Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model I: Asymptotic Construction and Stability
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چکیده
منابع مشابه
Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model I: Asymptotic Construction and Stability
Two pulse solutions play a central role in the phenomena of self-replicating pulses in the one-dimensional (1-D) Gray–Scott model. In the present work (part I of two parts), we carry out an existence study for solutions consisting of two symmetric pulses moving apart from each other with slowly varying velocities. This corresponds to a “mildly strong” pulse interaction problem in which the inhi...
متن کاملSlowly Modulated Two-Pulse Solutions in the Gray--Scott Model II: Geometric Theory, Bifurcations, and Splitting Dynamics
In this second paper, we develop a geometrical method to systematically study the singular perturbed problem associated to slowly modulated two-pulse solutions. It enables one to see that the characteristics of these solutions are strongly determined by the flow on a slow manifold and, hence, also to identify the saddle-node bifurcations and bifurcations to classical traveling waves in which th...
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Gray-Scott model has been studied extensively in recent years. However, most of the results are in one dimensional. In this paper, we study the 2-D Gray-Scott system. We rst construct two single pulse solutions and then we establish the stability and instability of such solutions in terms of the parameters involved.
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Two pulse solutions play a central role in the phenomena of self-replicating pulses in 1-D reaction-diiusion systems. In this work, we focus on the 1-D Gray-Scott model as a prototype. We carry out an existence and stability study for solutions consisting of two pulses moving apart from each other with slowly varying velocities. In the various parameter regimes, critical maximum wave speeds are...
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Localized patterns have been observed in many reaction-diffusion systems. One well-known such system is the two-component Gray-Scott model, which has been shown numerically to exhibit a rich variety of localized spatiotemporal patterns including, standing spots, oscillating spots, self-replicating spots, etc. This thesis concentrates on analyzing the localized pattern formation in this model th...
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ژورنال
عنوان ژورنال: SIAM Journal on Applied Mathematics
سال: 2000
ISSN: 0036-1399,1095-712X
DOI: 10.1137/s0036139999354923