The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar e...The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two-by-two reaction-diffusion systems. The authors' interest is the extension of the Fisher/KPP equation to a two-species reaction which represents reactant concentration and temperature when used for flame propagation,and bacterial population and nutrient concentration when used in biology.The authors study circumstances in which instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, which means a one-dimensional wave propagating in two space dimensions.Then, explicit analytical formulas give explicitely the range of paramaters for instability.展开更多
基金supported by the FONDECYT Grant(No.1130126)the ECOS Project(No.C11E07)the Fondo Basal CMM and the French"ANR Blanche"Project Kibord(No.ANR-13-BS01-0004)
文摘The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two-by-two reaction-diffusion systems. The authors' interest is the extension of the Fisher/KPP equation to a two-species reaction which represents reactant concentration and temperature when used for flame propagation,and bacterial population and nutrient concentration when used in biology.The authors study circumstances in which instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, which means a one-dimensional wave propagating in two space dimensions.Then, explicit analytical formulas give explicitely the range of paramaters for instability.
