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Localized front structures in FitzHugh-Nagumo equations. (English) Zbl 1415.34048

Summary: We are interested in various types of localized waves in FitzHugh-Nagumo equations. Variational methods have been successfully worked out to establish the existence of traveling and standing waves. Starting with a simple planar traveling front, an ordered method is employed to demonstrate different front propagation between two stable equilibria. If these two stable equilibria are in the same energy level, a saddle-focus condition ensures that there are infinite number of standing waves with multiple fronts.

MSC:

34B08 Parameter dependent boundary value problems for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35K57 Reaction-diffusion equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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