Travelling fronts, pulses, and pulse trains in a 1D discrete reaction–diffusion system

Following an earlier work (briefly reviewed below) we investigate the temporal stability of an exact travelling front solution, constructed in the form of an integral expression, for a one-dimensional discrete Nagumo-like model without recovery. Since the model is a piecewise linear one with an on-site reaction function involving a Heaviside step function, a straightforward linearisation around the front solution presents problems, and we follow an alternative approach in estimating a ‘stability multiplier’ by looking at the variational problem as a succession of linear evolution of the perturbations, punctuated with ‘kicks’ of small but finite duration. Stability depends crucially on perturbations located at specific sites relative to the moving front (the ‘significant perturbations’, see below). Comparison is made with results of numerical integration of the reaction–diffusion system, whereby it appears likely that the travelling front is temporally stable for all relevant parameter values characterising the model. We modify the system by introducing a slow variation of a relevant recovery parameter and perform a leading order singular perturbation analysis to construct a pulse solution in the resulting model. In addition, we obtain a 1-parameter family of periodic pulse trains for the system, modelling re-entrant pulses in a one-dimensional ring of excitable cells.