Scattering and separators in dissipative systems

Nishiura, Yasumasa; Teramoto, Takashi; Ueda, Kazuo

doi:10.1103/physreve.67.056210

Cited by 71 publications

(75 citation statements)

References 21 publications

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“…For the Gray-Scott system we refer to [54] (numerical simulation), [41], [42] (asymptotic expansion), [8], [9], [10] (rigorous proofs for one dimension), [69], [70], [76], [77] (rigorous study of multi-spots for higher dimensions), [7], [45], [46], [48], [49], [25], [26], [62] (rigorous study of instability mechanisms of multi-spots).…”

Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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Abstract. In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reactiondiffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activatorsubstrate systems (such as the Gray-Scott system or the Schnakenberg model).The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature.We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator.Existence of multi-spots is proved using tools from nonlinear functional analysis such as LiapunovSchmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis.Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots.Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis.The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.

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Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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“…A related type of Hopf bifurcation, followed by a monotonic drift instability as a reaction-time constant is increased, has been analyzed in [15] and [27] for hyperbolic tangent-type interfaces associated with a two-component reaction-diffusion system with bistable nonlinearities. Alternatively, for a three-component reaction-diffusion system it was shown numerically in [35] that the Hopf bifurcation occurs after the onset of a monotonic drift instability as a reaction-time parameter is increased.…”

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

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“…However, as will be clear later on, the dynamics of the 2-front solutions (and more general N -front solutions) are much richer in the 3-component model, see also [5,21]. Moreover, to analyze scattering and other more complex phenomena observed in [17], the analysis developed here for the full 3-component model will be required. Also, as is clear from the above discussion, this model has been extensively studied via numerical simulations.…”

Section: Introductionmentioning

confidence: 99%

Pinned fronts in heterogeneous media of jump type

et al. 2010

Self Cite

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In this paper, we analyze the impact of a (small) heterogeneity of jump type on the most simple localized solutions of a three-component FitzHugh-Nagumo-type system. We show that the heterogeneity can pin a 1-front solution, which travels with constant (non-zero) speed in the homogeneous setting, to a fixed, explicitly determined, distance from the heterogeneity. Moreover, we establish the stability of this heterogeneous pinned 1-front solution. In addition, we analyze the pinning of 1-pulse, or 2-front, solutions. The paper is concluded with simulations in which we consider the dynamics and interactions of N -front patterns in domains with M heterogeneities of jump type (N = 3, 4, M ≥ 1).

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Scattering and separators in dissipative systems

Cited by 71 publications

References 21 publications

Stationary multiple spots for reaction–diffusion systems

Stationary multiple spots for reaction–diffusion systems

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Pinned fronts in heterogeneous media of jump type

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