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
(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 instabilityFor 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
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
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 α≤μ, 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 α≤μ.
(Ⅱ) At the H fixed point
(9) |
and
(10) |
When
(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
(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 α≤μ; (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 frontsBy 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→∂t+ε2∂T. The derivative of spatial is ∂x→∂x+ε∂X, and the reaction diffusion operators are decoupled as ∂xx→ ∂xx+ε∂xX+ε2∂XX. ε is the small control parameter, chosen as
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
(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 ConclusionIn 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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