Defects in Oscillatory Media: Toward a Classification

Sandstede, Björn; Scheel, Arnd

doi:10.1137/030600192

Cited by 115 publications

(186 citation statements)

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“…Our analysis relates this wavenumber selection mechanism to a geometric transversality criterion for a heteroclinic solution. A similar connection has been noticed in [23,18]: in oscillatory media, localized structures that emit wave trains typically select wavenumbers in the far-field. In fact, such sources of wave trains can be viewed as heteroclinic orbits in a spatial dynamics description, and transversality of such heteroclinic orbits implies wavenumber selection in the far field.…”

Section: Discussionmentioning

confidence: 80%

Grain boundaries in the Swift–Hohenberg equation

Hărăguş¹,

Scheel²

2012

Eur. J. Appl. Math

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We study the existence of grain boundaries in the Swift-Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ODEs in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

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Section: Discussionmentioning

confidence: 80%

Grain boundaries in the Swift–Hohenberg equation

Hărăguş¹,

Scheel²

2012

Eur. J. Appl. Math

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“…Extension to the Robin boundary condition, which is more complicated mathematically, has been studied by Sherratt (submitted). A general classification of periodic travelling waves generated by boundary conditions is possible based on their group velocity far from the boundary; see Sandstede & Scheel (2004) for details of this, and Kollár & Scheel (2007, §1.3) for a brief summary. Figure 3b illustrates the generation of a periodic travelling wave by Dirichlet boundary conditions for the predator-prey model (3.2a) and (3.2b).…”

Section: Periodic Travelling Wave Generation By Boundaries With Hostimentioning

confidence: 99%

Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

Sherratt

Smith

2008

J. R. Soc. Interface.

158

123

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Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reactiondiffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reactiondiffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

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“…Without loss of generality we assume that µ = 1, which can be achieved by rescaling the above equation. It is shown, for instance in [BN85, PSAK95, Doe96, KR00, Leg01,SS04a], that the qCGL equation exhibits a family of defect solutions known as sources (see equation (1.2)). We are interested here in establishing nonlinear stability of these solutions, under suitable spectral stability assumptions.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

et al. 2014

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In an appropriate moving coordinate frame, source defects are time-periodic solutions to reactiondiffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for the Green's function, which allow one to close a nonlinear iteration scheme.

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Defects in Oscillatory Media: Toward a Classification

Cited by 115 publications

References 50 publications

Grain boundaries in the Swift–Hohenberg equation

Grain boundaries in the Swift–Hohenberg equation

Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

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