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Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane

  • Pascal Chossat
  • Grégory Faye
Article

Abstract

In this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc \(\mathbb{D }\). Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of \(\mathbb{D }\) also called H-planforms in reference with the “planforms” introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of \(\mathbb{D }\) and periodic in the “transverse” direction. We highlight our theoretical results with a selection of numerical simulations.

Keywords

Swift-Hohenberg equation Pattern formation Poincaré disk Equivariant bifurcation Traveling wave 

Notes

Acknowledgments

GF is grateful to James Rankin and David Lloyd for their helpful comments on the functionality of AUTO.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.JAD LaboratoryUniversity of Nice Sophia Antipolis & CNRSNice Cedex 02France
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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