Front propagation in both directions and coexistence of traveling fronts and pulses

TL;DR

This paper investigates the possibility of bidirectional front propagation and the coexistence of traveling fronts and pulses in reaction-diffusion systems, providing positive answers through analysis of a FitzHugh-Nagumo model.

Contribution

It demonstrates that reaction-diffusion waves can propagate in both directions and that traveling fronts and pulses can coexist, using a specific FitzHugh-Nagumo model.

Findings

Reaction-diffusion waves can propagate bidirectionally.

Traveling fronts and pulses can coexist in the same system.

Positive answers to two previously unexplored questions.

Abstract

For scalar reaction-diffusion equations, a traveling wave is a front which transforms a higher energy state to a lower energy state. The same is true for a system of equations with a gradient structure. At the core of this phenomenon, the wave propagation results in an invasion of an equilibrium state from a different one. This paper is aimed at two questions: (i) Can reaction-diffusion waves exhibit front propagation in both directions between two distinct equilibrium states? (ii) The co-existence of traveling fronts and pulses; a subject rarely studied. Working on a FitzHugh-Nagumo model with the same physical parameters, we give positive answers to both questions.