论文标题
单位可反应扩散方程中的最小波速：通过多项式优化的急剧界限
Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization
论文作者
论文摘要
当且仅当波速足够高时，许多单一的反应扩散方程就可以接收一维行进波。除了一些简单的情况外，这些最小波速的值尚不确切。我们提出了在最小波速下查找上限和下限的方法。他们依靠为杂斜连接对应于行进波的动力系统构建捕获边界。该方法的简单版本可以通过分析进行，但通常会在最小波速上产生过度保守的界限。当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中实现。对于标量反应扩散方程，我们提出了一种通用方法，然后将其应用于最小波速未知的文献中的示例。然后，使用文献中的立方自催化模型来说明我们对多组分反应扩散系统的扩展。在所有三个示例和许多不同的参数值中，多项式优化计算可提供彼此0.1％以内的上限和下限。对于标量RD方程式，在分析上得出上限。
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 RD equations.
