Pattern Selection In A Diffusion-Reaction System With Global Or Long-Range Interaction

  • Moshe Sheintuch
  • Olga Nekhamkina
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)


We review recent results on pattern selection in the one- or two-dimensional reaction-diffusion system xt - Δx = f(x, y, λ), y t = εg(x, y), subject to global (〈x〉 = x 0) or long-range interaction; the source functions may be realistic kinetic functions or simple cubic or quintic f(x) functions for which the system admits inversion symmetry. This review discusses: (i) physical sources of such interactions and experimental observations in catalytic and electrochemical systems; (ii) the main emerging patterns and their classification according to their symmetry; (iii) the bifurcation between patterns; (iv) patterns when f(x) = 0 is tristable and can sustain several fronts.

The rich class of patterns simulated in a ribbon can be classified as stationary-front solutions (including oscillating fronts and antiphase oscillations) and moving pulse solutions (unidirectional, back-and-forth and source-points). Patterns on a disk may be classified as circular (including oscillatory or moving target patterns), rotating (stationary or moving spiral wave) and other patterns.


Pattern Selection Spiral Wave Inversion Symmetry Target Pattern Bistable System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    CROSS M.C. AND P.C. HOHENBERG, 1993, Rev. Mod. Phys., 65, 851.CrossRefGoogle Scholar
  2. [2]
    ELMER, F.J., 1988, Physica D, 30, 321.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    ELMER, F.J., 1992, Z. Physik B, 87, 377.CrossRefGoogle Scholar
  4. [4]
    ERTL, G. AND R. IMBIHL, 1995, Chemical Reviews, 97, 697.Google Scholar
  5. [5]
    FITZHUGH, R, 1961, Biophys. J., 1, 445.CrossRefGoogle Scholar
  6. [6]
    FLäTGEN, G. AND KRISCHER, K., 1995, Phys. Rev. E, 51, 3997.CrossRefGoogle Scholar
  7. [7]
    GRAHAM, M.D., LANE, S.L. AND LUSS, D., 1993, J. Phys. Chem., 97, 7564.CrossRefGoogle Scholar
  8. [8]
    HAGBERG A. AND E. MERON, 1994a, Nonlinearity, 7, 805.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    HAGBERG A. AND E. MERON, 1994b, Phys. Rev. Lett., 72, 2494.CrossRefGoogle Scholar
  10. [10]
    HAIM, D., LEV, O., L.M. PISMEN AND M.SHEINTUCH, 1992, Chem. Engng Sci., 47, 3907.CrossRefGoogle Scholar
  11. [11]
    IMBIHL, R., 1998, This volume.Google Scholar
  12. [12]
    KERNER, B.S. AND V.V. OSIPOV, 1994, Autosolitons, A New Approach to Problems of Self-Organization and Turbulence, (Kluwer Academia).Google Scholar
  13. [13]
    KRISHER, K. AND A. MIKHAILOV, 1994, Phys. Rev. Lett., 73, 3165.CrossRefGoogle Scholar
  14. [14]
    KURAMOTO, Y., 1984, Chemical Oscillations, Waves and Turbulence, (Springer, Berlin).zbMATHCrossRefGoogle Scholar
  15. [15]
    LANE, S.L. AND LUSS, D., 1993, Phys. Rev. Lett., 70, 830.CrossRefGoogle Scholar
  16. [16]
    LANE, S.L., GRAHAM, M.D. AND LUSS, D., 1993, A.I.Ch.E. J., 39, 1497.CrossRefGoogle Scholar
  17. [17]
    LOBBAN, L. AND Luss, D., 1989, J. Phys. Chem., 93, 6530.CrossRefGoogle Scholar
  18. [18]
    MERTENS, F., IMBIHL, R. AND MIKHAILOV, A., 1994, J. Phys. Chem., 101, 9903.CrossRefGoogle Scholar
  19. [19]
    MIDDYA, U., M. SHEINTUCH, M.D. GRAHAM AND D. LUSS, 1993a, Physica D, 63, 393.zbMATHCrossRefGoogle Scholar
  20. [20]
    MIDDYA, U., GRAHAM, M.D., Luss, D., AND M. SHEINTUCH, 1993b, J. Chem. Phys., 98, 2823.CrossRefGoogle Scholar
  21. [21]
    MIDDYA, U., D. LUSS, AND M. SHEINTUCH, 1994a, J. Chem. Phys., 100, 3568.CrossRefGoogle Scholar
  22. [22]
    MIDDYA, U., D. LUSS, AND M.SHEINTUCH, 1994b, J. Chem. Phys., 101, 4688.Google Scholar
  23. [23]
    MIDDYA, U. AND D. LUSS, 1995, J. Chem. Phys., 102, 5029.CrossRefGoogle Scholar
  24. [24]
    NEKHAMKINA, O. AND M. SHEINTUCH, 1998, Physica A, 249, 134.CrossRefGoogle Scholar
  25. [25]
    NOSZTICZIUS, Z., HORSTHEMKE, W., McCORMICK, W.D. AND SWINNEY, H.L., 1987, Nature, 329, 619.CrossRefGoogle Scholar
  26. [26]
    OTTERSTEDT, R.D., PLATH, P.J., JAEGER, N.I. AND HUDSON, J.L., 1996, J. Chem. Soc., Faraday Trans., 92, 2933.CrossRefGoogle Scholar
  27. [27]
    PHILIPPOU, G., M. SOMANI AND D. LUSS, 1993, Chem. Engng Sci., 48, 2325.CrossRefGoogle Scholar
  28. [28]
    PURWINS, H.-G., 1998, This volume.Google Scholar
  29. [29]
    SHEINTUCH, M., 1989, Chem. Engng Sci., 44, 1081.CrossRefGoogle Scholar
  30. [30]
    SHEINTUCH, M., 1997, Physica D, 102, 125.zbMATHCrossRefGoogle Scholar
  31. [31]
    SHEINTUCH, M. AND O. NEKHAMKINA, 1997, J. Chem. Phys., 107, 8165.CrossRefGoogle Scholar
  32. [32]
    SHEINTUCH, M. AND O. NEKHAMKINA, 1998, J. Chem. Phys., 109.Google Scholar
  33. [33]
    SOMANI, M., LIAUW, M.A. AND LUSS, D., 1996, Chem. Engng Sci., 51, 4259.CrossRefGoogle Scholar
  34. [34]
    SOMANI, M., LIAUW, M.A. AND LUSS, D., 1997, Chem. Engng Sci., 52, 2331.CrossRefGoogle Scholar
  35. [35]
    TURING, A.M., 1952, Phil. Trans. R. Soc. B, 237, 37, 99.Google Scholar
  36. [36]
    VESER, G., MERTENS, F., MIKHAILOV, A.S AND IMBIHL, R., 1993, Phys. Rev. Lett., 71, 935.CrossRefGoogle Scholar
  37. [37]
    WILLEBRAND, H., HUNTLER, T., NIEDERNOSTHEIDE F.J., DOHMEN, R. AND H.-G. PURWINS, 1992, Phys. Rev. A, 45,8766.CrossRefGoogle Scholar
  38. [38]
    YAMAMOTO, S.Y., SURKO, C.M., MAPLE, M.B. AND PINA, P.K., 1995, Phys. Rev. Lett., 74, 4071.CrossRefGoogle Scholar
  39. [39]
    ZHABOTINSKI, A. M., M. DOLNIK AND I. EPSTEIN, 1995, J. Chem. Phys., 103, 10306.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Moshe Sheintuch
    • 1
  • Olga Nekhamkina
    • 1
  1. 1.Department of Chemical EngineeringTechnion, Israel Institute of TechnologyHaifaIsrael

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