Two-dimensional patterns in reaction–diffusion systems: an analytical tool for the experimentalist

NICOLAS GIOVAMBATTISTA; MAURICIO BELLINI; ROBERTO DEZA

INVERSE PROBLEMS

IOP Publishers

Lugar: London; Año: 2000 vol. 16 p. 811 - 819

0266-5611

We develop a scheme whereby two-dimensional patterns with observed qualitative 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 models and, as such, it might be a valuable phenomenological aid in modelling. The usefulness of the proposed scheme relies on the judicious choice of some ingredients by the user (as dictated by his knowledge on the physical system) after which one simply resorts to known solutions of one-dimesional time-independent Schr¨odinger 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 features. We illustrate the procedure by retrieving the functional dependence on space of a RD equation (with variable diffusion coefficient) that sustains static spirals as solutions. Finally, we extend the method to simple time-dependent patterns such as outgoing stationary travelling waves. We illustrate the procedure with steadily rotating spiral solutions.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 models and, as such, it might be a valuable phenomenological aid in modelling. The usefulness of the proposed scheme relies on the judicious choice of some ingredients by the user (as dictated by his knowledge on the physical system) after which one simply resorts to known solutions of one-dimesional time-independent Schr¨odinger 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 features. We illustrate the procedure by retrieving the functional dependence on space of a RD equation (with variable diffusion coefficient) that sustains static spirals as solutions. Finally, we extend the method to simple time-dependent patterns such as outgoing stationary travelling waves. We illustrate the procedure with steadily rotating spiral solutions.