Higher-dimensional localized patterns in excitable media

Abstract

The excitable reaction-diffusion equation model of the form ετu t=ε 2▿ 2u+ƒ(u)–v , v t=▿ 2v+u−γv is considered. When ƒ(u) is assumed to be of McKean's piecewise linear type, the interfacial approach can be applied to the stability of various localized patterns in higher-dimensional spaces. It is shown that a band-shaped localized pattern is destabilized into a zig-zag mode or a varicose mode and that a disk-shaped localized pattern is destabilized into static modes and, when τ is small, into an oscillatory mode like a “breather motion”. Numerical simulations are performed to confirm such destabilizations for more general nonlinear functions ƒ(u).