Interfaces between rolls in the Swift-Hohenberg equation. (English) Zbl 1169.35032

Haragus, Mariana; Scheel, Arnd

Int. J. Dyn. Syst. Differ. Equ. 1, No. 2, 89-97 (2007).

Authors’ abstract: We study the existence of interfaces between stripe or roll solutions in the Swift-Hohenberg equation. We prove the existence of two different types of interfaces: corner-like interfaces, also referred to as knee solutions, and step-like interfaces. The analysis relies upon a spatial dynamics formulation of the existence problem and an equivariant centre-manifold reduction. In this setting, the interfaces are found as heteroclinic and homoclinic orbits of a reduced system of ODEs.

Reviewer: Nils Ackermann (México)

MSC:

35K55	Nonlinear parabolic equations
35K57	Reaction-diffusion equations
37L10	Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35Q53	KdV equations (Korteweg-de Vries equations)
37G05	Normal forms for dynamical systems
76E06	Convection in hydrodynamic stability
34C37	Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

interfaces; roll solutions; Swift-Hohenberg equation; zigzag instability; equivariant centre manifold reduction; normal form