Phase Ordering Dynamics in the Complex Ginzburg-Landau Equation

  • Sushanta Dattagupta
  • Sanjay Puri
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 71)

Abstract

There has been intense interest in problems of pattern formation in biological systems and chemical reactions [327–331]. In general, many physical systems exhibit fascinating spatio-temporal dynamics, resulting from the emergence and interaction of spatially-extended structures, e.g., vortices, spirals, etc. In this context [327–333], much attention has focused on the complex Ginzburg-Landau (CGL) equation, which has the general form:
$$\frac{\partial }{{\partial t}}\psi (r,t)\; = \;\psi + (1 + ia){\nabla ^2}\psi - (1 + i\beta ){\text{|}}\psi {{\text{|}}^{\text{2}}}\psi $$
(6.1)
.

Keywords

Correlation Function Defect Core Domain Growth Random Initial Condition Spiral Line 
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. 327.
    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence ( Springer-Verlag, Berlin 1984 )MATHCrossRefGoogle Scholar
  2. 328.
    M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)ADSCrossRefGoogle Scholar
  3. 329.
    W. van Saarloos, in Spatiotemporal Patterns in Nonequilibrium Systems, ed. by P.E. Cladis and P. Palffy-Muhoray ( Addison-Wesley, Reading, MA 1994 ), p. 19Google Scholar
  4. 330.
    L.M. Pismen, Vortices in Nonlinear Fields (Oxford University Press, Oxford 1999 )Google Scholar
  5. 331.
    I.S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 332.
    S.C. Müller, T. Plesser and B. Hess, Physica D 24, 71 (1987)ADSCrossRefGoogle Scholar
  7. 333.
    G.S. Skinner and H.L. Swinney, Physica D 48, 1 (1991)ADSMATHCrossRefGoogle Scholar
  8. 334.
    R.W. Walden, P. Kolodner, A. Passner and C.M. Surko, Phys. Rev. Lett. 55, 496 (1985)ADSCrossRefGoogle Scholar
  9. 335.
    E. Moses and V. Steinberg, Phys. Rev. A 34, 693 (1986)ADSCrossRefGoogle Scholar
  10. 336.
    F.T. Arecchi, G. Giacomelli, P.L. Ramazza and S. Residori, Phys. Rev. Lett. 65, 2531 (1990); Phys. Rev. Lett. 67, 3749 (1991)ADSCrossRefGoogle Scholar
  11. 337.
    P.S. Hagan, SIAM J. Appl. Math 42, 762 (1982)MATHGoogle Scholar
  12. 338.
    I.S. Aranson, L. Aranson, L. Kramer and A. Weber, Phys. Rev. A 46, R2992 (1992); A. Weber, L. Kramer, I.S. Aranson and L. Aranson, Physica D 61, 279 (1992)ADSMATHCrossRefGoogle Scholar
  13. 339.
    H. Chate, Nonlinearity 7, 185 (1994)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 340.
    H. Chate, in Spatiotemporal Patterns in Nonequilibrium Systems, ed. by P.E. Cladis and P. Palffy-Muhoray ( Addison-Wesley, Reading, MA 1994 ), p. 33Google Scholar
  15. 341.
    H. Chate and P. Manneville, Physica A 224, 348 (1996); P. Manneville and H. Chate, Physica D 96, 30 (1996)MATHCrossRefGoogle Scholar
  16. 342.
    S.K. Das, S. Puri and M.C. Cross, Phys. Rev. E 64, 046206 (2001)Google Scholar
  17. 343.
    S. Puri, S.K. Das and M.C. Cross, Phys. Rev. E 64, 056140 (2001)Google Scholar
  18. 344.
    S.K. Das and S. Puri, Phys. Rev. E 65, 046123 (2002)Google Scholar
  19. 345.
    S. Puri, Phys. Lett. A 164, 211 (1992)ADSCrossRefGoogle Scholar
  20. 346.
    M. Suzuki, Prog. Theor. Phys. 56, 477 (1976)ADSCrossRefGoogle Scholar
  21. 347.
    K. Kawasaki, M.C. Yalabik and J.D. Gunton, Phys. Rev. A 17, 455 (1978)ADSCrossRefGoogle Scholar
  22. 348.
    S. Puri and C. Roland, Phys. Lett. A 151, 500 (1990)ADSCrossRefGoogle Scholar
  23. 349.
    M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics ( SIAM Press, Philadelphia 1981 )Google Scholar
  24. 350.
    S. Puri, Int. J. Mod. Phys. B 4, 1483 (1990)ADSMATHCrossRefGoogle Scholar
  25. 351.
    E.P. Gross, Nuovo Cimento 20, 454 (1961); L. Pitaevskii, Sov. Phys. JETP 13, 451 (1961)Google Scholar
  26. 352.
    Bose-Einstein Condensation,ed. by A. Griffin, D.W. Snoke and S. Stringari (Cambridge University Press, Cambridge 1995)Google Scholar
  27. 353.
    I.S. Aranson, L. Kramer and A. Weber, Phys. Rev. E 47, 3231 (1993); Phys. Rev. E 48, R9 (1993)CrossRefGoogle Scholar
  28. 354.
    G. Huber, P. Alstrom and T. Bohr, Phys. Rev. Lett. 69, 2380 (1992)ADSCrossRefGoogle Scholar
  29. 355.
    V.N. Biktashev, in Nonlinear Waves II, ed. by A.V. Gaponov-Grekhov and M.I. Rabinovich ( Springer-Verlag, Heidelberg 1989 ), p. 87Google Scholar
  30. 356.
    L.M. Pismen and A.A. Nepomnyashchy, Phys. Rev. A 44, R2243 (1991)ADSCrossRefGoogle Scholar
  31. 357.
    T. Bohr, G. Huber and E. Ott, Europhys. Lett. 33, 589 (1996); Physica D 106, 95 (1997)MathSciNetMATHGoogle Scholar
  32. 358.
    P.G. Kevrekidis, A.R. Bishop and K.O. Rasmussen, Phys. Rev. E 65, 016122 (2001)Google Scholar
  33. 359.
    B.A. Malomed, Phys. Rev. E 50, R3310 (1994)MathSciNetADSGoogle Scholar
  34. 360.
    H. Sakaguchi and B.A. Malomed, Physica D 118, 250 (1998)ADSMATHCrossRefGoogle Scholar
  35. 361.
    G.F. Mazenko, Phys. Rev. E 64, 016110 (2001)Google Scholar
  36. 362.
    G.F. Mazenko, Phys. Rev. Lett. 78, 401 (1997)ADSCrossRefGoogle Scholar
  37. 363.
    C. Brito, I.S. Aranson and H. Chate, Phys. Rev. Lett. 90, 068301 (2003)Google Scholar
  38. 364.
    A.T. Winfree, S. Caudle, G. Chen, P. McGuire and Z. Szilagyi, Chaos 6, 617 (1996)ADSCrossRefGoogle Scholar
  39. 365.
    F. Siegert and C.J. Weijer, Physica D 49, 224 (1991)ADSCrossRefGoogle Scholar
  40. 366.
    R.A. Gray and J. Jalife, Int. J. of Bifurcation and Chaos 6, 415 (1996)ADSCrossRefGoogle Scholar
  41. 367.
    L.M. Pismen and J. Rubinstein, Physica D 47, 353 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  42. 368.
    M. Gabbay, E. Ott and P.N. Guzdar, Phys. Rev. Lett. 78, 2012 (1997); Physica D 118, 371 (1998); Phys. Rev. E 58, 2576 (1998)MathSciNetADSGoogle Scholar
  43. 369.
    A.T. Winfree, Physica D 84, 126 (1995)CrossRefGoogle Scholar
  44. 370.
    F. Fenton and A. Karma, Phys. Rev. Lett. 81, 481 (1998); Chaos 8, 20 (1998)ADSMATHGoogle Scholar
  45. 371.
    J. Koplik and H. Levine, Phys. Rev. Lett. 71, 1375 (1993); Phys. Rev. Lett. 76, 4745 (1996)CrossRefGoogle Scholar
  46. 372.
    S. Puri, unpublishedGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Sushanta Dattagupta
    • 1
  • Sanjay Puri
    • 2
  1. 1.S.N. Bose National Centre for Basic SciencesSalt Lake KolkataIndia
  2. 2.School of Physical SciencesJawaharlal Nehru UniversityNew DelhiIndia

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