Many monostable reaction–diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction–diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction–diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction–diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

中文翻译：

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单稳态反应扩散方程中的最小波速：多项式优化的锐界

许多单稳态反应扩散方程允许一维行波当且仅当波速足够高时。除了一些简单的情况外，这些最小波速的值并不准确。我们提出了寻找最小波速上限和下限的方法。他们依赖于为异宿连接对应于行波的动力系统构建捕获边界。这种方法的简单版本可以通过分析进行，但通常会给出关于最小波速的过于保守的界限。当正在研究的反应扩散方程具有多项式非线性时，我们的方法可以使用多项式优化在计算上实现。对于标量反应-扩散方程，我们提出了一种通用方法，然后将其应用于文献中最小波速未知的示例。然后使用文献中的立方自催化模型说明了我们的方法对多组分反应扩散系统的扩展。在所有三个示例中，并且具有许多不同的参数值，多项式优化计算给出的上限和下限彼此相差在 0.1% 以内，因此几乎是尖锐的。标量反应扩散方程的上限也是通过分析推导出来的。多项式优化计算给出的上限和下限彼此相差在 0.1% 以内，因此几乎是尖锐的。标量反应扩散方程的上限也是通过分析推导出来的。多项式优化计算给出的上限和下限彼此相差在 0.1% 以内，因此几乎是尖锐的。标量反应扩散方程的上限也是通过分析推导出来的。