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Traveling spike autosolitons in the Gray-Scott model. (English) Zbl 0986.34023

Summary: The authors develop singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons (self-sustained solitary waves) in the Gray-Scott model of an autocatalytic reaction. They found that when the inhibitor diffusion is sufficiently slow, the ultrafast traveling spike autosolitons are realized in a wide range of parameters. When the diffusion of the inhibitor is sufficiently fast, the slower traveling spike autosolitons with the diffusion precursor are realized. The authors asymptotically calculate the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the Gray-Scott model. They also show that in certain parameter regions the traveling spike autosolitons coexist with static.

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92E20 Classical flows, reactions, etc. in chemistry
80A30 Chemical kinetics in thermodynamics and heat transfer
80A32 Chemically reacting flows
35Q51 Soliton equations
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