Front Interactions in a Three-Component System
Front Interactions in a Three-Component System
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DOI：
10.1137/080744785
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发表时间：
2010-04
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影响因子：
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通讯作者：
P. Heijster;A. Doelman;T. Kaper;K. Promislow
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文献类型：
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作者：
P. Heijster;A. Doelman;T. Kaper;K. Promislow
C.P.Schenk等人，Phys.中介绍的三组分反应-扩散系统。Rev.Lett，78(1997)，pp.3781-3784)已经成为模式形成的范例模型。它表现出丰富多样的锋面、脉冲和光斑的动力学。波前相互作用和脉冲相互作用的类型从弱相互作用到强相互作用，在弱相互作用中，局域结构只通过它们指数级的小尾巴相互作用，在强相互作用中，它们被湮灭或碰撞，并且在局域结构之间的区域中，所有成分都远离平衡。位于这两个极端之间的是半长线相互作用机制，在该机制中，锋面的激活剂成分在相邻锋面之间接近平衡，但两个抑制物成分在那里远离平衡，因此它们的浓度分布驱动着锋面的演化。在这篇文章中，我们主要关注在半长型区域内动态演化的N-阵面解。主要结果是使用重整化群方法严格地推导出控制锋面位置的N个耦合常微分系统。与关于N阵面解的线性化相关的算子具有N个小的特征值，并且N阵面解可以分解为由相关特征函数所跨越的空间中的分量和投影到该空间的补集上的分量。这种分解是在一系列时间迭代执行的。前一种投影产生前沿位置的常量，而后一种投影与我们显示的在重整化群方法的每次迭代中保持在适当范数中的余数相关。我们的结果也有助于将重整化群方法的应用从最初发展的弱相互作用区扩展到半强相互作用区。我们提出的第二组结果是对这一常微分方程组的详细分析，提供了在N=1、2、3、4的情况下可能的前沿相互作用的分类，以及前沿解如何与(A.Doelman，P.van Heijster和T.J.Kaper，J.Dyam)先前研究的稳定脉冲解相互作用。《微分方程式》，21(2009)，第73-115页；P.van Heijster，A.Doelman，和T.J.Kaper，Phys.D.237(2008年)，第3335-3368页)。此外，我们还给出了N阵面相互作用一般情况下的一些结果。
The three-component reaction-diffusion system introduced in (C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781-3784) has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only through their exponentially small tails, to strong interactions, in which they annihilate or collide and in which all components are far from equilibrium in the domains between the localized structures. Intermediate to these two extremes sits the semistrong interaction regime, in which the activator component of the front is near equilibrium in the intervals between adjacent fronts but both inhibitor components are far from equilibrium there, and hence their concentration profiles drive the front evolution. In this paper, we focus on dynamically evolving N -front solutions in the semistrong regime. The primary result is use of a renormalization group method to rigorously derive the system of N coupled ODEs that governs the positions of the fronts. The operators associated with the linearization about the N -front solutions have N small eigenvalues, and the N -front solutions may be decomposed into a component in the space spanned by the associated eigenfunctions and a component projected onto the complement of this space. This decomposition is carried out iteratively at a sequence of times. The former projections yield the ODEs for the front positions, while the latter projections are associated with remainders that we show stay small in a suitable norm during each iteration of the renormalization group method. Our results also help extend the application of the renormalization group method from the weak interaction regime for which it was initially developed to the semistrong interaction regime. The second set of results that we present is a detailed analysis of this system of ODEs, providing a classification of the possible front interactions in the cases of N =1 , 2, 3, 4, as well as how front solutions interact with the stationary pulse solutions studied earlier in (A. Doelman, P. van Heijster, and T. J. Kaper,J. Dynam. Differential Equations, 21 (2009), pp. 73-115; P. van Heijster, A. Doelman, and T. J. Kaper, Phys. D, 237 (2008), pp. 3335-3368). Moreover, we present some results on the general case of N -front interactions.