Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction–diffusion system

We establish the existence of standing waves with one pulse, multiple spikes and transition layers in the nonlinear reaction–diffusion system ut=f(u,w)+uxx,wt=(epsilon)^2g(u,w)+wxx,x(element of)R, where (epsilon)>0 is a small parameter, f(u,w)=0 has three branches u=h1(w), u=h2(w) and u=h3(w) on an interval of w and g(u,w)=0 intersects u=h1(w) once and u=h3(w) at most once. We use a different method to prove the existence of a single pulse. The existence of multiple spikes and transition layers is a new result. The new method is a topological shooting plan consisting of various shooting arguments. The main technical advance of this plan is that it provides a setup for proving the existence of not only a single pulse, but also multiple spikes and transition layers. The proof of the existence of spikes and transition layers involves more delicate estimates in order to assure the existence of certain types of solutions necessary in the topological shooting arguments. A two-dimensional topological shooting principle and another elementary topological principle are applied in the shooting arguments.