Two-dimensional patterns in reaction-diffusion systems: an analytical toolfor the experimentalist

We develop a scheme whereby two-dimensional patterns with observed qualitat ive features (like, for example, spirals) can be obtained (by construction) as solutions of a single reaction-diffusion (RD) equation with a spatially variable diffusion coefficient. The latter can be eventually thought of as an effective one-component system to be derived from more fundamental mode ls and, as such, it might be a valuable phenomenological aid in modelling. The usefulness of the proposed scheme relies on the judicious choice of som e ingredients by the user (as dictated by his knowledge on the physical sys tem) after which one simply resorts to known solutions of one-dimesional ti me-independent Schrodinger equations to generate patterns that are similar to the ones under analysis. This kind of semi-inverse method tells what the reaction term and the diffusion coefficient should look like as functions of space in order that the solution has some sought-for qualitative feature s. We illustrate the procedure by retrieving the functional dependence on s pace of a RD equation (with variable diffusion coefficient) that sustains s tatic spirals as solutions. Finally, we extend the method to simple time-de pendent patterns such as outgoing stationary travelling waves. We illustrat e the procedure with steadily rotating spiral solutions.