A family of travelling wave solutions to the Fisher–KPP equation with speeds $c=\pm 5∕\sqrt{6}$ can be expressed exactly using Weierstra$ß$ 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 ending 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 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.

Fisher-KPP方程带速度的行波解 $C=\pm 5∕\sqrt{6}$ 可以用Weierstra精确表达$ß$椭圆函数。著名的解决方案$C=5∕\sqrt{6}$在远场中衰减到零的情况是例外的，因为它可以简单地用指数函数表示。该解决方案具有以下特性：相平面轨迹是从鞍点开始并在原点处结束的异质轨道。对于$C=-5∕\sqrt{6}$，也有一条从鞍点开始的轨迹，但是通常不理会这种解决方案，因为它在有限的范围内爆炸，因此是非物理的 $ž$。我们将这种特殊的轨迹重新解释为Fisher-Stefan型移动边界问题的精确的前锋旅行解决方案，该问题是人口从一个空旷的空间撤退而不是前进。通过数值模拟完整的运动边界问题，我们证明了时间依赖型解决方案如何在较长时间内演变为这种精确的旅行解决方案。还讨论了此类后退行波与细胞迁移和细胞增殖数学模型的相关性。