Turing instability and traveling fronts for a reaction-diffusion model with cross-diffusion in plasmas
LING Heyang, PENG Yahong     
School of Science, Donghua University, Shanghai 201620, China
Received date: 2016-10-26
Foundation item: Supported by the National Natural Science Foundation of China(11571227)
Corresponding author: LING Heyang(1993-), female, native of Anshan, Liaoning province, master degree candidate of Donghua University, the research direction is partial differential equation.E-mail:heyanglingling@163.com
Abstract: The model of turbulence shear flow interaction has been widely studied in the magnetically confined plasmas, and an important problem is to understand how Turing pattern forms.The influence of cross-diffusion on a magnetically confined plasmas model is concerned.Stability at the equilibrium points are analyzed by linear analysis, it is shown that the diffusion term is the key mechanism of pattern formation. Then through the expansion of multiple scales, time scale and spatial scale are separated into fast scale and slow scale. The result of weakly nonlinear analysis indicates that the solution of real Ginzburg-Landau equation describes the amplitude equation.
Key words: reaction-diffusion     Turing instability     amplitude equation     weakly nonlinear expansion     traveling fronts     plasmas    
等离子体中带有交叉扩散的反应-扩散模型的图灵不稳定性和行波
凌赫阳 , 彭亚红     
东华大学 理学院, 上海 201620
摘要: 磁约束等离子体中湍流、剪切流相互作用的模型得到了广泛研究,其中图灵斑图的形成成为研究重点.本文研究交叉扩散对一类磁约束等离子体模型的影响.首先通过线性化分析,研究了系统平衡点的稳定性,发现交叉扩散是导致斑图形成的关键因素.其次,通过将时间、空间尺度展开成快慢尺度后,用弱的非线性分析的方法,得到斑图的振幅方程由实Ginzburg-Landau方程确定.
关键词: 反应-扩散     图灵失稳     振幅方程     弱非线性展开     行波     等离子体    
0 Introduction

The reaction-diffusion systems describe well the interaction between two populations[1]. And the systems are now used in many fields, such as ecology[2-5], social systems[6-7], turbulent flow in plasmas[8-9], semiconductors[10-12], tumor growth[13] and so on. Self-and cross-diffusion terms are the potential mechanisms of pattern formation. In [14], Gambino and her colleagues studied Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion and they also investigated the Lengyel-Epstein System, Brusselator System and Activator-Inhibitor System[15-17].

The model we studied is from [8]. Diamond et al[18] considered the model of the plasma. For the convenience study of turbulence-shear flow interaction, Carreras et al presented the following improved version in [8]:

(1)

Here is the turbulence fluctuation level, where n0 is the equilibrium plasma density and is the fluctuating density; 〈Vθ〉′ represents the poloidal flow, where 〈〉 denotes poloidal, toroidal average over a magnetic flux surface and the prime denotes radial derivative. x is the radial coordinate and γ0 is the linear growth rate of the underlying instability in absence of shear flow. α1 is the turbulence saturation coefficient, α2 is the shear flow turbulence suppression coefficient, D0 is the turbulent diffusivity, is the magnetic pumping coefficient and α3 is the Reynolds stress coefficient. By introducing a series of nondimensional transform ε=(α1/γ0)E, , Da=[D0/(γ0a2)][γ0/α1], then the system (1) is rewritten as follows

(2)

Del-Castillo-Negrete and Carreras investigated the formation of shear flows with radial structure of the system (2) in [8].

Compared with (2), the following system has rich dynamics

(3)

The definitions for μ and α are the same as which in (2). The analysis of (3) is based on (2), Del-Castillo-Negrete and Carreras studied the turbulence-shear flow interaction in plasmas in [9]. They focused on the three nontrivial fixed points and presented the propagation and segregation of traveling fronts.

In this paper, we are more interested in the system (3).The Turing instability and traveling fronts by weakly nonlinear analysis will be investigated.

1 Linear analysis and Turing instability

For the convenience of analyzing, the system (3) is rewritten as follows

(4)

where

here u(x, t), v(x, t) are the substitutions of ε(x, t) and σ(x, t) in (3) respectively, the parameters di, Di(i=1, 2), α and μ are all positive.

The initial conditions and boundary conditions must be added to the above system. As we are interested in the pattern formation in this paper, the homogenous Neuman boundary conditions and will be imposed, which is the weakest constrict on pattern formation.

In this part, the possibility of the appearance of the pattern for the system (4) will be investigated. For the system (4), there are four equilibrium points (u*, v*)=(0, 0), (1, 0) and , with as discussed in [2]. (0, 0) is the trivial solution, the others are nontrivial solutions. In the plasma physical, the (1, 0) is fixed as L fixed point, corresponds to a state of low confinement. While corresponds to a high confinement state, called H fixed point. It is easy to prove that the point (0, 0) is unstable. The investigation of is similar to . In this paper, we will focus on the discussion of L fixed point (1, 0) and H fixed point .

The general form of the linearized system of the system (4) in the neighborhood of (u*, v*) is

(5)

where

(6)

Looking for the solution of system (4) with the form exp(ikx+λt), we obtain the following dispersion relation

where

(7)

with

(8)

In the following, we will give the conditions for the formation of the patterns, the critical values of the bifurcation parameter as well as the critical wave number. The stable and unstable conditions will also be investigated.

(Ⅰ) At the L fixed point (1, 0), (6)~(8) can be written as

with

and

For the reaction system of (4), when αμ, it is easy to prove that L fixed point is stable. When diffusion terms are added, then k2(tr(DL)-tr(KL)) > 0, h(k2) > 0 for any k≠0, the full reaction-diffusion system (4) is still stable at (1, 0) under the condition of αμ.

(Ⅱ) At the H fixed point with , (6) and (8) are given as follows

(9)

and

(10)

When the H fixed point is stable for the kinetics while unstable when diffusion terms are considered. In fact, from (9), we have tr(KH) < 0, tr(DH) > 0 and det (DH) > 0. There exist k > 0 such that h(k2) < 0. Therefore, the marginal stable condition is obtained at k=kc

(11)

where

which demand that qH is negative. From (10), we know the appearance of the cross-diffusion term D2 destroy the stability of H fixed point.

qH < 0 implies the following condition

In what follows, we use the cross-diffusion coefficient D2 as bifurcation parameter. In order to seek the critical values at which bifurcation occurs.Define

and let . Substituting the expression of D2 into (11), we have the following equation of ξ,

(12)

Then the critical value for bifurcation is

(13)

where ξ+ is the positive root of Eq.(12).

Based on the above analysis, the following theorem can be summarized.

Theorem 1   (ⅰ) The L fixed point is stable for the system (4) under the condition αμ; (ⅱ) Assume the conditions (C) and D2 > D2c hold, the H fixed point is spatially unstable for the system (4), where D2c is given by (13).

2 Weakly nonlinear analysis and traveling fronts

By the analysis of Section 1, we know that the H fixed point is Turing unstable and the cross-diffusion term D2 is the key mechanism of pattern formation. However, when the domain is large, one can observe a typical phenomenon that is propagation of the pattern through the physical domain in the form of a traveling wave. In order to quantitatively describe the amplitude of pattern in space, we shall introduce weakly nonlinear analysis[19] and the time scale and spatial scale should be considered. Time scale will be separated into fast scale t and slow time T=ε2t, and time derivative decoupled as ∂t→∂t2T. The derivative of spatial is ∂x→∂x+εX, and the reaction diffusion operators are decoupled as ∂xx→ ∂xx+εxX+ε2XX. ε is the small control parameter, chosen as , representing the dimensionless distance from the threshold.

By separating the linear part and the nonlinear part, the original system (4) can be rewritten as follows

(14)

where w is defined in (5) and linear operator is defined as

where KH and DHD2 are given by (9). For x=(xu, xv), y =(yu, yv) the definition of bilinear operators on vector couple (x, y) are as follows

By using the asymptotic analysis, D2 and w can be expanded to

(15)
(16)

Then the linear operator LD2 can be expanded as

(17)

and

(18)
(19)

Substituting (15)-(19) into (14), according to the order of ε, collecting a series of terms of the equations for wi

(20a)
(20b)

where

and

Under the Neumann boundary condition, the solution of the linear equation (20a) is given by

(21)

The amplitude of the pattern is A(X, T) and the vector ρ is given by the following way

Substituting (21) into the right side of F and through calculating, F can be simplified as

where

Through the Fredholm alternative, (20b) admits a solution if and only if 〈 F, ϕ 〉=0, where 〈·, ·〉 is the scalar product in and ϕ ∈ker(KH-kc2DHD2c)+. Here

(22)

It is obvious to see that the Fredholm alternative is immediately satisfied. The solution of (20b) is then computed according to the parameters of the full system:

(23)

where the vectors w2i satisfies the linear systems as follows:

with

Substituting (21) and (23) into the right side of G, then it can be simplified as the following expression

with

where

The condition 〈 G, ϕ 〉=0 leads to the equation for amplitude A as follows

(24)

where

and ϕ is given by (22).

The envelope evolution and the progressing of the pattern can be described by the real Ginzburg-Landau equation (24)[20-22], which the exact solution in R is

with

This result describes that the traveling front envelop a pattern which diffuses in a spatial domain.

3 Conclusion

In this paper, the model of turbulence shear flow interaction is studied, the cross-diffusion of turbulent flow and shear flow induces the formation of the pattern, and gradually diffuses into the large spatial region. In the reaction-diffusion system, the nonlinear cross-diffusion term is the source of the formation of the pattern. By the linear analysis, it is concluded that diffusion leads to instability, and then the pattern forms. Then by weakly nonlinear analysis, we know that the amplitude of pattern which diffuses as traveling waves in the large spatial region. Of course, the further work is to give numerical simulation that will show a good agreement with the solution prescribed by the weakly nonlinear expansion.

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西安工程大学、中国纺织服装教育学会主办
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文章信息

LING Heyang, PENG Yahong.
凌赫阳, 彭亚红.
Turing instability and traveling fronts for a reaction-diffusion model with cross-diffusion in plasmas
等离子体中带有交叉扩散的反应-扩散模型的图灵不稳定性和行波
Basic Sciences Journal of Textile Universities, 2017, 30(3): 318-324, 330.
纺织高校基础科学学报, 2017, 30(3): 318-324, 330

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收稿日期: 2016-10-26

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