Geometric desingularisation of the sharp-to-smooth travelling wave transition
Abstract
We study travelling front solutions of a family of degenerate Fisher-KPP equations ut=(un ux)x+u(1−un), where n is a positive integer. At the minimal wave speed cmin=1/(√(1+n)), these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to c=1/(√(1+n))+ε, with ε sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.
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BibTeX
@misc{sattar2026geometric,
title = {{Geometric desingularisation of the sharp-to-smooth travelling wave transition}},
author = {Zak Sattar},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.23489},
eprint = {2608.23489},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23489},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric desingularisation of the sharp-to-smooth travelling wave transition
AU - Zak Sattar
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.23489
UR - https://arxiv.org/abs/2608.23489
AB - We study travelling front solutions of a family of degenerate Fisher-KPP equations u_t=(u^n u_x)_x+u(1−u^n), where n is a positive integer. At the minimal wave speed c_min=1/(√(1+n)), these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to c=1/(√(1+n))+ε, with ε sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.
ER -