PLANAR STANDING WAVEFRONTS IN THE FITZHUGH-NAGUMO EQUATIONS

This article is devoted to the investigation of standing waves for the FitzHugh-Nagumo equations, a well-known reaction-diffusion model of activator-inhibitor type for exhibiting Turing patterns. Similar to the Allen-Cahn equation, a balanced condition for the potential induced from the reaction terms is imposed in studying the existence of planar standing wavefronts. Furthermore, the diffusion rates of activator and inhibitor must be in an appropriate range to ensure the existence of such waves. For the standing front with a symmetry property, an application of the comparison argument yields a uniqueness result. Moreover, the asymptotic stability of wavefronts up to a phase shift is analyzed.