Advertisement

A Quantitative Description of the Relaxation of Textured Patterns

  • Gemunu H. Gunaratne
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)

Abstract

A characterization of textured patterns, referred to as the disorder function δ(β), is used to study the dynamics of patterns generated in the Swift-Hohenberg equation (SHE). The evolution of random initial states under the SHE exhibits two stages of relaxation. The initial phase, where local striped domains emerge from a noisy background, is quantified by a power law decay δ(β) ~t -1/2 Beyond a sharp transition a slower power law decay of δ(β), which corresponds to the coarsening of striped domains, is observed. The transition between the phases advances as the system is driven further from the onset of patterns, and suitable scaling of time and δ(β) leads to the collapse of distinct curves.

Keywords

Domain Wall Texture Pattern Target Pattern Random Initial State Disorder Function 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Q. OUYANG AND H. L. SwiNNEY, Nature, (London), 352, 610, (1991).CrossRefGoogle Scholar
  2. [2]
    M. S. HEUTMAKER AND J. P. GOLLUB, Phys. Rev. A, 35, 242, (1987).CrossRefGoogle Scholar
  3. [3]
    E. BODENSCHATZ, J. R. DE BRUYN, G. AHLERS, AND D. S. CANNEL, Phys. Rev. Lett., 67, 3078, (1991).CrossRefGoogle Scholar
  4. [4]
    R. E. ROSENWEIG, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985.Google Scholar
  5. [5]
    F. MELO, P. UMBANHOWER, AND H. L. SWINNEY, Phys. Rev. Lett., 72, 172, (1993).CrossRefGoogle Scholar
  6. [6]
    M. C. CROSS AND P. C. HOHENBERG, Rev. Mod. Phys., 65(3), 851, (1993).CrossRefGoogle Scholar
  7. [7]
    P. K. JAKOBSEN, J. V. MALONEY, AND A. C. NEWELL, Phys. Rev. A, 45, 8129, (1992).CrossRefGoogle Scholar
  8. [8]
    M. BESTEHORN, Phys. Rev. E, 48, 3622, (1993).MathSciNetCrossRefGoogle Scholar
  9. [9]
    M. GOLUBITSKY, I. STEWART, AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 2, 1988.zbMATHCrossRefGoogle Scholar
  10. [10]
    K. R. ELDER, J. VIñALS, AND M. GRANT, Phys. Rev. A, 46, 7618, (1990).CrossRefGoogle Scholar
  11. [11]
    Q. OUYANG AND H. L. SWINNEY, Chaos, 1, 411, (1991).zbMATHCrossRefGoogle Scholar
  12. [12]
    M. C. CROSS AND D. I. MEIRON, Phys. Rev. Lett., 75, 2152, (1995).CrossRefGoogle Scholar
  13. [13]
    J. J. CHRISTENSEN AND A. J. BRAY, preprint (cond-mat/9804034).Google Scholar
  14. [14]
    Q. Hou, S. SASA, AND N. GOLDENFELD, Physica A, 239, 219, (1997).CrossRefGoogle Scholar
  15. [15]
    G. H. GUNARATNE, R. E. JONES, Q. OUYANG, AND H. L. SWINNEY, Phys. Rev. Lett., 75, 3281, (1995).CrossRefGoogle Scholar
  16. [16]
    G. H. GUNARATNE, D. K. HOFFMAN, AND D. J. KOURI, Phys. Rev. E, 57, 5146, (1998).CrossRefGoogle Scholar
  17. [17]
    A. KUDROLLI, B. PIER, AND J. P. GOLLUB, Superlattice Patterns on Surface Waves, to appear in Physica D.Google Scholar
  18. [18]
    S. L. JUDD AND M. SILBER, Simple and Superlattice Turing Patterns in Reaction-Diffusion Systems: Bifurcation, Bistability, and Parameter Collapse, Northwestern University preprint.Google Scholar
  19. [19]
    W. S. EDWARDS AND S. FAUVE, Phys. Rev. E, 47, R788, (1993).CrossRefGoogle Scholar
  20. [20]
    D. LEVINE AND P. J. STEINHART, Phys. Rev. Lett., 53, 2477, (1984).CrossRefGoogle Scholar
  21. [21]
    L. LANDAU AND E. LIFSHITZ, Fluid Mechanics, Pergamon Press, Oxford, 1959.Google Scholar
  22. [22]
    I. MELBOURNE, J. Nonlin. Sci. 8, 1, (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    M. C. CROSS, Phys. Rev. A, 25, 1065, (1982).CrossRefGoogle Scholar
  24. [24]
    D. K. HOFFMAN, M. ARNOLD, AND D. J. KOURI, J. Chem. Phys., 96, 6539, (1992).CrossRefGoogle Scholar
  25. [25]
    D. K. HOFFMAN, G. H. GUNARATNE, D. Z. ZHANG, AND D. J. KOURI, University of Houston preprint.Google Scholar
  26. [26]
    Y. POMEAU AND P., MANNEVILLE, Phys. Lett., 75A, 296, (1980).MathSciNetGoogle Scholar
  27. [27]
    Y. Tu AND M. C. CROSS, Phys. Rev. Lett., 75, 2152, (1995).CrossRefGoogle Scholar
  28. [28]
    R. E. JONES, Characterizing Disorder in Labyrinthine Patterns, Ph. D. Thesis, University of Houston, (1997).Google Scholar
  29. [29]
    J. SWIFT AND P. C. HOHENBERG, Phys. Rev. A, 15, 319, (1977).CrossRefGoogle Scholar
  30. [30]
    G. H. GUNARATNE, A. RATNAWEERA, AND K. TENNEKONE, Emergence of Order in Textured Patterns, University of Houston preprint.Google Scholar
  31. [31]
    P. GRAY AND S. K. SCOTT, Chem. Eng. Sci., 38, 29, (1983); J. Phys. Chem., 89, 22, (1985).Google Scholar
  32. [32]
    W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY AND W. T. VETTERING, Numerical Recipes — The Art of Scientific Computing, Cambridge University Press, Cambridge, 1988.Google Scholar
  33. [33]
    A. D. RUTENBERG AND A. J. BRAY, Phys. Rev. E, 51, 5499, (1995).CrossRefGoogle Scholar
  34. [34]
    K. KAWASAKI, Phys. Rev. A, 31, 3880, (1985).CrossRefGoogle Scholar
  35. [35]
    R. LOFT AND T. A. DEGRAND, Phys. Rev. B, 35, 8528, (1987).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Gemunu H. Gunaratne
    • 1
    • 2
  1. 1.Department of PhysicsThe University of HoustonHouston
  2. 2.The Institute of Fundamental StudiesKandySri Lanka

Personalised recommendations