Turning points and traveling waves in FitzHugh–Nagumo type equations

Consider the following FitzHugh–Nagumo type equation where f(u,w)=u(u−a(w))(1−u) for some smooth function a(w) and g(u,w)=u−w. By allowing a(w) to cross zero and one, the corresponding traveling wave equation possesses special turning points which result in very rich dynamics. In this work, we examine the existence of fronts, backs and pulses solutions; in particular, the co-existence of different fronts will be discussed.