论文标题
平面界面(在)奇异扰动的2组分反应扩散方程(在)稳定性的标准
Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction-diffusion equations
论文作者
论文摘要
我们考虑了一类奇异的2组分反应 - 扩散方程,这些方程允许双向行进的前溶液,表现为锋利,缓慢的慢速,在稳定的均匀静止状态之间的接口。在许多示例系统中,例如旱地生态系统中的荒漠化阵线模型,这些方面可以表现出不稳定的不稳定性,界面会破坏指义模式。由于这种模式的外观,我们提出了两个版本的2D稳定性标准,该标准沿着这些行进的缓慢快速慢的前面的界面(横向)长波长扰动。前部是使用几何奇异扰动技术构建的,该技术通过在快速问题中通过异质轨道连接两个不同的双曲线慢歧管上的慢速轨道。相关的稳定性标准以系统的非线性和前部的慢速慢结构表示。我们通过在许多示例系统/模型中明确研究旅行前线的存在和横向(在)稳定性中说明并进一步阐述了一般设置。我们通过分析性建立了在几种旱地生态系统模型中侵入裸露的土壤/植被界面,以防止横向长波长的扰动,并在数值上恢复指法植被模式,以抵抗入侵的沙漠化阵线。
We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability by which the interface destabilizes into fingering patterns. Motivated by the appearance of such patterns, we propose two versions of a 2D stability criterion for (transversal) long wavelength perturbations along the interface of these traveling slow-fast-slow fronts. The fronts are constructed using geometric singular perturbation techniques by connecting slow orbits on two distinct normally hyperbolic slow manifolds through a heteroclinic orbit in the fast problem. The associated stability criteria are expressed in terms of the nonlinearities of the system and the slow-fast-slow structure of the fronts. We illustrate and further elaborate the general set-up by explicitly working out the existence and transversal (in)stability of traveling fronts in a number of example systems/models. We analytically establish the instability of invading bare soil/vegetation interfaces against transversal long wavelength perturbations in several dryland ecosystem models and numerically recover fingering vegetation patterns counter-invading an invading desertification front.