We analyse shock wave behaviour in a hyperbolic diffusion system with a general forcing term which is qualitatively not dissimilar to a logistic growth term. The amplitude behaviour is interesting and depends critically on a parameter in the forcing term. We also develop a fully nonlinear acceleration wave analysis for a hyperbolic theory of diffusion coupled to temperature evolution. We consider a rigid body and we show that for three-dimensional waves there is a fast wave and a slow wave. The amplitude equation is derived exactly for a one-dimensional (plane) wave and the amplitude is found for a wave moving into a region of constant temperature and solute concentration. This analysis is generalized to allow for forcing terms of Selkov–Schnakenberg, or Al Ghoul-Eu cubic reaction type. We briefly consider a nonlinear acceleration wave in a heat conduction theory with two solutes present, resulting in a model with equations for temperature and each of two solute concentrations. Here it is shown that three waves may propagate.

中文翻译：

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温度和扩散模型中的不连续波

我们使用一般强迫项来分析双曲扩散系统中的冲击波行为，该强迫项在质量上与逻辑增长项没有什么不同。振幅行为很有趣，并且关键取决于强迫项中的参数。我们还为与温度演化耦合的扩散双曲理论开发了完全非线性加速波分析。我们考虑刚体，并证明三维波有快波和慢波。振幅方程是针对一维（平面）波精确导出的，并且针对移动到恒定温度和溶质浓度区域的波找到振幅。该分析被推广为允许 Selkov-Schnakenberg 或 Al Ghoul-Eu 立方反应类型的强制项。我们简要考虑存在两种溶质的热传导理论中的非线性加速波，从而产生一个包含温度方程和两种溶质浓度的模型。这里表明三个波可以传播。