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Dissipative Solitons in Reaction-Diffusion Systems

Part of the Lecture Notes in Physics book series (LNP,volume 661)

Abstract

A major goal of natural science is to understand the formation of spatiallyextended patterns in all kinds of physical, chemical, biological and other systems. In many cases, it is advantageous to interpret the overall pattern under consideration in terms of a superposition of certain spatially well-localized elementary patterns that we may refer to as “particles”. In the simplest case, all these particles are of the same kind and the complex behavior of the extended pattern can be described in terms of simple individual properties of the particles and their interaction. A clear illustrative example for this approach is the concept of atoms. In this case, the elementary pattern or particle is the atom and the complex spatially-extended pattern is, e.g., the crystal.

Keywords

  • Bifurcation Point
  • Reaction Function
  • Interaction Function
  • Physical Review Letter
  • Goldstone Mode

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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REFERENCES

  1. J. Abshagen, A. Schulz and G. Pfister, The Taylor-Couette flow: A paradigmatic system for instabilities, pattern formation, and routes to chaos, Lecture Notes in Physics, Eds.: S. Parisi, S. Müller and W. Zimmermann (Springer, Berlin, 1996).

    Google Scholar 

  2. T. Ackemann and W. Lange, Applied Physics B, 72, 21 (2001).

    Google Scholar 

  3. H.-G. Purwins, Yu. A. Astrov, I. Brauer and M. Bode, Localized Patterns in Planar Gas-Discharge Systems, RIMS Project 2000: Reaction-Diffusion Systems: Theory and Application. Interfaces, Pulses and Waves in Nonlinear Dissipative Systems, 28–31 August 2000, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, pp. 1–8, (2001).

    Google Scholar 

  4. H.-G. Purwins, Yu. Astrov and I. Brauer, Self-Organized Quasi Particles and Other Patterns in Planar Gas-Discharge Systems, The 5th Experimental Chaos Conference. Orlando, Florida, USA 28 June – 1 July 1999, Eds.: M. Ding, W. L. Ditto, L. M. Pecora and M. L. Spano, (World Scientific, Singapore, 2001) pp. 3–13.

    Google Scholar 

  5. E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors, Cambridge Nonlinear Science Series, V. 10, (Cambridge University Press, Cambridge, 2001).

    Google Scholar 

  6. E. Schöll, F. J. Niedernostheide, J. Parisi, W. Prettl and H.-G. Purwins, Formation of Spatio-Temporal Structures in Semiconductors, In: Evolution of Spontaneous Structures in Dissipative Continuous Systems, Eds.: F. H. Busse and S. C. Müller, pp. 446–494, 1998.

    Google Scholar 

  7. M. Bode and H.-G. Purwins, Physica D 86, 53 (1995).

    Google Scholar 

  8. V. V. Bel’kov, J. Hirschinger, V. Novák, F. J. Niedernostheide, S. D. Ganichev and W. Prettl, Nature, 397, 398 (1999).

    Google Scholar 

  9. K. Aoki, Nonlinear Dynamics and Chaos in Semiconductors, (Institute of Physics Publishing, Bristol and Philadelphia, 2001).

    Google Scholar 

  10. H. H. Rotermund, S. Jakubith, A. von Oertzen and G. Ertl, Physical Review Letters, 66, 3083 (1991).

    Google Scholar 

  11. Yu. A. Astrov and H.-G. Purwins, Physics Letters A, 283, 349 (2001).

    Google Scholar 

  12. R. J. Field and R. M. Noyes, Journal of Chemical Physics 60, 1877 (1974).

    Google Scholar 

  13. R. FitzHugh, Biophysical Journal 1, 445 (1961).

    Google Scholar 

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

    Google Scholar 

  15. I. Prigogine and R. Lefever, Journal of Chemical Physics 48, 1695 (1968).

    Google Scholar 

  16. A. M. Turing, Philosophical Transactions of the Royal Society of London Series B – Biological Sciences 237, (1952).

    Google Scholar 

  17. B. Schäpers, M. Feldmann, T. Ackemann and W. Lange, Physical Review Letters, 85, 748 (2000).

    Google Scholar 

  18. K. M. Mayer, J. Parisi and R. P. Huebener, Zeitschrift für Physik B – Condensed Matter 71, 171 (1988).

    Google Scholar 

  19. A. L. Hodgkin and A. F. Huxley, Journal of Physiology 117, 500 (1952).

    Google Scholar 

  20. D. Barkley, Physica D, 49, 61 (1991).

    Google Scholar 

  21. B. S. Kerner and V. V. Osipov, Uspekhi Fizicheskikh Nauk 157, 201 (1989).

    Google Scholar 

  22. B. S. Kerner and V. V. Osipov, Autosolitons. A New Approach to Problems of Self-Organization and Turbulence, Fundamental Theories of Physics, V. 61, (Kluwer Acad. Publ., Dordrecht, 1994).

    Google Scholar 

  23. F.-J. Niedernostheide, M. Arps, R. Dohmen, H. Willebrand and H.-G. Purwins, Physica Status Solidi B 172, 249 (1992).

    Google Scholar 

  24. H. Meinhardt, Models of Biological Pattern Formation, (Academic Press, London, 1982).

    Google Scholar 

  25. J. Nagumo, S. Arimoto and S. Yoshizawa, Proceedings of the institute of radio engineers 50, 2061 (1962).

    Google Scholar 

  26. J. D. Murray, Mathematical Biology, (Springer, Berlin, 1993).

    Google Scholar 

  27. Q. Ouyang and H. L. Swinney, Nature, 352, 610 (1991).

    CrossRef  Google Scholar 

  28. C. I. Christov and M. G. Velarde, Physica D, 86, 323 (1995). .

    Google Scholar 

  29. J. J. Tyson, The Belousov-Zhabotinsky Reaction, Lecture Notes in Biomathematics, Volume 10, (Springer, Berlin, 1976).

    Google Scholar 

  30. M. A. Vorontsov and W. B. Miller, Self-Organization in Optical Systems and Application in Information Technology, 2-nd edition, (Springer, Berlin, 1998).

    Google Scholar 

  31. F. T. Arecchi, S. Boccaletti and P. Ramazza, Physics Reports, 328, 1,

    Google Scholar 

  32. O. Lioubashevski, H. Arbell, and J. Fineberg, Physical Review Letters, 76, 3959 (1996).

    Google Scholar 

  33. P. B. Umbanhowar, F. Melo, H. L. Swinney, Nature, 382, 793 (1996).

    Google Scholar 

  34. C. Crawford and H. Riecke, Physica D, 129, 83 (1999).

    Google Scholar 

  35. T. Ohta, Physica D, 151, 61 (2001).

    Google Scholar 

  36. Yu. A. Astrov, Physical Review E, 67, 035203 (2003).

    Google Scholar 

  37. C. P. Schenk, P. Schütz, M. Bode and H.-G. Purwins, Physical Review E, 57, 6480 (1998).

    Google Scholar 

  38. P. Coulett, C. Riera and C. Tresser, Physical Review Letters, 84, 3069 (2000).

    Google Scholar 

  39. J. Smoller, Shock Waves and Reaction Diffusion Equations, Second Edition, (Springer, New York, 1994).

    Google Scholar 

  40. S. Koga and Y. Kuramoto, Progress of Theoretical Physics, 63, 106 (1980).

    Google Scholar 

  41. T. Ohta, M. Mimura and R. Kobayashi, Physica D, 34, 115 (1989).

    Google Scholar 

  42. R. Schmeling, Experimentelle und numberische Untersuchungen von Strukturen in einem Reaktions-Diffusions-System anhand eines elektrischen Netzwerkes, Institut für Angewandte Physik, Westfälische Wilhelms-Universität Münster, Dissertation, 1994.

    Google Scholar 

  43. Mimura, M. and Nagayama, M., Methods and Applications of Analysis oder Tohoko Mathematical Publications, 8, 239 (1997).

    Google Scholar 

  44. Mimura, M. and Nagayama, M. and Ikeda, H. and Ikeda, T., Hiroshima Mathematical Journal, 30, 221 (1999).

    Google Scholar 

  45. R. J. Field and M. Burger, Oscillations and traveling waves in chemical systems, (Wiley, New York, 1985).

    Google Scholar 

  46. Y. Nishiura and D. Ueyama, Physica D, 130, 73 (1999).

    Google Scholar 

  47. A. Gierer and H. Meinhardt, Kibernetik, 12, 30 (1972).

    Google Scholar 

  48. G. Bordiougov and H. Engel, Physical Review Letters, 90, 148302 (2003).

    Google Scholar 

  49. C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Physical Review Letters, 78, 3781 (1997).

    Google Scholar 

  50. M. Bode, A. W. Liehr, C. P. Schenk and H.-G. Purwins, Physica D, 161, 45 (2002).

    Google Scholar 

  51. Physica D, 165, 127 (2002).

    Google Scholar 

  52. C. P. Schenk, Numberische und analytische Untersuchung solitärer Strukturen in zwei- und dreikomponentigen Reaktions-Diffusions-Systemen, Institut für Angewandte Physik, Westfälische Wilhelms-Universität Münster, Dissertation, 1999.

    Google Scholar 

  53. A. W. Liehr, A. S. Moskalenko, M. C. Röttger, J. Berkemeier and H.-G. Purwins, Replication of Dissipative Solitons by Many-Particle Interaction, In: High Performance Computing in Science and Engineering ‘02. Transactions of the High Performance Computing Center Stuttgart (HLRS) 2002, Eds.: E. Krause and W. Jäger, pp. 48–61 (Springer, Berlin, 2003).

    Google Scholar 

  54. A. S. Moskalenko, A. W. Liehr and H.-G. Purwins, Europhysics Letters, 63, 361 (2003).

    Google Scholar 

  55. S. V. Gurevich, H. U. Bödeker, A. S. Moskalenko, A. W. Liehr and H.-G. Purwins, Drift Bifurcation of Dissipative Solitons due to a Change of Shape, Proceedings of the International Conference of Physics and Control St. Petersburg, pp. 601–606, (IEEE, 2003).

    Google Scholar 

  56. S. V. Gurevich, H. U. Bödeker, A. S. Moskalenko, A. W. Liehr and H.-G. Purwins, Physica D, 199, 115 (2004).

    Google Scholar 

  57. H. Haken, Synergetics, An Introduction, 3rd edition, Springer Series Synergetics, (Springer, Berlin, Heidelberg, New York, 1983).

    Google Scholar 

  58. R. Friedrich, Zeitschrift für Physik B – Condensed Matter, 90, 373 (1993).

    Google Scholar 

  59. M. Bode, Physica D, 106, 270 (1997).

    Google Scholar 

  60. E. Ammelt, Yu. A. Astrov and H.-G. Purwins, Physical Review E, 58, 7109 (1998).

    Google Scholar 

  61. A. Moskalenko, Dynamische gebundene Zustände und Drift-Rotations-Dynamik von dissipativen Solitonen, Institut für Angewandte Physik, Westfälische Wilhelms-Universität Münster, 2002.

    Google Scholar 

  62. Y. P. Raizer, Gas Discharge Physics, 2-nd edition, (Springer, Berlin, 1997).

    Google Scholar 

  63. M. S. Benilov, Physical Review E, 48, 5901 (1992).

    Google Scholar 

  64. M. S. Benilov, Physical Review A, 45, 506 (1993).

    Google Scholar 

  65. Y. B. Golubovskii, I. A. Porokhova, A. Dinklage and C. Wilke, Journal of Physics D – Applied Physics, 33, 517 (2000).

    Google Scholar 

  66. Yu. Astrov, E. Ammelt, S. Teperick and H.-G. Purwins, Physics Letters A, 211, 184 (1996).

    Google Scholar 

  67. Yu. A. Astrov, Yu. A. Logvin, Physical Review Letters, 79, 2983 (1997).

    Google Scholar 

  68. E. L. Gurevich, A. S. Moskalenko, A. L. Zanin, Yu. A. Astrov and H.-G. Purwins, Physics Letters A, 307, 299 (2003).

    Google Scholar 

  69. I. Brauer, M. Bode, E. Ammelt and H.-G. Purwins, Physical Review Letters, 84, 4104 (2000).

    Google Scholar 

  70. C. Strümpel, Yu. A. Astrov and H.-G. Purwins, Physical Review E, 65, 066210 (2002).

    Google Scholar 

  71. C. Strümpel, H.-G. Purwins and Yu. A. Astrov, Physical Review E, 63, 026409 (2001).

    Google Scholar 

  72. H. U. Bödeker, M. C. Röttger, A. W. Liehr, T. D. Frank, R. Friedrich and H.-G. Purwins, Physical Review E, 67, 056220 (2003).

    Google Scholar 

  73. H. U. Bödeker, A. W. Liehr, M. C. Röttger, T. D. Frank, R. Friedrich and H.-G. Purwins, New Journal of Physics, 6, 62 (2004).

    Google Scholar 

  74. A. W. Liehr, H. U. Bödeker, M. C. Röttger, T. D. Frank, R. Friedrich and H.-G. Purwins, New Journal of Physics, 5, 89.1 (2003).

    Google Scholar 

  75. U. Erdmann, W. Ebeling, L. Schimansky-Geier and F. Schweitzer, European Physical Journal B, 15, 105 (2000).

    Google Scholar 

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Purwins, HG., Bödeker, H., Liehr, A. Dissipative Solitons in Reaction-Diffusion Systems. In: Akhmediev, N., Ankiewicz, A. (eds) Dissipative Solitons. Lecture Notes in Physics, vol 661. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10928028_11

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