Folds, canards and shocks in advection–reaction–diffusion models
Folds, canards and shocks in advection–reaction–diffusion models
复制标题
平流-反应-扩散模型中的褶皱、鸭翼和激波
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
G. Pettet
中科院分区：
文献类型：
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作者：
M. Wechselberger;G. Pettet
由平流-反应-扩散（ARD）耦合方程模拟的战术驱动的细胞运动通常表现为平滑的行波，波形中较少出现尖锐的界面。利用几何奇异摄动技术研究了耦合ARD模型中具有光滑和尖锐界面的行波的存在性。特别地，我们证明了在适当的lisamadard变换下的行波分析揭示了在波形中观察类激波界面的一般折叠条件。这种几何方法进一步解释了双曲偏微分方程理论中众所周知的跳跃和熵条件（Rankine-Hugoniot和Lax条件）。我们的分析还表明，鸭形解是奇异摄动问题中的一类特殊解，在构造具有光滑和尖锐界面的行波中起着重要的作用。
Tactically driven cell movement modelled by coupled advection–reaction–diffusion (ARD) equations typically exhibit smooth travelling waves, and less frequently sharp interfaces in the wave form. We study the existence of travelling waves with smooth and sharp interfaces in coupled ARD models by using geometric singular perturbation techniques. In particular, we show that a travelling wave analysis under an appropriate Liénard transformation reveals a generic fold condition to observe shock-like interfaces in the wave form. This geometric approach further explains automatically well-known jump and entropy conditions for shocks in hyperbolic PDE theory (Rankine–Hugoniot and Lax conditions). Our analysis also shows that canards, a special class of solutions within singular perturbation problems, play an important role in the construction of travelling waves with smooth and sharp interfaces.