We follow up an earlier work (briefly reviewed below) to investigate thetemporal stability of an exact travelling front solution, constructed in theform of an integral expression, for a one-dimensional discrete Nagumo-likemodel without recovery. Since the model is a piecewise linear one with anon-site reaction function involving a Heaviside step function, astraightforward linearisation around the front solution presents problems, andwe follow an alternative approach in estimating a `stability multiplier' bylooking at the variational problem as a succession of linear evolution of theperturbations, punctuated with `kicks' of small but finite duration. Theperturbations get damped during the linear evolution, while the kicks amplifyonly the perturbations located at specific sites (the `significantperturbations', see below) with reference to the propagating front. Comparisonis made with results of numerical integration of the reaction-diffusion systemwhereby it appears likely that the travelling front is temporally stable forall parameter values characterising the model for which it exists. We modifythe system by introducing a slow variation of a relevant recovery parameter andperform a leading order singular perturbation analysis to construct a pulsesolution in the resulting model. In addition, we obtain (in the leading order)a 1-parameter family of periodic pulse trains for the system, modellingre-entrant pulses in a one-dimensional ring of excitable cells.
展开▼
