Waves in a reaction-transport system with memory, long-range interactions, and transmutations.

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Complex systems often exhibit wave propagation phenomena.
• Understanding wave dynamics in unstable states is crucial for predicting system behavior.
• Integral equations with memory and long-range interactions present significant theoretical challenges.

Purpose of the Study:

• To develop a general theory for wave propagation into unstable states.
• To provide a framework for analyzing transport and transmutation processes.
• To derive formulas for wave propagation speed in the long-time, large-distance limit.

Main Methods:

• Utilizing continuous-time random walk (CTRW) theory for transport and transmutation.
• Employing hyperbolic scaling and Hamilton-Jacobi formalism.
• Deriving analytical and numerical methods for propagation speed calculation.

Main Results:

• A general theory for wave propagation in systems with memory, long-range interactions, and transmutations.
• Formulas for wave propagation speed applicable to arbitrary waiting-time, jump-length, and transmutation probability densities.
• Analytic results for Markovian reaction-transport equations in both weakly and strongly coupled cases.

Conclusions:

• The developed theory provides a robust framework for studying complex wave phenomena.
• The CTRW approach effectively models transport and transmutation processes.
• The derived formulas enable accurate prediction of wave propagation speeds in diverse systems.