IngentaConnect Two-dimensional patterns in reaction-diffusion systems: an analyt...

Authors: Giovambattista N.¹; Bellini M.²; Deza R.²

Source: Inverse Problems, Volume 16, Number 3, 2000 , pp. 811-819(9)

Publisher: Institute of Physics Publishing

Abstract:

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?dinger 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.

Language: English

Document Type: Miscellaneous

Affiliations: 1: Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA 2: Departamento de F?sica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, De?n Funes 3350, 7600 Mar del Plata, Argentina

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