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Dislocations in an Anisotropic Swift-Hohenberg Equation

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Abstract

We study the existence of dislocations in an anisotropic Swift-Hohenberg equation. We find dislocations as traveling or standing waves connecting roll patterns with different wavenumbers in an infinite strip. The proof is based on a bifurcation analysis. Spatial dynamics and center-manifold reduction yield a reduced, coupled-mode system of differential equations. Existence of traveling dislocations is then established by showing that this reduced system possesses robust heteroclinic orbits.

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References

  1. Boyer D.: Numerical study of domain coarsening in anisotropic stripe patterns. Phys. Rev. E 69, 066111 (2004)

    Article  ADS  Google Scholar 

  2. Cross M., Hohenberg P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)

    Article  ADS  Google Scholar 

  3. Ercolani N., Indik R., Newell A., Passot T.: The geometry of the phase diffusion equation. J. Nonlinear Sci. 10, 223–274 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Ercolani, N., Indik, R., Newell, A., Passot, T.: Global description of patterns far from onset: a case study. In: Complexity and nonlinearity in physical systems (Tucson, AZ, 2001). Phys. D 184, 127–140 (2003)

  5. Ercolani N., Venkataramani S.: A variational theory for point defects in patterns. J. Nonlinear Sci. 19, 267–300 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Haragus, M., Iooss, G.: Local bifurcations, center manifolds, and normal forms in infinite dimensional dynamical systems. Universitext. London:Springer-Verlag London, Ltd., Les Ulis: EDP Sciences, 2011

  7. Kamaga C., Ibrahim F., Dennin M.: Dislocation dynamics in an anisotropic stripe pattern. Phys. Rev. E 69, 066213 (2004)

    Article  ADS  Google Scholar 

  8. Peach M., Koehler J.S.: The forces exerted on dislocations and the stress fields produced by them. Phys. Rev. 80, 436–439 (1950)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Pismen, L.: Patterns and interfaces in dissipative dynamics. With a foreword by Y. Pomeau. Springer Series in Synergetics. Berlin:Springer-Verlag 2006

  10. Qian H., Mazenko G.: Growth of order in an anisotropic Swift-Hohenberg model. Phys. Rev. E 73, 036117 (2006)

    Article  ADS  Google Scholar 

  11. Roxin A., Riecke H.: Rotating convection in an anisotropic system. Phys. Rev. E 65, 046219 (2002)

    Article  ADS  Google Scholar 

  12. van den Berg G., van der Vorst R.: A domain-wall between single-mode and bimodal states. Diff. Int. Eqs. 13, 369–400 (2000)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Université de Franche-Comté, Laboratoire de Mathématiques, 16 route de Gray, 25030, Besançon Cedex, France

    Mariana Haragus

  2. University of Minnesota, School of Mathematics, 206 Church St. S.E., Minneapolis, MN, 55455, USA

    Arnd Scheel

Authors
  1. Mariana Haragus
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  2. Arnd Scheel
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Correspondence to Arnd Scheel.

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Communicated by P. Constantin

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Haragus, M., Scheel, A. Dislocations in an Anisotropic Swift-Hohenberg Equation. Commun. Math. Phys. 315, 311–335 (2012). https://doi.org/10.1007/s00220-012-1569-x

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  • DOI: https://doi.org/10.1007/s00220-012-1569-x

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