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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 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 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.

We present the microscopic equation for the growing interface with quenched noise for the model first presented by Buldyrev et al. [Phys. Rev. A 45, R8313 (1992)]. The evolution equation for the height, the mean height, and the roughness are reached in a simple way. The microscopic equation allows us to express these equations in two contributions: the contact and the local one. We compare this two contributions with the ones obtained for the Tang and Leschhorn model [Phys. Rev A 45, R8309 (1992)] by Braunstein et al. [Physica A 266, 308 (1999)]. Even when the microscopic mechanisms are quiet different in both model, the two contribution are qualitatively similar. An interesting result is that the diffusion contribution, in the Tang and Leschhorn model, and the contact one, in the Buldyrev model, leads to an increase of the roughness near the criticality., Comment: 10 pages and 4 figures. To be published in Phys. Rev. E