Bistable pulsating waves with periodic advection: homogenization and sharp speed asymptotics
Abstract
We study bistable pulsating waves for reaction-diffusion equations with general periodic advection in arbitrary space dimension, allowing the diffusion matrix to be nonsymmetric. Assuming that the homogenized equation admits a traveling wave with nonzero speed in a given direction, we construct moving pulsating waves for all sufficiently small spatial periods L and prove their convergence to the homogenized wave as L→0+. The existence range and convergence are uniform in the propagation direction when the homogenized speeds never vanish. We also prove uniqueness of the wave speed for arbitrary periods, profile uniqueness for moving waves, and stationary-wave uniqueness in the standing case under continuity of the competing profile. For spatially homogeneous reactions, we derive the expansion cL=c0+Lc1+O(L2) and an explicit formula for c1. Examples with constant diffusion and zero-mean periodic advection show that c1 can have either sign, demonstrating that the heterogeneity in advection may accelerate or decelerate propagation relative to the homogenized limit and that the general O(L) speed estimate is sharp.
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BibTeX
@misc{ding2026bistable,
title = {{Bistable pulsating waves with periodic advection: homogenization and sharp speed asymptotics}},
author = {Weiwei Ding and Linfeng Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39444},
eprint = {2609.39444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39444},
}
RIS
TY - UNPB
TI - Bistable pulsating waves with periodic advection: homogenization and sharp speed asymptotics
AU - Weiwei Ding
AU - Linfeng Xu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39444
UR - https://arxiv.org/abs/2609.39444
AB - We study bistable pulsating waves for reaction-diffusion equations with general periodic advection in arbitrary space dimension, allowing the diffusion matrix to be nonsymmetric. Assuming that the homogenized equation admits a traveling wave with nonzero speed in a given direction, we construct moving pulsating waves for all sufficiently small spatial periods L and prove their convergence to the homogenized wave as L→0^+. The existence range and convergence are uniform in the propagation direction when the homogenized speeds never vanish. We also prove uniqueness of the wave speed for arbitrary periods, profile uniqueness for moving waves, and stationary-wave uniqueness in the standing case under continuity of the competing profile. For spatially homogeneous reactions, we derive the expansion c_L=c_0+Lc_1+O(L^2) and an explicit formula for c_1. Examples with constant diffusion and zero-mean periodic advection show that c_1 can have either sign, demonstrating that the heterogeneity in advection may accelerate or decelerate propagation relative to the homogenized limit and that the general O(L) speed estimate is sharp.
ER -