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

ABSTRACT

We develop a theory of wave propagation into an unstable state for a system of integral equations with memory, long-range interactions, and transmutations. In particular we use continuous-time random walk theory to describe the transport and transmutation processes. We use a hyperbolic scaling and Hamilton-Jacobi formalism to derive formulas for the speed of propagation of the traveling wave generated by the system in the long-time large-distance limit. Our theory is valid for arbitrary waiting-time, jump-length and, transmutation probability density functions and the propagation speed can generally be found numerically. However, we illustrate our theory by considering an example where analytic results are possible--that is, for a system of Markovian reaction-transport equations. We derive formulas to determine the propagation speed in both the so-called weakly coupled and strongly coupled cases.