Standing Wave Solutions for a System Derived from the Fitzhugh–Nagumo Equations for Nerve Conduction

We focus our attention on the reaction-diffusion system uₜ = D₁ Δ u + f(u) - v,v_t = D_2 Δ v + ε (u - γ v)$ where $f(u) = u(1 - u)(u - a),0 < a < 1/2$ and $D_1 $, $D_2 $, $ε $, $γ $ are positive constants. The parameter $γ $ is chosen large so that the associated dynamic equations $(D_1 = D_2 = 0)$ have three constant solutions two of which are stable. The authors establish necessary and sufficient conditions that the Dirichlet problem for this system possesses two nontrivial time independent solutions.