Localized Structures In Pattern-Forming Systems

  • Hermann Riecke
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)


A number of mechanisms that lead to the confinement of patterns to a small part of a translationally symmetric pattern-forming system are reviewed: nonadiabatic locking of fronts, global coupling and conservation laws, dispersion, and coupling to additional slow modes via gradients. Various connections with experimental results are made.


Solitary Wave Hopf Bifurcation Nematic Liquid Crystal Homoclinic Orbit Nonlinear Schrodinger Equation 
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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  1. [1]
    A. ALEKSEEV, S. BOSE, P. RODIN, AND E. SCHöLL, Stability of current filaments in a bistable semiconductor system with global coupling, Phys. Rev. E, 57, (1998), pp.2640-2649.Google Scholar
  2. [2]
    I. ARANSON, K. GORSHKOV, A. LOMOV, AND M. RABINOVICH, Stable particle-like solutions of multidimensional nonlinear fields, Physica D, 43, (1990), p. 435.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    B. BAXTER AND C. ANDERECK, Formation of dynamical domains in a circular Couette system, Phys. Rev. Lett., 57, (1986), p. 3046.CrossRefGoogle Scholar
  4. [4]
    D. BENSIMON, P. KOLODNER, AND C. SURKO, Competing and coexisting dynamical states of traveling-wave convection in an annulus, J. Fluid Mech., 217, (1990), p. 441.CrossRefGoogle Scholar
  5. [5]
    D. BENSIMON, B. SHRAIMAN, AND V. CROQUETTE, Nonadiabatic effects in convection, Phys. Rev. A, 38, (1988), p. 5461.CrossRefGoogle Scholar
  6. [6]
    U. BISANG AND G. AHLERS, Thermal fluctuations, subcritical bifurcation, and nucleation of localized states in electroconvection, Phys. Rev. Lett., 80, (1998), p. 3061.CrossRefGoogle Scholar
  7. [7]
    M. BODE AND H.-G. PURWINS, Pattern formation in reaction-diffusion systems-dissipative solitons in physical systems, Physica D, 86, (1995), p. 53.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    H. BRAND AND R. DEISSLER, Confined states in phase dynamics, Phys. Rev. Lett., 63, (1989), p. 508.CrossRefGoogle Scholar
  9. [9]
    B. CAROLI, C. CAROLI, AND S. FAUVE, On the phenomenology of tilted domains in lamellar eutectic growth, J. Phys. I (Paris), 2 (1992), pp. 281–290.Google Scholar
  10. [10]
    A. CHAMPNEYS, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Physica D, 112, (1998), p. 158.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    P. COULLET, R. GOLDSTEIN, AND G. GUNARATNE, Parity-breaking transitions of modulated patterns in hydrodynamic systems, Phys. Rev. Lett., 63, (1989), p. 1954.CrossRefGoogle Scholar
  12. [12]
    P. COULLET AND G. IOOSS, Instabilities of one-dimensional patterns, Phys. Rev. Lett., 64, (1990), p. 866.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    C. CRAWFORD AND H. RIECKE, Oscillon-type structures and their interaction in a Swift-Hohenberg equation, Physica D, submitted.Google Scholar
  14. [14]
    H. CUMMINS, L. FOURTUNE, AND M. RABAUD, Successive bifurcations in directional viscous fingering, Phys. Rev. E, 47, (1993), p. 1727.CrossRefGoogle Scholar
  15. [15]
    M. DENNIN, G. AHLERS, AND D. CANNELL, Chaotic localized states near the onset of electroconvection, Phys. Rev. Lett., 77, (1996), p. 2475.CrossRefGoogle Scholar
  16. [16]
    J. EGGERS AND H. RIECKE, A continuum model for vibrated sand, preprint, (1998).Google Scholar
  17. [17]
    C. ELPHICK AND E. MERON, Localized structures in surface-waves, Phys. Rev. A, 40, (1989), p. 3226.CrossRefGoogle Scholar
  18. [18]
    S. FAUVE, S. DOUADY, AND O. THUAL, Comment on ”Parity-breaking transitions of modulated patterns in hydrodynamic systems”, Phys. Rev. Lett., 65, (1990), p. 385.CrossRefGoogle Scholar
  19. [19]
    J.-M. FLESSELLES, A. SIMON, AND A. LIBCHABER, Dynamics of one-dimensional interfaces: An experimentalist’s view, Adv. Phys., 40, (1991), p. 1.CrossRefGoogle Scholar
  20. [20]
    G. GRANZOW AND H. RIECKE, Phase diffusion in localized spatio-temporal amplitude chaos, Phys. Rev. Lett., 77, (1996), p. 2451.CrossRefGoogle Scholar
  21. [21]
    A. GROISMAN AND V. STEINBERG, Solitary vortex pairs in viscoelastic Couette flow, Phys. Rev. Lett., 78, (1997), p. 1460.CrossRefGoogle Scholar
  22. [22]
    V. HAKIM AND Y. POMEAU, On stable localized structures and subcritical instabilities, Eur. J. Mech. B Suppl, 10, (1991), p. 137.MathSciNetzbMATHGoogle Scholar
  23. [23]
    J. HEGSETH, J. VINCE, M. DUBOIS, AND P. BERGé, Pattern domains in Rayleigh-Bénard slot convection, Europhys. Lett., 17, (1992), p. 413.CrossRefGoogle Scholar
  24. [24]
    H. HERRERO AND H. RIECKE, Bound pairs of fronts in a real Ginzburg-Landau equation coupled to a mean field, Physica D, 85, (1995), pp. 79–92.zbMATHCrossRefGoogle Scholar
  25. [25]
    L. HOCKING AND K. STEWARTSON, On the nonlinear response of a marginally unstable plane-parallel flow to a two-dimensional disturbance, Proc. R. Soc. Lond. A, 326, (1972), p. 289.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    S. KOGA AND Y. KURAMOTO, Localized patterns in reaction diffusion systems, Progr. Theor. Phys., 63, (1980), pp. 106–112.CrossRefGoogle Scholar
  27. [27]
    H. KOKUBO, M. SANO, B. JANIAUD, AND Y. SAWADA, Generation mechanism of a localized target pattern in electrohydrodynamic convection., J. Phys. Soc. Jpn., 63, (1994), p. 895.CrossRefGoogle Scholar
  28. [28]
    P. KOLODNER, Stable, unstable, and defected confined states of traveling-wave convection, Phys. Rev. E, 50, (1994), p. 2731.CrossRefGoogle Scholar
  29. [29]
    L. KRAMER AND W. ZIMMERMANN, On the Eckhaus instability for spatially periodic patterns, Physica D, 16, (1985), p. 221.zbMATHCrossRefGoogle Scholar
  30. [30]
    O. LIOUBASHEVSKI, H. ARBELL, AND J. FINEBERG, Dissipative solitary states in driven surface-waves., Phys. Rev. Lett., 76, (1996), p. 3959.CrossRefGoogle Scholar
  31. [31]
    T. MAHR AND I. REHBERG, Parametrically excited surface waves in magnetic fluids: observation of domain structures, Phys. Rev. Lett., 80, (1998), p. 89.CrossRefGoogle Scholar
  32. [32]
    B. MALOMED AND A. NEPOMNYASHCHY, Kinks and solitons in the generalized Ginzburg-Landau equation, Phys. Rev. A, 42, (1990), p. 6009.CrossRefGoogle Scholar
  33. [33]
    E. MERON, Pattern formation in excitable media, Phys. Rep., 218, (1992), p. 1.MathSciNetCrossRefGoogle Scholar
  34. [34]
    E. MOSES, J. FINEBERG, AND V. STEINBERG, Multistability and confined traveling-wave patterns in a convecting binary mixture, Phys. Rev. A, 35, (1987), p. 2757.CrossRefGoogle Scholar
  35. [35]
    M. NEUFELD, R. FRIEDRICH, AND H. HAKEN, Order-parameter equation and model equation for high Prandtl number Rayleigh-Bénard convection in a rotating large aspect ratio system, Z. Phys. B, 92, (1993), p. 243.CrossRefGoogle Scholar
  36. [36]
    J. NIEMELA, G. AHLERS, AND D. CANNELL, Localized traveling-wave states in binary-fluid convection, Phys. Rev. Lett., 64, (1990), p. 1365.CrossRefGoogle Scholar
  37. [37]
    L. PISMEN, Modulated solitons at a singular Hop] bifurcation, Phys. Rev. A, 35, (1987), p. 1873.Google Scholar
  38. [38]
    L. PISMEN AND H. RIECKE, unpublished.Google Scholar
  39. [39]
    Y. POMEAU, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23, (1986), p. 3.CrossRefGoogle Scholar
  40. [40]
    S. POPP, O. STILLER, E. KUZNETSOV, AND L. KRAMER, The cubic complex Ginzburg-Landau equation for a backward bifurcation, Physica D, 114, (1998), p. 81.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    D. RAITT AND H. RIECKE, Domain structures in fourth-order phase and Ginzburg-Landau equations, Physica D, 82, (1995), pp. 79–94.zbMATHCrossRefGoogle Scholar
  42. [42]
    H. RIECKE —, Parametric forcing of waves with a nonmonotonic dispersion relation: domain structures in ferrofluids, Phys. Rev. E, 55, (1997), pp. 5448–5454.CrossRefGoogle Scholar
  43. [43]
    H. RIECKE, Stable wave-number kinks in parametrically excited standing waves, Europhys. Lett., 11, (1990), p. 213.CrossRefGoogle Scholar
  44. [44]
    —, Ginzburg-Landau equation coupled to a concentration field in binary-mixture convection, Physica D, 61, (1992), pp.253–259.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    —, Self-trapping of traveling-wave pulses in binary mixture convection, Phys. Rev. Lett., 68, (1992), p. 301.CrossRefGoogle Scholar
  46. [46]
    —, Solitary waves under the influence of a long-wave mode, Physica D, 92, (1996), pp. 69–94.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    H. RIECKE AND G. GRANZOW, Localization of waves without bistability: Worms in nematic electroconvection, Phys. Rev. Lett., 81, (1998), p. 333.CrossRefGoogle Scholar
  48. [48]
    H. RIECKE AND H.-G. PAAP, Parity-breaking and Hopf bifurcation in axisymmetric Taylor vortex flow, Phys. Rev. A, 45, (1992), p. 8605.CrossRefGoogle Scholar
  49. [49]
    H. ROTERMUND AND G. ERTL, Solitons in a surface-reaction., Phys. Rev. Lett., 66, (1991), p. 3083.CrossRefGoogle Scholar
  50. [50]
    D. ROTHMAN, Oscillons, spiral waves, and stripes in a model of vibrated sand, Phys. Rev. E, 57, (1998), pp. 1239–1242.CrossRefGoogle Scholar
  51. [51]
    H. SAKAGUCHI, Localized oscillation in a cellular pattern., Prog. Theor. Phys., 87, (1992), p. 1049.MathSciNetCrossRefGoogle Scholar
  52. [52]
    H. SAKAGUCHI AND H. BRAND, Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation, Physica D, 97, (1996), p. 274.CrossRefGoogle Scholar
  53. [53]
    H. BRAND —, Stable localized squares in pattern-forming nonequilibrium systems, Europhys. Lett., 38, (1997), p. 341.CrossRefGoogle Scholar
  54. [54]
    B. SANDSTEDE, C. JONES, AND J. ALEXANDER, Existence and stability of n-pulses on optical fibers with phase-sensitive amplifiers, Physica D, 106, (1997), p. 167.MathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    L. SCHIMANSKY-GEIER, C. ZüLICKE, AND E. SCHOLL, Domain formation due to Ostwald ripening in bistable systems far from equilibrium, Z. Phys. B, 84, (1991), p. 433.CrossRefGoogle Scholar
  56. [56]
    O. THUAL AND S. FAUVE, Localized structures generated by subcritical instabilities, J. Phys., (Paris), 49, (1988), p. 1829.CrossRefGoogle Scholar
  57. [57]
    M. TREIBER AND L. KRAMER, Coupled complex Ginzburg-Landau equations for the weak electrolyte model of electroconvection, preprint, (1997).Google Scholar
  58. [58]
    L. TSIMRING AND I. ARANSON, Localized and cellular patterns in a vibrated granular layer, Phys. Rev. Lett., 79, (1997), p. 213.CrossRefGoogle Scholar
  59. [59]
    Y. Tu, Worm structure in modified Swift-Hohenberg equation for electroconvection, Phys. Rev. E, 56, (1997), p. 3765.CrossRefGoogle Scholar
  60. [60]
    J. TYSON AND J. P. KEENER, Singular perturbation-theory of traveling waves in excitable media, Physica D, 32, (1988), pp. 327–361.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [61]
    P. UMBANHOWAR, F. MELO, AND H. SWINNEY, Localized excitations in a vertically vibrated granular layer, Nature, 382, (1996), p. 793.CrossRefGoogle Scholar
  62. [62]
    W. VAN SAARLOOS AND P. HOHENBERG, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D, 56, (1992), p. 303.MathSciNetCrossRefGoogle Scholar
  63. [63]
    S. VENKATARAMANI AND E. OTT, Spatio-temporal bifurcation phenomena with temporal period doubling: patterns in vibrated sand, Phys. Rev. Lett., 80, (1998), p. 3495.CrossRefGoogle Scholar
  64. [64]
    M. WEINSTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, (1985), p. 472.MathSciNetzbMATHCrossRefGoogle Scholar
  65. [65]
    R. WIENER AND D. MCALISTER, Parity-breaking and solitary waves in axisymmetric Taylor vortex flow, Phys. Rev. Lett., 69, (1992), p. 2915.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Hermann Riecke
    • 1
  1. 1.Department of Engineering Sciences and Applied MathematicsNorthwestern UniversityEvanstonUSA

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