Application of p-Adic Wavelets to Model Reaction-Diffusion Dynamics in Random Porous Media

Fourier and more generally wavelet analysis over the fields of p-adic numbers are widely used in physics, biology and cognitive science, and recently in geophysics. In this note we present a model of the reaction-diffusion dynamics in random porous media, e.g., flow of fluid (oil, water or emulsion) in a a complex network of pores with known topology. Anomalous diffusion in the model is represented by the system of two equations of reaction-diffusion type, for the part of fluid not bound to solid's interface (e.g., free oil) and for the part bound to solid's interface (e.g., solids-bound oil). Our model is based on the p-adic (treelike) representation of pore-networks. We present the system of two p-adic reaction-diffusion equations describing propagation of fluid in networks of pores in random media and find its stationary solutions by using theory of p-adic wavelets. The use of p-adic wavelets (generalizing classical wavelet theory) gives a possibility to find the stationary solution in the analytic form which is typically impossible for anomalous diffusion in the standard representation based on the real numbers.