Coherent and incoherent structures in systems described by the 1D CGLE: experiments and identification

Hecke, Martin van

doi:10.1016/s0167-2789(02)00687-5

Cited by 38 publications

(35 citation statements)

References 48 publications

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“…This argument therefore establishes a beautiful connection between the intuition given by the group velocity and rigorous counts of the codimension of defects. Equation (1.16) has been studied thoroughly in the literature (see, for instance, [41,2,10,21,22] for references). In particular, the CGL has been shown to admit sources (the so-called Nozaki-Bekki holes), sinks, and transmission defects (commonly referred to as homoclons [21]).…”

Section: Examplesmentioning

confidence: 99%

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Defects in Oscillatory Media: Toward a Classification

Sandstede

Scheel

2004

SIAM J. Appl. Dyn. Syst.

115

180

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Abstract. We investigate, in a systematic fashion, coherent structures, or defects, which serve as interfaces between wave trains with possibly different wavenumbers in reaction-diffusion systems. We propose a classification of defects into four different defect classes which have all been observed experimentally. The characteristic distinguishing these classes is the sign of the group velocities of the wave trains to either side of the defect, measured relative to the speed of the defect. Using a spatial-dynamics description in which defects correspond to homoclinic and heteroclinic connections of an ill-posed pseudoelliptic equation, we then relate robustness properties of defects to their spectral stability properties. Last, we illustrate that all four types of defects occur in the one-dimensional cubicquintic Ginzburg-Landau equation as a perturbation of the phase-slip vortex.

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Section: Examplesmentioning

confidence: 99%

“…Interfaces between wave trains with almost identical wavenumbers were, for instance, studied in great detail in [26]. Defects in the cubic-quintic and in coupled complex Ginzburg-Landau equations were investigated by van Saarloos and coworkers [41,23,22] and by Doelman [10].…”

Section: Introductionmentioning

confidence: 99%

Defects in Oscillatory Media: Toward a Classification

Sandstede

Scheel

2004

SIAM J. Appl. Dyn. Syst.

115

180

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“…We continued the system (4) in the kinetic bifurcation parameter λ 0 as a boundary value problem with periodic boundary conditions subject to the constraint (16). In tandem with this, we also continue the linear spreading speed equations as outlined in Appendix A for the speeds c sec and c inv that correspond to Cases I (separated) and II (attached) of the criterion respectively.…”

Section: Equal Diffusion Coefficientsmentioning

confidence: 99%

“…In section 3 we consider coherent structures in the complex Ginzburg-Landau (CGL) equation [14][15][16][17][18], of which the λ-ω system is a special case. The unstable state in this case is the origin, which corresponds to the coexistence state in predator-prey systems, and coherent structures represent travelling fronts that connect the steady state to wave trains.…”

Section: Introductionmentioning

confidence: 99%

Wave train selection behind invasion fronts in reaction–diffusion predator–prey models

Merchant

Nagata

2010

Physica D: Nonlinear Phenomena

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Wave trains, or periodic travelling waves, can evolve behind invasion fronts in oscillatory reaction-diffusion models for predator-prey systems. Although there is a one-parameter family of possible wave train solutions, in a particular predator invasion a single member of this family is selected. Sherratt (1998) has predicted this wave train selection, using a λ-ω system that is a valid approximation near a supercritical Hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal. Away from a Hopf bifurcation or if the diffusion coefficients differ somewhat, these predictions lose accuracy. We develop a more general wave train selection prediction for a two-component reaction-diffusion predator-prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front. This prediction retains accuracy farther away from a Hopf bifurcation, and can also be applied when predator and prey diffusion coefficients are unequal. We illustrate the selection prediction with its application to three models of predator invasions.

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“…These bands are separated by localized defects known as sources and sinks, with the asymptotic wavetrains propagating away from sources and towards sinks. 1 Sources and sinks have been studied extensively in the complex Ginzburg-Landau equation; for reviews, see [56,1,55]. For a discussion in the context of general oscillatory reaction-diffusion systems, see [37].…”

Section: Source-sink Solutionsmentioning

confidence: 99%

Absolute Stability of Wavetrains Can Explain Spatiotemporal Dynamics in Reaction-Diffusion Systems of Lambda-Omega Type

Smith¹,

Rademacher²,

Sherratt³

2009

SIAM J. Appl. Dyn. Syst.

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Abstract. The lambda-omega class of reaction-diffusion equations has received considerable attention because they are more amenable to mathematical investigation than other oscillatory reaction-diffusion systems and include the normal form of any reaction-diffusion system with scalar diffusion close to a standard supercritical Hopf bifurcation. Despite this, detailed studies of the dynamics predicted by numerical simulations have mostly been restricted to regions of parameter space in which stable wavetrains (periodic traveling waves) are selected by the initial or boundary conditions; we use the term "stability" to denote spectral stability on the real line. Here we consider the emergent spatiotemporal dynamics on large bounded domains, with Dirichlet conditions at one boundary and Neumann conditions at the other. Previous studies have established a parameter threshold below which stable wavetrains are generated by the Dirichlet boundary condition. We use numerical continuation techniques to analyze the spectral stability of wavetrain solutions, and we identify a second stability threshold, above which the selected wavetrain is absolutely unstable. In addition, we prove that the onset of absolute stability always occurs through a complex conjugate pair of branch points in the absolute spectrum, which greatly simplifies the detection of this threshold. In the parameter region in which the spectra of the selected waves indicate instability but absolute stability, our numerical simulations predict so-called "source-sink" dynamics: bands of visibly regular periodic traveling waves that are separated by localized defects. Beyond the absolute stability threshold our simulations predict irregular spatiotemporal behavior.

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Coherent and incoherent structures in systems described by the 1D CGLE: experiments and identification

Cited by 38 publications

References 48 publications

Defects in Oscillatory Media: Toward a Classification

Defects in Oscillatory Media: Toward a Classification

Wave train selection behind invasion fronts in reaction–diffusion predator–prey models

Absolute Stability of Wavetrains Can Explain Spatiotemporal Dynamics in Reaction-Diffusion Systems of Lambda-Omega Type

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