2004/12/10 by Priyadarshi Majudar, Majudar, Priyadarshi, Avijit Lahiri +1
Computer Science · Physics and Astronomy · #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Spectroscopy and Quantum Chemical Studies #nlin.PS #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.nlin/0412030
26 pages, 13 figures
arxiv created 2004/12/10 · openalex publication_date 2004/12/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We follow up an earlier work (briefly reviewed below) to 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. The perturbations get damped during the linear evolution, while the kicks amplify only the perturbations located at specific sites (the `significant perturbations', see below) with reference to the propagating front. 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 parameter values characterising the model for which it exists. 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 (in the leading order) a 1-parameter family of periodic pulse trains for the system, modelling re-entrant pulses in a one-dimensional ring of excitable cells.