Propagation reversal for bistable differential equations on trees

TL;DR

This paper investigates traveling wave solutions in bistable differential equations on infinite k-ary trees, revealing how changes in diffusion parameters can reverse wave propagation direction.

Contribution

It provides a novel analysis of wave propagation and reversal phenomena on tree structures, extending classical lattice results to more complex graph topologies.

Findings

Wave solutions are pinned at low diffusion levels.

Increasing diffusion causes waves to travel with non-zero speed.

Strong diffusion leads to propagation direction reversal.

Abstract

We study traveling wave solutions to bistable differential equations on infinite $k$-ary trees. These graphs generalize the notion of classical square infinite lattices and our results complement those for bistable lattice equations on $\mathbb{Z}$. Using comparison principles and explicit lower and upper solutions, we show that wave-solutions are pinned for small diffusion parameters. Upon increasing the diffusion, the wave starts to travel with non-zero speed, in a direction that depends on the detuning parameter. However, once the diffusion is sufficiently strong, the wave propagates in a single direction up the tree irrespective of the detuning parameter. In particular, our results imply that changes to the diffusion parameter can lead to a reversal of the propagation direction.