Stationary Wave Solutions of a System of Reaction-Diffusion Equations Derived from the FitzHugh–Nagumo Equations

We consider an extension of the FitzHugh–Nagumo model, namely the system \[ u_t = D_1 uₓₓ + f(u) - w, w_t = D_2 wₓₓ + ε (u - γ w) \] where ε > 0,γ > 0,D₁ > 0,D₂ > 0 and $f(u)$ is cubic. We allow γ to be large which implies that there are three constant solutions. We show that over an appropriate range of parameters the system has time independent pulse solutions and an infinite number of periodic solutions. Depending on the particular choice of parameters, we show that the pulse solution leads to either the first constant solution or the third constant solution.