D. Pelinovsky

Traveling monotonic fronts in the discrete Nagumo equation

Journal of Dynamics in Differential Equations 23, 167–183 (2011)

We give an alternative proof of the theorem, which states that no propagation failure occurs for the discrete Nagumo equation with “translationally invariant” stationary monotonic fronts. The theorem was recently proved with the use of the invariant manifolds for lattice differential equations by Hupkes, Pelinovsky, and Sandstede. The alternative proof relies on the analysis of the advance-delay operator associated with the translationally invariant stationary front. This operator exhibits an infinite-dimensional kernel spanned by Fourier harmonics of front’s translations, which are accounted when the stationary front is continued into the traveling one.

lattice differential equations, differential advance-delay operators, propagation failure, discrete kink, singular perturbation theory