Pushed and pulled fronts in a discrete reaction-diffusion equation

King, John R.; O'Dea, Reuben D.

Abstract

We consider the propagation of wave fronts connecting unstable and stable uniform solutions to a discrete reaction-diffusion equation on a one-dimensional integer lattice. The dependence of the wavespeed on the coupling strength µ between lattice points and on a detuning parameter (α) appearing in a nonlinear forcing is investigated thoroughly. Via asymptotic and numerical studies, the speed both of 'pulled' fronts (whereby the wavespeed can be characterised by the linear behaviour at the leading edge of the wave) and of 'pushed' fronts (for which the nonlinear dynamics of the entire front determine the wavespeed) is investigated in detail. The asymptotic and numerical techniques employed complement each other in highlighting the transition between pushed and pulled fronts under variations of µ and α.

Citation

King, J. R., & O'Dea, R. D. (2015). Pushed and pulled fronts in a discrete reaction-diffusion equation. Journal of Engineering Mathematics, https://doi.org/10.1007/s10665-015-9829-3

• Journal Article Type: Article
• Publication Date: Nov 7, 2015
• Deposit Date: Nov 10, 2015
• Publicly Available Date: Nov 10, 2015
• Journal: Journal of Engineering Mathematics
• Print ISSN: 0022-0833
• Electronic ISSN: 1573-2703
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.1007/s10665-015-9829-3
• Keywords: Discrete Reaction-Diffusion Equation, Liouville-Green, Matched-Asymptotic Analysis, Travelling Waves
• Public URL: https://nottingham-repository.worktribe.com/output/767313
• Publisher URL: http://link.springer.com/article/10.1007/s10665-015-9829-3
• Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s10665-015-9829-3