Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains

Area of Science:

• Nonlinear dynamics
• Mathematical modeling
• Computational physics

Background:

• Reaction-diffusion systems are fundamental in modeling complex phenomena.
• FitzHugh-Nagumo models offer simplified yet insightful representations of neuronal activity.
• Tristable systems exhibit richer dynamics than bistable ones, including multiple stable states.

Purpose of the Study:

• To investigate wave propagation in a tristable piecewise-linear reaction-diffusion system.
• To analytically construct basic nonlinear waves (fronts, pulses, wave trains).
• To identify and characterize novel wave solutions unique to tristable systems.

Main Methods:

• Analytical construction of wave solutions based on bistable system results.
• Approximation of a quintic FitzHugh-Nagumo model.
• Numerical simulations to verify wave stability and behavior.

Main Results:

• Description of basic nonlinear waves with oscillatory tails.
• Identification of a novel bright-dark pulse solution with a zigzag profile.
• Demonstration of stable bright-dark pulses in systems with asymmetric thresholds.
• Observation of pulse splitting and multiwave propagation in systems with symmetric thresholds.

Conclusions:

• The tristable system supports complex wave phenomena beyond those in bistable systems.
• Bright-dark pulses are a unique feature, stable under asymmetric conditions.
• The multiwave regime highlights the rich dynamics and parameter-dependent behaviors of tristable systems.