Journal ArticleDOI
Instability and Stability of Rolls in the Swift–Hohenberg Equation
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In this article, the stability analysis of spatially periodic patterns under general nonperiodic perturbations is studied and a condition on the amplitude and the wave number of the rolls is derived.Abstract:
We develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations. In particular, it enables us to detect sideband instabilities. We treat in all detail the stability question of roll solutions in the two–dimensional Swift–Hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability. Moreover, we characterize the set of those wave vectors which give rise to unstable perturbations.
Dedicated to Professor K. Kirchgassner on the occasion of
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Citations
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Book ChapterDOI
Stability of Travelling Waves
TL;DR: In this paper, an overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given, including the point and essential spectrum of the linearization about a travelling wave, the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies for the linear operator.
Journal ArticleDOI
Localized Hexagon Patterns of the Planar Swift–Hohenberg Equation
TL;DR: It is found that stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift–Hohenberg equation exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound.
Journal ArticleDOI
Snakes, ladders, and isolas of localized patterns
TL;DR: It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls.
Book ChapterDOI
The Ginzburg-Landau Equation in Its Role as a Modulation Equation
TL;DR: The Ginzburg-Landau Equation (GLe) as discussed by the authors is a modulation equation that is a normal form of a bifurcation equation in the context of weakly unstable systems when continuous spectrum moves over the imaginary axis.
Book ChapterDOI
Spatio-Temporal Dynamics of Reaction-Diffusion Patterns
Bernold Fiedler,Arnd Scheel +1 more
TL;DR: In this article, a survey of parabolic partial differential equations from a dynamical systems point of view is presented, where the success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Book
An introduction to partial differential equations
Michael Renardy,Robert C. Rogers +1 more
TL;DR: In this paper, a classification of characteristics and classification of characteristics is presented, along with a discussion of conservation laws and shocks, conservation laws, maximum principles, distribution, and function spaces.
Book
Instabilities and fronts in extended systems
TL;DR: In this article, a systematic account of the mathematics connected with infinite space domains is given, with a coherent description of several problems in which instabilities occur, notably the Eckhaus instability and the formation of fronts in the Swift-Hohenberg equation.
Journal ArticleDOI
Attractors for modulation equations on unbounded domains-existence and comparison
Alexander Mielke,Guido Schneider +1 more
TL;DR: In this paper, the authors studied the long-time behavior of nonlinear parabolic PDEs defined on unbounded cylindrical domains and developed an abstract theorem based on the interaction of a uniform and a localizing norm which allowed them to define global attractors for some dissipative problems.