Rigorous Computation of Non-uniform Patterns for the 2-Dimensional Gray-Scott Reaction-Diffusion Equation
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Rigorous Computation of Non-uniform Patterns for the 2-Dimensional Gray-Scott Reaction-Diffusion Equation Roberto Castelli1
Received: 22 July 2016 / Accepted: 26 April 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from the homogenous states is depicted, also showing coexistence of multiple solutions at the same parameter values. Combining analytical estimates and rigorous computations, the solutions are sought as fixed points of a operator in a suitable Banach space. To address the curse of dimensionality, a variation of the existing technique is presented, necessary to enable successful computations in reasonable time. Keywords Rigorous numerics · 2-Dimensional Gray-Scott reaction diffusion equation · Contraction mapping theorem · Pattern dynamics Mathematics Subject Classification 35K57 · 65G20 · 65G40
1 Introduction Formation and self-organisation of patterns are fascinating phenomena that occur in several natural systems, as for instance in semiconductors, ferroelectric and magnetic materials, in combustion systems, in biological processes and chemical reactions. See [30] for a survey. Since the seminal work of Turing [28], who first proposed a reaction-diffusion system to account for morphogenesis, systems of reaction-diffusion equations are widely considered to describe and analyse the formation and the dynamics of patterns. A prototype is the Gray-Scott system, a model for the interaction of a pair of reactions involving cubic autocatalysis [12]. It consists in the following reaction-diffusion system Ut = d1 U − U V 2 + F (1 − U ), x ∈ Ω (1) x∈Ω Vt = d2 V + U V 2 − (F + κ)V
B R. Castelli
[email protected]
1
Dept. of Mathematics, VU University Amsterdam, de Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
R. Castelli
where U and V are the concentrations of chemical reactants U and V , d1 , d2 are the diffusion coefficients, F is the feed rate and (F + κ) the removal rate of V . Experiments and numerical investigation reported by Pearson [24] and in [16, 25] reveal a rich and complex structure in the solutions for the Gray-Scott equation, including self-replicating patterns, oscillating patterns, spots annihilation, irregular patterns, spatiotemporal chaos. Since then, the behaviour and the dynamics of the patterns in the GrayScott system has been extensively studied through experimental observations, numerical techniques and analytic approach. In two dimensional domains, particular solutions like single spot solutions, ring-like solitons and stripe patterns have been found and analysed in [15, 19, 20, 31]. Also, an hybrid analytic-numerical approach is adopted in [6] to de
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