Manchester Geometry Seminar 1999/2000

The problem of wavefront propagation for the n-dimensional reaction-diffusion system involving Kolmogorov-Petrovskii-Piskunov kinetics and the diffusion transport with a finite velocity has been considered. By using a scaling procedure we have given an asymptotic derivation of the equation governing the evolution of a reaction front in the long-time large-distance limit. It has been found that this equation is identical in form to the relativistic Hamilton-Jacobi equation.

The effects of an anisotropic diffusion have been examined in the framework of Hamilton-Jacobi theory. We have found that in the large scale asymptotic limit the Hamiltonian dynamical system associated with the reaction-diffusion system has a structure identical to that of general relativity. We have shown that the function determining the position of the reaction front and its speed can be interpreted as the action functional for a particle in both gravitational and electromagnetic fields. The metric tensor of the 4-dimensional Riemannian space of general relativity has been determined through the diffusivity tensor, while the speed of light corresponds to the finite speed of ''diffusion'' waves. The analogy with general relativity theory has allowed us to find the explicit formula for the reaction front position.