Nucleation in the FitzHugh–Nagumo system: Interface–spike solutions

We find nucleation solutions of N interfaces and K spikes to the one-dimensional FitzhHugh–Nagumo system. Each spike sits asymptotically in the middle between two interfaces. We use the Lyapunov–Schmidt reduction method, in which the problem is split into a finite-dimensional problem related to the translation of the K spikes and an infinite-dimensional complement problem. However the complement problem remains near degenerate due to the translation of the N interfaces. To overcome this difficulty we move the interfaces by a small distance and solve the complement problem with the help of a Newton iteration argument.