, C. K. R. T. (1996). Bifurcations and edge oscillations in the semiconductor Fabry-Pérot interferometer (preprint).

  • 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 

  • 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 

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, Rhode Island, 02912

    Christopher K. R. T. Jones

  2. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio, 43210

    Jonathan E. Rubin

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  1. Christopher K. R. T. Jones
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  2. Jonathan E. Rubin
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Jones, C.K.R.T., Rubin, J.E. Existence of Standing Pulse Solutions to an Inhomogeneous Reaction–Diffusion System. Journal of Dynamics and Differential Equations 10, 1–35 (1998). https://doi.org/10.1023/A:1022651311294

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