Front propagation in both directions and coexistence of traveling fronts and pulses

For a scalar reaction-diffusion equation, a traveling wave is a special type of solution which as a function of time transforms a higher energy equilibrium state to a lower energy equilibrium state. Thus such wave propagation can be viewed as an invasion of one equilibrium state by a different one. Similar phenomena have been obtained for a variety of systems of reaction-diffusion equations possessing a gradient structure. Our main goal in this paper, is to study the following two questions. If all physical parameters of a reaction-diffusion system are fixed, Can reaction-diffusion waves exhibit front propagation in both directions between two distinct equilibrium states, so that both the high and low energy states can be invaded by one another? Can traveling fronts and pulses co-exist? By a traveling front, we mean the equilibrium states are different, i.e. we are dealing with a heteroclinic solution of the equation or system. By a (traveling) pulse, we mean the equilibrium states are the same, i.e. we have a homoclinic solution. The phenomena mentioned in (i) or (ii) do not occur in many reaction-diffusion systems. However, working on a FitzHugh-Nagumo model, we give positive answers to both questions with each wave having its own distinct speed and propagation direction. In fact (i) and (ii) can occur simultaneously so that there exist both a pulse and two fronts traveling in opposite directions connecting the same distinct equilibrium states for the same set of physical parameters.