A Critical Value for the Viscosity Coefficient: Triggering Oscillatory Traveling Waves in the Pseudo-parabolic Fisher-KPP Equation

Abstract

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 $$u_t-\tau u_{xxt}=D{u}_{xx}+u(1-u).$$ We completely characterize how the parameter ratio $\tau /D$ acts as a critical switch: it preserves the monotonic structure of classical traveling waves when $\tau /D\le 1$, yet it triggers a qualitative shift to oscillatory traveling waves when $\tau /D>1$. Numerical simulations support these findings. The threshold $\tau /D=1$ 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.

arXiv comment41 pages, 3 figures. Corresponding author: Chunhua Jin ([email protected])