Scaling and crossover dynamics in the hyperbolic reaction-diffusion equations of initially separated components

Area of Science:

• Chemical kinetics
• Reaction-diffusion systems
• Mathematical modeling

Background:

• Front propagation is crucial in various chemical and physical processes.
• Understanding reaction-diffusion dynamics is essential for predicting system behavior.
• Hyperbolic and parabolic models offer different descriptions of diffusion and reaction.

Purpose of the Study:

• To investigate front propagation dynamics using hyperbolic reaction-diffusion equations.
• To compare the behavior of hyperbolic systems with their parabolic counterparts.
• To analyze the influence of stoichiometric coefficients on front dynamics.

Main Methods:

• Utilizing one-dimensional hyperbolic reaction-diffusion equations.
• Employing the mean-field approximation for reaction rates.
• Applying perturbation techniques to analyze temporal behavior and scaling exponents.
• Comparing perturbation results with full numerical solutions.

Main Results:

• Hyperbolic systems exhibit linear scaling for front center and width, unlike the t(1/2) scaling in parabolic systems.
• Scaling laws derived from hyperbolic models are independent of stoichiometric coefficients (n and m).
• Identified and studied the crossover time between hyperbolic and parabolic regimes.
• Derived and numerically validated conditions for static and moving fronts.

Conclusions:

• Hyperbolic reaction-diffusion equations predict fundamentally different front propagation dynamics than parabolic equations.
• The linear scaling observed in hyperbolic systems offers a distinct perspective on early-stage reaction dynamics.
• The findings provide a more comprehensive understanding of front propagation in complex reaction systems.