Modulated Traveling Waves in Nonequilibrium Systems

  • M. Bestehorn
  • R. Friedrich
  • H. Haken
Part of the NATO ASI Series book series (NSSB, volume 225)


Time periodic behaviour may arise in systems far from equilibrium due to an instability of a stationary state. If such systems are additionally able to produce spatial patterns a rich variety of phenomena occurs which have recently attracted experimental as well as theoretical interest. Experimental systems under consideration are the Taylor-Couette experiment with counter rotating cylinders [1], convection in binary fluid mixtures [2], as well as higher instabilities arising in the Bénard experiment [3]. Since interesting spatio-temporal behaviour occurs already close to instability the description of the system can be formulated in terms of the synergetic concepts of order parameters and their dynamics [4] because other degrees of freedom of the systems are enslaved. The present paper gives an overview over theoretical results which have been obtained by an examination of the generalized Ginzburg-Landau equation which describes the behaviour of the system close to onset in terms of a suitably defined order parameter [5]. Section II presents this generalized Ginzburg-Landau equation. Section III deals with one-dimensional traveling wave patterns and indicates a mechanism by which modulated traveling waves are generated. Section IV is devoted to two-dimensional traveling wave patterns.


Wave Train Amplitude Equation Oscillatory Instability Convective Roll Realistic Boundary Condition 
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  1. [1]
    R. Tagg, W. S. Edwards, H. Swinney, and P. S. Marcus, Phys. Rev. A 39, 3734 (1989)ADSCrossRefGoogle Scholar
  2. [2]
    V. Steinberg, E. Moses, and J. Fineberg in the proceedings of the International Conference on ‘The Physics of Chaos and Systems Far From Equilibrium’, Monterey, Jan 10–14, 1986, in the Journal of Nuclear Phys. B 2, 109 (1987);Google Scholar
  3. [2a]
    P. Kolodner, A. Passner, H.L. Williams, and C. M. Surko, ibid., 97Google Scholar
  4. [3]
    V. Croquette, H. Williams, Phys. Rev. A 39, 2765 (1989)ADSCrossRefGoogle Scholar
  5. [3a]
    A. Chiffaudel, B. Perrin, and S. Fauve, Phys. Rev. A 39, 2761 (1989)ADSCrossRefGoogle Scholar
  6. [4]
    H. Haken, ‘Synergetics, An Introduction’, (3rd ed., Springer, Berlin 1983)zbMATHGoogle Scholar
  7. [4a]
    H. Haken, ‘Advanced Synergetics’, (2.print., Springer, Berlin 1987)Google Scholar
  8. [5]
    M. Bestehorn, R. Friedrich, H. Haken, Z.Phys. B 72, 265 (1988)ADSCrossRefGoogle Scholar
  9. [5a]
    M. Bestehorn, R. Friedrich, H. Haken, Z.Phys. B 75, 265 (1989)ADSCrossRefGoogle Scholar
  10. [5b]
    M. Bestehorn, R. Friedrich, H. Haken, Z.Phys. B 77, 151 (1989)ADSCrossRefGoogle Scholar
  11. [5c]
    M. Bestehorn, R. Friedrich, H. Haken, Physica D 37, 295 (1989)ADSCrossRefGoogle Scholar
  12. [6]
    J. Fineberg, E. Moses, and V. Steinberg, Phys. Rev. Lett. 61, 838 (1988)ADSCrossRefGoogle Scholar
  13. [6a]
    P. Kolodner and C. M. Surko, Phys. Rev. Lett. 61, 842 (1988)ADSCrossRefGoogle Scholar
  14. [7]
    R. Tagg, H. Swinney, poster presented conference on Advances in Turbulence, Los Alamos, May 16–20 (1988)Google Scholar
  15. [8]
    M. Cross, Phys. Rev. A 38, 3593 (1988)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • M. Bestehorn
    • 1
  • R. Friedrich
    • 1
  • H. Haken
    • 1
  1. 1.Institut für Theoretische Physik und SynergetikUniversität StuttgartStuttgart 80Germany

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