Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane

TL;DR

This paper explores pattern formation in the Swift-Hohenberg equation on the hyperbolic plane, revealing unique bifurcation behaviors and solutions distinct from Euclidean cases, supported by theoretical analysis and numerical simulations.

Contribution

It provides a comprehensive analysis of pattern formation on the hyperbolic plane, including bifurcation theory, existence of localized solutions, and discovery of traveling waves unique to hyperbolic geometry.

Findings

Complete description of H-planforms for octagonal lattice

Existence of stationary localized radial solutions

Hopf bifurcation to traveling waves invariant along horocycles

Abstract

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 Poincar\'e disc 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 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.