Abstract
Here, we extend the Fisher-Kolmogorov-Petrovsky-Piskunov equation to
capture the interplay of multiscale and multiphysics coupled processes.
We use a minimum of two coupled reaction-diffusion equations with
additional nonlocal terms that describe the coupling between scales
through mutual cross-diffusivities. This system of equations
incorporates the physics of interaction of thermo-hydro-chemo-mechanical
processes and can be used to understand a variety of localisation
phenomena in nature. Applying bifurcation theory to the system of
equations suggests that geological patterns can be interpreted as
physical representation of three classes of well-known instabilities:
Turing instability, Hopf bifurcation, and a chaotic regime of complex
soliton-like waves. For specific parameters, the proposed system of
equations predicts all three classes of instabilities encountered in
nature. The third class appears for small fluid release reactions rates
as a slow quasi-soliton wave for which our parametric diagram shows
possible transition into the Hopf- or Turing-style instability upon
dynamic evolution of coefficients.