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Numerical challenges for resolving spike dynamics for two one-dimensional reaction-diffusion systems. (English) Zbl 1141.35389

Summary: Asymptotic and numerical methods are used to highlight different types of dynamical behaviors that occur for the motion of a localized spike-type solution to the singularly perturbed Gierer-Meinhardt and Schnakenberg reaction-diffusion models in a one-dimensional spatial domain. Depending on the parameter range in these models, there can either be a slow evolution of a spike toward the midpoint of the domain, a sudden oscillatory instability triggered by a Hopf bifurcation leading to an intricate temporal oscillation in the height of the spike, or a pulse-splitting instability leading to the creation of new spikes in the domain. Criteria for the onset of these oscillatory and pulse-splitting instabilities are obtained through asymptotic and numerical techniques. A moving-mesh numerical method is introduced to compute these different behaviors numerically, and results are compared with corresponding results computed using a method of lines based software package.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K57 Reaction-diffusion equations
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