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Reaction-diffusion front crossing a local defect

By: Jean Guy Caputo
From: INSA Rouen
At: Instituto de Investigação Interdisciplinar, Anfiteatro
[2012-10-30] 14:00

The interaction of a reaction-diffusion front with a localized defect is studied numerically and analytically. We consider a quadratic and a cubic reaction term leading to one or two stable stationary states. Such models can describe the combustion of a solid (Zeldovich), the propagation of a nerve impulse in a neuron or the evolution of a gene in a population (Fisher). We present the qualitative differences in the dynamics of fronts in the bistable and monostable situations. A bistable front will keep its form. It's width and speed will be modulated by the defect. A monostable front can develop a secondary pulse as it approaches the defect. Also the bistable front can be pinned by a defect while the monostable front will always cross it. We develop a collective coordinate description of the front width and position based on conservation laws. This reduced model is in good agreement with the numerical solution of the full problem when the front is well defined. This analysis leads to quantitative estimates for the front pinning. It also enables to estimate the defect from the time series of the front position and width.