A Critical Value for the Viscosity Coefficient: Triggering Oscillatory Traveling Waves in the Pseudo-parabolic Fisher-KPP Equation
Abstract
The pseudo-parabolic term uxxt 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 ut − τuxxt = D uxx + 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 ≤ 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.
Related
- Nonlinear stability of composite waves of traveling wave and rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis
- Nonnegative Viscosity and Weak Solutions to Trudinger's Parabolic Equation
- Null Controllability of a Stochastic Parabolic Equation with a Space-Dependent Analytic Noise Coefficient
- Variable-Coefficient Parabolic Equations and Finite-Horizon Hamilton--Jacobi--Bellman Equations in Augmented Spectral Barron Spaces
- An exact traveling waves solution for a special class of semilinear equations
BibTeX
@misc{cao2026critical,
title = {{A Critical Value for the Viscosity Coefficient: Triggering Oscillatory Traveling Waves in the Pseudo-parabolic Fisher-KPP Equation}},
author = {Yang Cao and Haomeng Chen and Chunhua Jin and Jingxue Yin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34368},
eprint = {2609.34368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34368},
}
RIS
TY - UNPB
TI - A Critical Value for the Viscosity Coefficient: Triggering Oscillatory Traveling Waves in the Pseudo-parabolic Fisher-KPP Equation
AU - Yang Cao
AU - Haomeng Chen
AU - Chunhua Jin
AU - Jingxue Yin
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34368
UR - https://arxiv.org/abs/2609.34368
AB - 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 ≤ 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.
ER -