The Eckhaus and zigzag instability criteria in gradient/skew-gradient dissipative systems

Kuwamura, Masataka; Yanagida, Eiji

doi:10.1016/s0167-2789(02)00735-2

Cited by 9 publications

(16 citation statements)

References 29 publications

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“…In the case of a shadow reaction-diffusion system with the Neumann boundary condition in a one-dimensional interval, all the stable steady states are constant or monotone [N94,NPY01]. See [JM94,K05,KY03,L96,Y02a,Y02b,Y02c] for other stability and instability results. However, the shape of the stable steady states of a large class of reaction-diffusion systems in high-dimensional domains seems not to be known very much.…”

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confidence: 99%

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Miyamoto

2007

Quart. Appl. Math.

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Abstract. We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domainswhere f and g satisfy the following: g ξ < 0, and there is a function (u, ξ). This class of reaction-diffusion systems includes the FitzHughNagumo system and a special case of the Gierer-Meinhardt system. In the author's previous paper "An instability criterion for activator-inhibitor systems in a two-dimensional ball " (J. Diff. Eq. 229 (2006), 494-508), we obtain a necessary condition about the profile of u on the boundary of the domain for a steady state (u, ξ) to be stable when the domain is a two-dimensional ball. In this paper, we give a necessary condition about the profile of u in the domain when the domain is a two-dimensional ball, annulus or rectangle. Roughly speaking, we show that if (u, ξ) is stable for some τ > 0, then the shape of u is like a boundary one-spike layer even if D u is not small.

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confidence: 99%

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Miyamoto

2007

Quart. Appl. Math.

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“…We give an extension of the Krein-Lyubarskii theorem [16] that treats the behavior of multipliers. Moreover, it is shown that the Eckhaus instability criteria in dissipative systems with gradient/skew-gradient structure introduced in [10,17] are reformulated in the framework of our results.…”

Section: Remark 2 Formula (15) Is Formulated In the Framework Of Hamentioning

confidence: 82%

“…Here we consider a family of spatially periodic patterns in dissipative systems with gradient/skew-gradient structure introduced in [10,17], which covers the Swift-Hohenberg equation and some reaction-diffusion systems of activator-inhibitor type.…”

Section: The Eckhaus Instability In Dissipative Systemsmentioning

confidence: 99%

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Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

Kuwamura

Yanagida

2006

Journal of Differential Equations

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This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift-Hohenberg equation and reaction-diffusion systems of activator-inhibitor type.

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“…In the case of skew-gradient systems in a convex domain, several instability results for inhomogeneous steady states are obtained by [30]. See [12,13,31] for other instability results of skew-gradient systems. However, it seems that the instability criterion of the steady states of a wide class of reaction-diffusion systems, which clarify the relation between the stability and the profile, is not known except the one-dimensional case [19,24].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

An instability criterion for activator–inhibitor systems in a two-dimensional ball

Miyamoto

2006

Journal of Differential Equations

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Let B be a two-dimensional ball with radius R. Let (u(x, y), ξ ) be a nonconstant steady state of the shadow systemwhere f and g satisfy the following: f ξ (u, ξ ) < 0, g ξ (u, ξ ) < 0 and there is a function k(ξ ) such that g u (u, ξ ) = k(ξ )f ξ (u, ξ ). This system includes a special case of the Gierer-Meinhardt system and the FitzHugh-Nagumo system. We show that if Z[U θ (·)] 3, then (u, ξ ) is unstable for all τ > 0, where U(θ) := u(R cos θ, R sin θ) and Z[w(·)] denotes the cardinal number of the zero level set of w(·) ∈ C 0 (R/2π Z). The contrapositive of this result is the following: if (u, ξ ) is stable for some τ > 0, then Z[U θ (·)] = 2. In the proof of these results, we use a strong continuation property of partial differential operators of second order on the boundary of the domain.

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The Eckhaus and zigzag instability criteria in gradient/skew-gradient dissipative systems

Cited by 9 publications

References 29 publications

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

An instability criterion for activator–inhibitor systems in a two-dimensional ball

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