A geometric classification of traveling front propagation in the Nagumo equation with cut-off

Abstract

An important category of solutions to reaction-diffusion systems of partial differential equations is given by traveling fronts, which provide a monotonic connection between rest states and maintain a fixed profile when considered in a co-moving frame. Reactiondiffusion equations are frequently employed in the mean-field (continuum) approximation of discrete (many-particle) models; however, the quality of this approximation deteriorates when the number of particles is not sufficiently large. The (stochastic) effects of this discreteness have been modeled via the introduction of (deterministic) 'cut-offs' that effectively deactivate the reaction terms at points where the particle concentration is below a certain threshold. In this article, we present an overview of the effects of such a cutoff on the front propagation dynamics in a prototypical reaction-diffusion system, the classical Nagumo equation. Our analysis is based on the method of geometric desingularization ('blow-up'), in combination with dynamical systems techniques such as invariant manifolds and normal forms. Using these techniques, we categorize front propagation in the cutoff Nagumo equation in dependence of a control parameter, and we classify the corresponding propagation regimes ('pulled,' 'pushed,' and 'bistable') in terms of the bifurcation structure of a projectivized system of equations that is obtained from the original traveling front problem, after blow-up. In particular, our approach allows us to determine rigorously the asymptotics (in the cutoff parameter) of the correction to the front propagation speed in the Nagumo equation that is due to a cutoff. Moreover, it explains the structure of that asymptotics (logarithmic, superlinear, or sublinear) in dependence of the front propagation regime. Finally, it enables us to calculate the corresponding leadingorder coefficients in the resulting expansions in closed form.

Singularly Perturbed Oscillators with Exponential Nonlinearities

“…The former approach is typically preferred in analytical studies such as ours, while the latter tends to be preferable for applications and simulations. Analytically, we expect that the ZFK equation (1) with (3) and a cutoff can be treated using a combination of our approach with the geometric methods developed for problems with cutoffs in [11,33,35,36,37,38].…”

“…This distinguishes it from the problems considered in e.g. [11,33,35,36,37,38], where travelling waves that are identified using GSPT and geometric blow-up feature a non-smooth cutoff which renders the equations non-smooth for ϵ > 0.…”

Warning signs for wave speed transitions of noisy Fisher–KPP invasion fronts

“…where we have the choice to make the scheme fullyimplicit with = U k+1 or semi-implicit with = U k . Since the deterministic drift term F causes the stability problems if D(∆t) > (∆x) 2 [24, p. 64,67] it makes sense to chose the semi-implicit version. The algebraic problem to solve for U k+1 in ( 17) can be solved using standard techniques such as Newton's method.…”