Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model

doi:10.1016/S0096-3003(03)00295-9

Applied Mathematics and Computation

Volume 150, Issue 3, 17 March 2004, Pages 623-645

https://doi.org/10.1016/S0096-3003(03)00295-9 Get rights and content

Abstract

By using an approach developed by one of the authors, approximate solutions of the soft periodic boundary conditions for a two-cell reaction diffusion model have been obtained. The system is considered with reactant A and autocatalyst B. The reaction is taken cubic in the autocatalyst in the two-cell with linear exchange through A. The formal exact solution is obtained which is symmetric with respect to the mid-point of the container. Approximate solutions are found through the Picard iterative sequence of solutions constructed after the exact one. It is found that the solution obtained is not unique. When the initial conditions are periodic, the most dominant modes initiate to traveling waves in systems with moderate size. Symmetric configurations forming a parabolic one for large time are observed. In systems of large size, spatially symmetric chaos are produced which are stationary in time. Furthermore, it is found the symmetric pattern formation hold irrespective of the condition of linear instability against small spatial disturbance.

Introduction

Experimental studies of the pattern dynamics in Faraday waves [1], [2] reveal ordered time averages. The form of the average pattern is induced by the symmetry imposed by the boundary conditions. The numerical simulations [3] on the Kuramato–Sivashinsky equation under soft (free) boundary conditions show that the system has a parabolic averaged pattern. The stability of spatial patterns, near stationary state solutions, in diffusion reactions subjected to rigid (Neumann) boundary conditions has been studied currently in the literature, especially for reactions in the presence of a precursor [4], [5], [6], [7], [8], [9]. In the absence of the precursor, spatial patterns are expected to be stable. This holds for the present case. In these reactions self-oscillations are produced. Similar observations have been found in biological systems, in enzyme reactions (glycosis) [10]. Our aim here is to study the self-oscillations produced in a cubic two-cell reaction subjected to soft periodic boundary conditions. The coupling between the two-cell (regions) is assumed by allowing a linear diffusion interchange by the reactant. This is achieved practically through semi-permeable membrane interface between the two regions. Now, we consider the following model proposed in [11]. In these papers attention is focused to the initial value problem. In the region I, the reaction is given by cubic autocatalyst together with a linear decay to an inner product C: $A +2 B →3 B (rate =k_{1} ab^{2}),$ $B → C (rate =k_{2} b).$

In the region II, the reaction is given by a purely cubic autocatalysis, $A +2 B →3 B (rate =k_{1} ab^{2}).$

Also, it is assumed that A interchanges from one region to another through mass exchange rate k₃. The motivation for studying these problems is to investigate the self-oscillation produced by the reactions in systems of finite sizes. It will be shown that the size of the system plays a significant role in the number of oscillations produced in the concentrations of the reactant and the autocatalyst. Here, we shall confine ourselves with the one-dimensional model equations, which describe such autocatalytic reactions. They are given by (see also [11]): $a_{1τ} =D_{A} a_{1xx} −k_{1} a_{1} b_{1}^{2} +k_{3} (a_{2} −a_{1}),$ $b_{1τ} =D_{B} b_{1xx} +k_{1} a_{1} b_{1}^{2} −k_{2} b_{1},$ $a_{2τ} =D_{A} a_{2xx} −k_{1} a_{2} b_{2}^{2} +k_{3} (a_{1} −a_{2}),$ $b_{2τ} =D_{B} b_{2xx} +k_{1} a_{2} b_{2}^{2}, (x,t)∈(0,ℓ)×(0,∞).$

In , , , a₁, b₁, a₂ and b₂ are the concentrations of the reactant A and the autocatalyst B in the two regions I and II respectively. The coefficients D_A,: D_B are the diffusion coefficients of the species a and b in the two regions. The Dirichlet boundary conditions are taken $a_{i} (0,t)=a_{i} (L,t)=a_{0}, b_{i} (0,t)=b_{i} (L,t)=0.$

It is worth noting that the boundary values in (1.5) are the homogeneous steady state solutions of Eqs. , , , which will be found in the next section.

Section snippets

Linear stability analysis

The linear stability analysis of the homogenous steady state solution (hsss) of the system , , , against homogenous and non-homogenous disturbance is studied. In the absence of the diffusion term in , , , the (hsss) of these equations is obtained as the solution of the equations $f_{1} ≡−k_{1} a_{1} b_{1}^{2} +k_{3} (a_{2} −a_{1})=0,$ $f_{2} ≡k_{1} a_{1} b_{1}^{2} −k_{2} b_{1} =0,$ $f_{3} ≡−k_{1} a_{2} b_{2}^{2} +k_{3} (a_{1} −a_{2})=0,$ $f_{4} ≡k_{1} a_{2} b_{2}^{2} =0.$

These equations solve to $b_{1} =b_{2} =0, a_{1} =a_{2} =a_{0} .$

The linear stability analysis suggests evaluating the eigenvalues of the Jacobian matrix $∂ f_{1} ∂ a_{1} ∂ f_{1} ∂ b$

Nonlinear analysis

Here, we shall investigate the non-linear analysis of the system , , , , against non-homogeneous perturbations. To this end, we shall find approximate solution for this system by using the following method.

Approximate solutions for the Dirichlet problem

We distinguish two cases γ<1 and γ>1. First, we consider the case γ<1, and make the rescaling transformation t→γt into , , , . Thus Eq. (3.20) becomes $V →^{(0)} =∑_{n=1}^{∞} ψ → (n) sin n 2 L cos n ̄ L 2 −x e^{−(n̄)^{2}γt} .$

In the original variables, we have $α_{1}^{(0)} =1− 12 ∑_{n=1}^{∞} (a_{n} δ_{2}^{−} +c_{n} δ_{1}^{−}) sin n ̄ 2 L cos n ̄ L 2 −x e^{−n̄^{2}γt},$ $β_{1}^{(0)} = e^{−ktγ} ∑_{n=1}^{∞} b_{n} sin n ̄ 2 L cos n ̄ L 2 −x e^{−n̄^{2}γt},$ $α_{2}^{(0)} =1− 12 ∑_{n=1}^{∞} (a_{n} δ_{1}^{−} +c_{n} δ_{2}^{−}) sin n ̄ 2 L cos n ̄ L 2 −x e^{−n̄^{2}γt},$ $β_{2}^{(0)} =∑_{n=1}^{∞} d_{n} sin n ̄ 2 L cos n ̄ L 2 −x e^{−n̄^{2}γt},$ where δ_i⁻, i=1, 2 defined in (3.17) but t→γt.

We remark that all modes generated in the

Numerical results

The results , , , for α_i and β_i are displayed against x in Fig. 1, Fig. 2, Fig. 3, Fig. 4 for different values of t.

In Fig. 1, the concentration α₁ is displayed for a_n=0.02, b_n=0.001, c_n=0.04 and d_n=0.001. The other parameters are taken as L=100, k=0.1 and γ=0.2. Fig. 1a–d correspond to the values of t=0, 1, 5 and 100 respectively. In Fig. 2, Fig. 3, Fig. 4 the concentrations β₁⁽¹⁾, α₂ and β₂⁽¹⁾ are displayed against x for the same caption as in Fig. 1a–d. After these figures, we find that the

Conclusions

An approach for finding approximate solutions of reaction diffusion model equations is presented. It is based on finding the formal exact solution of the Dirichlet problem studied in this work. It is found that the solution of the one-dimensional diffusion model for periodic boundary conditions is symmetric with respect to the mid-point of the size of the system. This symmetric behavior is induced by the evenness of the order of the spatial operator and the periodic boundary conditions. It is a

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Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model

Abstract

Introduction

Section snippets

Linear stability analysis

Nonlinear analysis

Approximate solutions for the Dirichlet problem

Numerical results

Conclusions

Chaos, Solutions and Fractals

Time-averaging of chaotic spatiotemporal wave patterns

Phys. Rev. Lett.

Average patterns in Faraday waves

Phys. Rev. E

Average patterns of spatiotemporal chaos: a boundary effect

Phys. Rev. E

Nonlinear dynamics and pattern formation in the natural environment

Pattern sensitivity to boundary conditions in reaction-diffusion models

J. Math. Biol.

T-pattern formation in generalized Turing systems. I. Steady-state patterns in systems with mixed boundary conditions

J. Math. Biol.

Pattern formation in a coupled cubic autocatalator system

IMA J. Appl. Math.