Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system
The author studies the existence of standing waves with one pulse and transition layer in the nonlinear reaction-diffusion system: \(u_{t}=f(u,w)+u_{xx}\), \(w_{t}= \varepsilon^{2}g(u,w)+ w_{xx}\), \(x\in \mathbb{R}\), where \(\varepsilon>0\) is a small parameter, \(f(u,w)\) has three branches \(u=h_{i}(w),\;i=1,2,3\) on an interval of \(w\) and \(g(u,w)=0\) intersects \(u=h_{1}(w)\) once and \(u=h_{3}(w)\) at most once. The 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 layer.
scientific article; zbMATH DE number 2117238