Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion

Oscillatory reaction-diffusion fronts are described analytically in a piecewise-linear approximation of the FitzHugh-Nagumo equations with linear cross-diffusion terms, which correspond to a pursuit-evasion situation. Fundamental dynamical regimes of front propagation into a stable and into an unstable state are studied, and the shape of the waves for both regimes is explored in detail. We find that oscillations in the wave profile may either be negligible due to rapid attenuation or noticeable if the damping is slow or vanishes. In the first case, we find fronts that display a monotonic profile of the kink type, whereas in the second case the oscillations give rise to fronts with wavy tails. Further, the oscillations may be damped with exponential decay or undamped so that a saw-shaped pattern forms. Finally, we observe an unexpected feature in the behavior of both types of the oscillatory waves: the coexistence of several fronts with different profile shapes and propagation speeds for the same parameter values of the model, i.e., a multifront regime of wave propagation.