The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime
The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime
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DOI：
10.1016/j.physd.2005.02.009
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发表时间：
2005-03-15
影响因子：
4
通讯作者：
Wei, JC
中科院分区：
文献类型：
--
作者：
Kolokolnikov, T;Ward, MJ;Wei, JC
在半强尖峰相互作用下，研究了有限区域上一维Gray-Scott模型中尖峰模式的存在性、稳定性和脉冲分裂行为。这种制度的特点是本地化的反应的组件之一，在某些尖峰位置附近，而另一个组件表现出更全球的空间变化在整个域。匹配渐近展开的方法，然后被用来构建k-穗平衡的一个特定的核心问题。这个核心问题进行了研究数值和渐近。对于每一个整数k >= 1，它表明，有两个分支的k-尖峰平衡，满足在一个鞍结分歧值。对于第二组分的扩散系数D的小值，这些鞍结分叉点发生在近似相同的值处。渐近和数值方法相结合，用于分析这些分支的k-尖峰平衡的稳定性与小特征值和振荡不稳定性的尖峰配置文件相关的漂移不稳定性。这样，Ei等[S. Ei，Y Nishiura，K.上田，2（n）分裂还是边分裂？耗散系统分裂的一种方式，Jpn。J.Ind.Appl.Math.18（2001）181-205]中被认为对反应扩散系统中的脉冲分裂行为是必不可少的。通过验证这些条件，一个简单的分析标准的脉冲分裂的发生，然后制定和确认与完整的Gray-Scott模型的数值模拟。这一准则验证了[A]中报道的基于数值和拓扑参数的猜想。Doelman，R.A. Gardner，T. J. Kaper，ID Gray-Scott模型中奇异模式的稳定性分析：匹配渐近方法，Physica D 122（1-4）（1998）1-36]。分析结果与以前获得的脉冲分裂行为的结果进行了比较。(c)2005 Elsevier B. V.保留所有权利。
The existence, stability, and pulse-splitting behavior of spike patterns in the one-dimensional Gray-Scott model on a finite domain is analyzed in the semi-strong spike-interaction regime. This regime is characterized by a localization of one of the components of the reaction near certain spike locations, while the other component exhibits a more global spatial variation across the domain. The method of matched asymptotic expansions is then used to construct k-spike equilibria in terms of a certain core problem. This core problem is studied numerically and asymptotically. For each integer k >= 1, it is shown that there are two branches of k-spike equilibria that meet at a saddle-node bifurcation value. For small values of the diffusivity D of the second component, these saddle-node bifurcation points occur at approximately the same value. A combination of asymptotic and numerical methods is used to analyze the stability of these branches of k-spike equilibria with respect to both drift instabilities associated with the small eigenvalues and oscillatory instabilities of the spike profile. In this way, the key bifurcation and spectral conditions of Ei et al. [S. Ei, Y Nishiura, K. Ueda, 2(n) splitting or edge splitting? A manner of splitting in dissipative systems, Jpn. J. Ind. Appl. Math. 18 (2001) 181-205] believed to be essential for pulse-splitting behavior in a reaction-diffusion system are verified. By having verified these conditions, a simple analytical criterion for the occurrence of pulse-splitting is then formulated and confirmed with full numerical simulations of the Gray-Scott model. This criterion verifies a conjecture based on numerics and topological arguments reported in [A. Doelman, R.A. Gardner, T.J. Kaper, Stability analysis of singular patterns in the ID Gray-Scott model: a matched asymptotics approach, Physica D 122 (1-4) (1998) 1-36]. The analytical results are compared with previously obtained results for pulse-splitting behavior. (c) 2005 Elsevier B.V. All rights reserved.