AI 中文总结
研究具有部分耦合扩散的反应扩散系统的行波解，在第一个分量不扩散时进行理论分类，两个分量都扩散时进行数值研究，应用于捕食者 - 猎物和流行病学等模型。
AI 中文摘要
本工作致力于研究反应扩散系统的行波解，其中第二个分量\(v\)的扩散率取决于第一个分量\(u\)。此类系统出现在捕食者 - 猎物模型（捕食者\(v\)积极追捕猎物\(u\)）或流行病学模型（如狂犬病，疾病导致行为不稳定）中，这导致\(v\)方程的最高阶项出现拟线性耦合。理论上，当第一个分量不扩散时，我们对行波解进行了完全分类，此时问题可简化为非线性标量方程。对于两个分量都扩散的情况进行了数值研究。
英文摘要
This work is devoted to the study of traveling wave solutions of reaction-diffusion systems, where the diffusion rate of~$v$, the second component, depends on~$u$, the first component. Such systems arise in prey-predator models, where the predator~$v$ is actively hunting its prey~$u$, or in epidemiological models where the disease induces erratic behavior, for example rabies. This results in a quasilinear coupling in the highest-order term of the equation for~$v$. From the theoretical point of view, we fully classify traveling wave solutions when the first component does not diffuse, in which case the problem can be reduced to a nonlinear scalar equation. The case where both components diffuse is investigated numerically.
