On the speed of amplitude transition waves in reaction-diffusion systems of λ−ω type

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Abstract
λ–;ω systems are a class of simple examples of two coupled reaction-diffusion equations whose kinetics have a stable limit cycle. The author considers the evolution of such systems from simple initial data in which a small homogeneous perturbation is applied to the unique steady state in a localized region of the domain. Using a combination of analytical and numerical methods, it is shown that the system evolves to two sets of transition wave fronts in the solution amplitude, one moving outwards from the perturbation, and the other moving inwards. An expression is derived for the speed of the outward moving wave, and it is shown that this in turn determines the speed of the inward moving wave. Between these transition fronts, the solution has the form of periodic plane waves, whose amplitude is the solution of a simple algebraic equation. In some cases these periodic plane waves are unstable as reaction-diffusion solutions, in which case they degenerate into irregular spatiotemporal oscillations.

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