线性耦合系统的WKBJ逼近：反应扩散系统的渐近性
WKBJ approximation for linearly coupled systems: asymptotics of reaction-diffusion systems
摘要：
渐近分析已成为研究反应扩散方程和斑图形成的常用方法，特别是当考虑对原始模型的推广时，例如空间异质性，其中即使对线性化方程也通常不可能找到解析解。WKBJ方法，一个更强大的渐近方法研究耗散现象捕获的线性方程组，最近被应用到图灵模型在一个异质环境。它证明了预期的修改，在一个均匀的设置，如本地模式和本地图灵条件获得的结果。在这种情况下，我们试图推广的标量WKBJ理论的多组分系统。我们更广泛的数学方法的结果在一般的逼近定理系统的常微分方程。我们讨论的情况下，指数和振荡行为的第一次处理之前的一般情况。随后，我们证明了一个典型的图灵系统的近似定理中使用的谱特性，从而表明这样的近似是合理的。请注意，我们的方法是通过显示解决方案接近（使用合适的权重函数来测量误差）Airy函数的线性组合。
英文摘要：
Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations to the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The WKBJ method, one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar WKBJ theory to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence suggesting that such an approximation is reasonable. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy functions.
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