We examine travelling wave solutions of the reaction–diffusion equation, , with a Stefan-like condition at the edge of the moving front. With only a few assumptions on R(u) and D(u), a variety of new semi-infinite travelling waves arise in this reaction–diffusion Stefan model. While other reaction–diffusion models can admit semi-infinite travelling waves for a unique wavespeed, we show that semi-infinite travelling waves in the reaction–diffusion Stefan model exist over a range of wavespeeds. Furthermore, we determine the necessary conditions on R(u) and D(u) for which semi-infinite travelling waves exist for all wavespeeds. Using asymptotic analysis in various distinguished limits of the wavespeed, we obtain approximate solutions of these travelling waves, agreeing with numerical simulations with high accuracy.

我们研究了反应扩散方程的行波解，在移动前沿的边缘具有类 Stefan 条件。只需对R ( u ) 和D ( u )做一些假设，在这个反应-扩散 Stefan 模型中就会出现各种新的半无限行波。虽然其他反应 - 扩散模型可以接受具有独特波速的半无限行波，但我们表明反应 - 扩散 Stefan 模型中的半无限行波存在于一定的波速范围内。此外，我们确定了R ( u ) 和D ( u) 对于所有波速都存在半无限行波。在波速的各种不同极限下使用渐近分析，我们获得了这些行波的近似解，与高精度的数值模拟一致。