发表机构
Dalian University of Technology; South China Normal University（大连理工大学; 华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究伪抛物型Fisher-KPP方程，发现参数比τ/D=1为临界值：当τ/D>1时，行波由单调变为振荡，解释了饱和度超调现象，揭示了伪抛物项的双重作用。
AI 中文摘要
伪抛物项 $u_{xxt}$ 是高阶黏性的典型例子，它提供有效的正则化，同时捕捉精细的物理机制。它如何改变原型方程的动力学仍然是一个基本问题，尚未完全理解。本文的目标是通过研究伪抛物型Fisher-KPP方程 $$ u_t - \tau u_{xxt} = D u_{xx} + u(1-u) $$ 来证明这一项引起行波结构的急剧转变。我们完全刻画了参数比 $\tau/D$ 如何作为一个临界开关：当 $\tau/D \leq 1$ 时，它保持经典行波的单调结构；而当 $\tau/D > 1$ 时，它引发向振荡行波的定性转变。数值模拟支持这些发现。因此，阈值 $\tau/D=1$ 捕捉了伪抛物项的双重作用：当它小时，它起到保持结构的黏性作用；但当它大时，它反映了占主导地位的毛细管滞后，产生振荡，从而解释了饱和度超调现象，这一现象与经典扩散模型相矛盾，但在两相流中被广泛观察到。
英文摘要
The pseudo-parabolic term $u_{xxt}$ serves as a canonical example of higher-order viscosity that provides effective regularization while capturing the refined physical mechanism. How it alters the dynamics of prototype equations remains a fundamental issue, which is not yet fully understood. The goal of this paper is to show that this term induces a sharp transition in traveling wave structure, via studying the pseudo-parabolic Fisher-KPP equation $$ u_t - τu_{xxt} = D u_{xx} + u(1-u). $$ We completely characterize how the parameter ratio $τ/D$ acts as a critical switch: it preserves the monotonic structure of classical traveling waves when $τ/D \leq 1$, yet it triggers a qualitative shift to oscillatory traveling waves when $τ/D > 1$. Numerical simulations support these findings. The threshold $τ/D=1$ thus captures the dual role of the pseudo-parabolic term: it acts as a structure-preserving viscosity when small, but when large it reflects dominant capillary hysteresis, generating the oscillations that explain the saturation overshoot, a phenomenon that contradicts classical diffusion models yet is widely observed in two-phase flow.
Comments41 pages, 3 figures. Corresponding author: Chunhua Jin (jinchhua@126.com)
