Effects of boundary conditions on spatially periodic states
type: Item
instance of: scholarly article
title: Effects of boundary conditions on spatially periodic states
zbMATH Open document ID: 0599.76099
DOI: 10.1016/0167-2789(84)90136-2
author: Lorenz Kramer
author: Pierre C. Hohenberg
published in: Physica D
publication date: 1984-01-01T00:00:00Z
review text: We investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension. For all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary (type-I solution). A linear stability analysis of the uniform state at threshold reveals that type-I solutions are often unstable. Then there exists in the full-Eckhaus-stable band, a static solution where the amplitude rises above its bulk value near the boundary (''type-II'' solution), or a limit-cycle solution where the amplitude near the boundary oscillates. These solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold.
Mathematics Subject Classification ID: 76Rxx
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 3968196
zbMATH Keywords: linear homogeneous boundary conditions
zbMATH Keywords: stationary solutions
zbMATH Keywords: cellular pattern formation in one dimension
zbMATH Keywords: reduced wavenumber band
zbMATH Keywords: static solution
zbMATH Keywords: type-I solution
zbMATH Keywords: linear stability analysis
zbMATH Keywords: uniform state
zbMATH Keywords: full-Eckhaus-stable band
zbMATH Keywords: type-II'' solution
zbMATH Keywords: limit-cycle solution
MaRDI profile type: Publication
full work available at URL: 0167-2789(84)90136-2
OpenAlex ID: W2072797269
cites work: Nonlinear waves: Dynamics and transport
cites work: The oscillatory instability of convection rolls in a low Prandtl number fluid
cites work: Towards a catastrophe theory for the mechanics of plasticity and fracture
cites work: Finite bandwidth, finite amplitude convection
cites work: Distant side-walls cause slow amplitude modulation of cellular convection
cites work: The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection
cites work: The Post-Buckling Problem for Thin Elastic Shells
cites work: Discrete numerical methods in physics and engineering
description: scientific article; zbMATH DE number 3968196