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

A theory of wave propagation into an unstable state for the complex system of integral equations was developed to analyze the stochastic transport involving non-Markovian random processes with long-range interactions and transmutations. A probabilistic approach based upon the continuous-time random walk (CTRW) theory was used to describe the transport and transmutation processes. A hyperbolic scaling and Hamilton-Jacobi formalism was used to derive formulas for the speed of propagation of the traveling wave generated by the system in the long-time large-distance limit. The results show that the theory is valid for arbitrary waiting-time, jump-length and transmutation probability density functions.