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

Area of Science:

• Mathematical modeling
• Nonlinear dynamics
• Chemical kinetics

Background:

• Reaction-diffusion systems exhibit complex spatiotemporal patterns.
• Oscillatory fronts are crucial in phenomena like pattern formation and signal propagation.
• Cross-diffusion introduces unique dynamics, often related to pursuit-evasion scenarios.

Purpose of the Study:

• To analytically describe oscillatory reaction-diffusion fronts.
• To investigate front propagation dynamics into stable and unstable states.
• To explore the impact of cross-diffusion on wave profiles and stability.

Main Methods:

• Piecewise-linear approximation of FitzHugh-Nagumo equations.
• Inclusion of linear cross-diffusion terms.
• Analytical investigation of fundamental dynamical regimes and wave shapes.

Main Results:

• Identified two primary front propagation regimes: kink-type (monotonic) and wavy-tailed (oscillatory).
• Observed that oscillation damping determines wave profile characteristics (exponential decay vs. undamped saw-shaped patterns).
• Discovered a multifront regime where multiple fronts with varying speeds and shapes coexist for identical model parameters.

Conclusions:

• The study provides a detailed analytical framework for understanding oscillatory reaction-diffusion fronts.
• Cross-diffusion significantly influences wave dynamics, leading to diverse propagation behaviors.
• The multifront regime represents a novel finding with implications for complex system dynamics.