Front dynamics for an anisotropic reaction-diffusion equation

The effects of anisotropic diffusion with a finite velocity on propagating fronts in a reaction-diffusion equation are examined within the framework of Hamilton-Jacobi theory. It is found that in the long-time large-distance asymptotic limit the Hamiltonian dynamical system associated with the reaction-diffusion equation has a structure identical to that of general relativity. It is shown that the function which determines the position of the reaction front and its speed can be interpreted as the action functional for a relativistic particle moving in both gravitational and electromagnetic fields. The diffusivity tensor determines the metric tensor of the four-dimensional Riemannian space of general relativity, while the speed of light corresponds to the finite speed of diffusion waves. The mass of the relativistic particle and scalar potential are found to be functions of the reaction rate coefficient and relaxation time. The analogy with general relativity theory allows us to find an explicit formula for the reaction front position.