Travelling Fronts, Pulses, and Pulse Trains in a 1D discrete Reaction-Diffusion System

Priyadarshi Majudar; Avijit Lahiri

Home/Archive/arXiv

arXiv · nlin/0412030

Travelling Fronts, Pulses, and Pulse Trains in a 1D discrete Reaction-Diffusion System

Priyadarshi Majudar·Avijit Lahiri

View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

nlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Priyadarshi Majudar, Avijit Lahiri. 2004-12-10. Travelling Fronts, Pulses, and Pulse Trains in a 1D discrete Reaction-Diffusion System. https://arxiv.org/abs/nlin/0412030

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

General conditions for Turing and wave instabilities in reaction-diffusion systems

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with $n$ components and a diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on a study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to $\infty$, gives a constructive method for choosing diffusion constants. The results are illustrated on a model for the dispersion of malaria, a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4 components

nlin.PS↗

arXiv2026

Multiprecision computation of bright and dark solitons in the discrete nonlinear Schrödinger equation

We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to $N=65{,}250$ grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.

nlin.PS↗

arXiv2026

Amplitude equations for wave bifurcations in reaction-diffusion systems

A wave bifurcation is the counterpart to a Turing instability in reaction-diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary $n$-component systems ($n\geq 3$) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction-diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

nlin.PS↗