Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Journal of Dynamics and Differential Equations Year : 2013

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

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.

Dates and versions

hal-00845612 , version 1 (17-07-2013)

Identifiers

Cite

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

Altmetric

Share

Gmail Facebook X LinkedIn More