TY - JOUR
T1 - Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations
AU - Schouten-Straatman, W.M.
AU - Hupkes, Hermen Jan
AU - Morelli, Leonardo
AU - Van Vleck, Erik
PY - 2020
Y1 - 2020
N2 - We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh–Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.
AB - We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh–Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.
U2 - 10.1007/978-3-030-35502-9_3
DO - 10.1007/978-3-030-35502-9_3
M3 - Article
SP - 55
EP - 112
JO - ICDEA 2018
JF - ICDEA 2018
ER -