Fisher-KPP方程的精确尖波前行波解
Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation
摘要：
速度为$c=\pm 5/\sqrt{6}$的Fisher-KPP方程的一族行波解可用Weierstrass椭圆函数精确表示。$c=5/\sqrt{6}$的公知解在远场中衰减到零，在可以简单地用指数函数表示的意义上是例外的。该解具有相平面轨迹是从鞍点开始到原点结束的异宿轨道的性质。对于$c=-5/\sqrt{6}$，也有一个从鞍点开始的轨迹，但是这个解通常被认为是非物理的，因为它对有限的$z$爆炸了。我们重新解释这个特殊的轨迹作为一个确切的尖锐的正面旅行解决方案的\textit{Fisher-Stefan}型移动边界问题，人口是后退，而不是前进到一个空的空间。通过数值模拟全移动边界问题，我们展示了如何时间依赖的解决方案演变到这个精确的旅行解决方案的大时间。还讨论了这种后退行波的细胞迁移和细胞增殖的数学模型的相关性。
英文摘要：
A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ends at the origin. For $c=-5/\sqrt{6}$, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite $z$. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a \textit{Fisher-Stefan} type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed.
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