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

Pascal Chossat 1 Grégory Faye 2
2 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
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 D. Different types of patterns are considered: spatially periodicstationarysolutions,radialsolutionsandtraveling waves,howeverthereare significantdifferencesintheresultswiththeEuclideancase.Weapplyequivariantbifurcation theory to the study of spatially periodic solutions on a given lattice of 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 D and periodic in the "transverse" direction. We highlight our theoretical results with a selection of numerical simulations.
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Submitted on : Wednesday, July 17, 2013 - 2:15:51 PM
Last modification on : Thursday, May 3, 2018 - 1:32:58 PM

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Pascal Chossat, Grégory Faye. Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane. Journal of Dynamics and Differential Equations, Springer Verlag, 2013, pp.1--47. ⟨10.1007/s10884-013-9308-3⟩. ⟨hal-00845612⟩

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