Stabilizing a homoclinic stripe
Stabilizing a homoclinic stripe
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稳定同宿条纹
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Juncheng Wei
中科院分区：
文献类型：
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作者：
T. Kolokolnikov;M. Ward;J. Tzou;Juncheng Wei
对于一大类具有大扩散率的反应扩散系统，众所周知，二维条纹（其横截面是一维同宿尖峰）是不稳定的并且会分裂成点。在这里，我们研究了两种可以稳定这种同宿条纹的效应。首先，我们考虑向模型添加各向异性。对于施纳肯伯格模型，我们表明，如果快速扩散变量（基底）具有足够的各向异性，则（无限）条纹可以稳定。得出两种类型的不稳定阈值：之字形（或弯曲）和破碎不稳定性。不稳定边界将参数空间细分为三个不同的区域：稳定条带、由于弯曲而导致的不稳定条带以及由于破裂不稳定性而导致的不稳定。数值实验表明，破裂不稳定性是超临界的，导致出现“斑点条纹”解决方案。最后，我们对陡峭山坡植被模式的 Klausmeier 模型进行了类似的分析，并检查从斑点到条纹的过渡。本文是主题“失衡物质的耗散结构：来自化学、光子学和生物学（第 2 部分）”的一部分。
For a large class of reaction–diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a ‘spotted-stripe’ solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.