, Volume 71, Issue 2, pp 253–262 | Cite as

Patterns in the Kardar-Parisi-Zhang equation

  • Hans C. FogedbyEmail author


We review a recent asymptotic weak noise approach to the Kardar-Parisi-Zhang equation for the kinetic growth of an interface in higher dimensions. The weak noise approach provides a many-body picture of a growing interface in terms of a network of localized growth modes. Scaling in 1d is associated with a gapless domain wall mode. The method also provides an independent argument for the existence of an upper critical dimension.


Scaling weak noise growth modes dynamical network upper critical dimension nonlinear Schrödinger equation domain walls solitons dispersion diffusive modes 


05.40.-a 02.50.-r 05.45.Yv 05.90.+m 


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  1. [1]
    A-L Barabasi and H E Stanley, Fractal concepts in surface growth (Cambridge University Press, 1995)Google Scholar
  2. [1a]
    J Krug and H Spohn, Solids far from equilibrium; Kinetic roughening of growing surfaces: Fractal concepts in surface growth (Cambridge University Press, 1992)Google Scholar
  3. [1b]
    J Krug, Adv. Phys. 46, 139 (1997)CrossRefGoogle Scholar
  4. [2]
    M Kardar, G Parisi and Y C Zhang, Phys. Rev. Lett. 56, 889 (1986)zbMATHADSCrossRefGoogle Scholar
  5. [2a]
    E Medina, T Hwa, M Kardar and Y C Zhang, Phys. Rev. A39, 3053 (1989)ADSMathSciNetGoogle Scholar
  6. [3]
    T Halpin-Healy and Y C Zhang, Phys. Rep. 254, 215 (1995)ADSCrossRefGoogle Scholar
  7. [4]
    D Forster, D R Nelson and M J Stephen, Phys. Rev. Lett. 36, 867 (1976); Phys. Rev. A16, 732 (1977)ADSCrossRefGoogle Scholar
  8. [5]
    M Kardar and Y C Zhang, Phys. Rev. Lett. 58, 2087 (1987)ADSCrossRefGoogle Scholar
  9. [5a]
    M Kardar, Nucl. Phys. B290, 582 (1987)ADSMathSciNetCrossRefGoogle Scholar
  10. [6]
    D A Huse, C L Henley and D S Fisher, Phys. Rev. Lett. 55, 2924 (1985)ADSCrossRefGoogle Scholar
  11. [6a]
    E Frey and U C Täuber, Phys. Rev. E50, 1024 (1994)ADSGoogle Scholar
  12. [6b]
    E Frey, U C Täuber and T Hwa, Phys. Rev. E53, 4424 (1996)ADSGoogle Scholar
  13. [6c]
    K J Wiese, J. Stat. Phys. 93, 143 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  14. [6d]
    F Colaiori and M A Moore, Phys. Rev. Lett. 86, 3946 (2001)ADSCrossRefGoogle Scholar
  15. [6e]
    E Katsav and M Schwartz, Physica A309, 69 (2002)ADSGoogle Scholar
  16. [6f]
    M Lässig, Phys. Rev. Lett. 80, 2366 (1998); Nucl. Phys. B448, 559 (1995)CrossRefGoogle Scholar
  17. [6g]
    M Lässig and H Kinzelbach, Phys. Rev. Lett. 78, 903 (1997)ADSCrossRefGoogle Scholar
  18. [6h]
    P Le Doussal and K J Wiese, Phys. Rev. E72, 035101 (2005)Google Scholar
  19. [7]
    H C Fogedby, Phys. Rev. Lett. 94, 195702 (2005); Phys. Rev. E73, 031104 (2006); Phys. Rev. E68, 026132 (2003); Phys. Rev. E59, 5065 (1999); Phys. Rev. E57, 49431 (1998); Phys. Rev. Lett. 80, 1126 (1998)Google Scholar
  20. [8]
    L Onsager and S Machlup, Phys. Rev. 91, 1505 (1953), ibid 1512zbMATHADSMathSciNetCrossRefGoogle Scholar
  21. [8a]
    P C Martin, E D Siggia and H A Rose, Phys. Rev. A8, 423 (1973)ADSGoogle Scholar
  22. [8b]
    R Baussch, H K Janssen and H Wagner, Z. Phys. B24, 113 (1976)ADSGoogle Scholar
  23. [9]
    H C Fogedby, J Hertz and A Svane, Europhys. Lett. 62, 795 (2003)ADSCrossRefGoogle Scholar
  24. [9a]
    H C Fogedby and V Poutkaradze, Phys. Rev. E66, 021103 (2002)Google Scholar
  25. [9b]
    H C Fogedby and Ralf Metzler, Phys. Rev. Lett. 98, 070601 (2007); Phys. Rev. E76, 061915 (2007)Google Scholar

Copyright information

© Indian Academy of Sciences 2008

Authors and Affiliations

  1. 1.Department of Physics and AstronomyAarhus UniversityAarhus CDenmark
  2. 2.Niels Bohr InstituteUniversity of CopenhagenCopenhagen ØDenmark

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