Advertisement

Defects and Transition to Disorder in Space-Time Patterns of Non-Linear Waves

  • A. Joets
  • R. Ribotta
Part of the NATO ASI Series book series (NSSB, volume 237)

Abstract

The traveling-wave convection found in a liquid crystal gives an example of nonlinear waves. The basic state is a uniform progressive wave which is a perfectly ordered structure in space-time. We present an experimental study of some elementary mechanisms that trigger the nucleation of defects in waves and we show that these defects mediate a transition to a chaotic state. It is found that a homogeneous progressive wave can become unstable against local perturbations of the phase which generally have a shock structure. The shocks give rise to topological singularities that are defects of the space-time ordering. It is shown that the structure and the role of these defects in the evolution to a space-time disordering are reminiscent of that of the dislocations and the grain boundaries in stationary convective structures. Some numerical simulations of defects nucleation by use of a Landau-Ginzburg equation, are also presented.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Benjamin, T. B. and Feir, J. E., 1967, The disintegration of wave trains on deep water, J. Fluid Mech., 27: 417.CrossRefGoogle Scholar
  2. Bernoff, A., 1988, Slowly varying fully nonlinear wavetrains in the Ginzburg—Landau equation, Physica D, 30: 363.MathSciNetCrossRefGoogle Scholar
  3. Brand, H.R., Lomdahl, P. and Newell, A., 1986, Evolution of the order parameter in situations with broken rotational symmetry, Phys. Let.A 118: 67.CrossRefGoogle Scholar
  4. Bretherton, C. S. and Spiegel, E. A., 1983, Intermittency through modulational instability, Phys. Lett., 96A: 152.CrossRefGoogle Scholar
  5. Busse, F. H., 1972, The oscillatory instability of convection rolls in a low Prandtl number, J. Fluid Mech., 52: 97.CrossRefGoogle Scholar
  6. Coullet, P., Elphick, C., Gil, L. and Lega, J., 1987, Topological defects of wave patterns, Phys. Rev. Lett., 59: 884.CrossRefGoogle Scholar
  7. Coullet, P. and Lega, J., 1988, Defect—mediated turbulence in wave patterns, Europhys. Lett., 7: 511.CrossRefGoogle Scholar
  8. Eckhaus, W., 1963, “Studies in nonlinear stability theory”, Springer, Berlin.MATHGoogle Scholar
  9. de Gennes, P. G., 1974, “The Physics of Liquid Crystals”, Clarendon, Oxford.MATHGoogle Scholar
  10. Joets, A. and Ribotta, R., 1986, Hydrodynamic transitions to chaos in the convection of an anisotropic fluid, J. Phys. (Paris), 47: 595.CrossRefGoogle Scholar
  11. Joets, A. and Ribotta, R., 1988, Propagative structures and localization in the convection of a liquid crystal, in: “Propagation in Systems far from Equilibrium”, J.E. Wesfreid, H.R. Brand, P. Manneville, G. Albinet, and N. Boccara, eds.,Springer, Berlin.Google Scholar
  12. Joets, A. and Ribotta, R., 1988, Localized time—dependent state in the convection of a nematic liquid crystal, Phys. Rev. Lett., 60: 2164.CrossRefGoogle Scholar
  13. Joets, A. and Ribotta, R., 1988, Propagative patterns in the convection of a nematic liquid crystal, in: “Proceedings of the 12th international liquid crystals conference, Aug. 1988, Freiburg), Liquid Crystals, London.Google Scholar
  14. Joets, A. and Ribotta, R., 1988, Structure of defects in nonlinear waves, preprintGoogle Scholar
  15. Joets, A. and Ribotta, R., 1988, Nucleation of defects in traveling—wave convection, preprintGoogle Scholar
  16. Kawasaki, K.and Ohta, T., 1982, Kink dynamics in one dimensional nonlinear systems, Physica, 116A: 573.MathSciNetCrossRefGoogle Scholar
  17. Keefe, L.R., 1986, Dynamics of perturbed wavetrain solutions to the Ginzburg—Landau equation, Phys. Fluids, 29: 3135.MathSciNetCrossRefGoogle Scholar
  18. Kuramoto, Y., 1978, Diffusion induced chaos in reaction systems, Prog. Theor. Phys. Suppl., 64: 346.CrossRefGoogle Scholar
  19. Kuramoto, Y., 1984, Phase dynamics of weakly unstable periodic structures, Prog. Theor. Phys. 71: 1182.MathSciNetCrossRefGoogle Scholar
  20. Moon, H. T., Huerre, P., Redekopp, L. G., 1983, Transitions to chaos in the Ginzburg—Landau equation, Physica, 7D: 135.MathSciNetMATHGoogle Scholar
  21. Newell, A.C., 1974, Envelope equations, Lect. Appl. Math., 15: 157.MathSciNetMATHGoogle Scholar
  22. Nozaki, K. and Bekki, N., 1983, Pattern selection and spatiotemporal transition to chaos in the Ginzburg—Landau equation, Phys. Rev. Lett., 51: 2171.CrossRefGoogle Scholar
  23. Nozaki, K. and Bekki, N., 1984, Exact solutions of the generalized Ginzburg—Landau equation, J. Phys. Soc. Jap., 53: 1581.Google Scholar
  24. Ribotta, R. and Joets, A., 1984, Defects and interactions with the structures in EHD convection in nematic liquid crystals, in “Cellular Structures in Instabilities”, J. E. Wesfreid and S. Zaleski, eds, Springer, Berlin.Google Scholar
  25. Ribotta, R., Joets, A., and Lin Lei, Oblique roll instability in an electroconvective anisotropic fluid, 1986, Phys. Rev. Lett., 56: 1595.CrossRefGoogle Scholar
  26. Ribotta, R., 1988, Solitons, defects and chaos in dissipative systems, in “Non—linear phenomena in materials science”, L. Kubin, G. Martin, eds., Trans Tech Publications, Switzerland.Google Scholar
  27. Stuart, J.T. and DiPrima, R.C., 1978, The Eckhaus and Benjamin—Feir resonance mechanisms, Proc. R. Soc. Lond. A, 362: 27.CrossRefGoogle Scholar
  28. Whitham, G.B., 1974, “Linear and Nonlinear Waves”, John Wiley zhaohuan Sons, New York.Google Scholar
  29. Yang, X. D., Joets, A. and Ribotta, R., 1986, Singularities in the transition to chaos of a convective anisotropic fluid, Physica, 23D: 235.Google Scholar
  30. Yang, X.D., Joets, A., and Ribotta, R., 1988, Localized instabilities and nucleation of dislocations in convective rolls, in “Propagation in Systems far from Equilibrium”, J.E. Wesfreid, H.R. Brand, P. Manneville, G. Albinet, and N. Boccara, eds., Springer, Berlin.Google Scholar
  31. Yang, X. D., Ribotta, R., 1988, Transition to chaos mediated by defects in convective stationary structures, in preparation.Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • A. Joets
    • 1
  • R. Ribotta
    • 1
  1. 1.Laboratoire de Physique des SolidesUniversité de Paris—SudOrsay CedexFrance

Personalised recommendations