Computer Tools for Bifurcation Analysis: General Approach with

1002 L. M. Pismen & B. Y. Rubinstein
k 1 , k 2 is written as:
∂ 1 a 1 =2a(k 1 ∇ 1 )a 1 + σa 1R 1
1+σ , (78)
and may be simplified by setting R 1 =0.
It appears that the term describing the mode–
mode interaction in the Ginzburg–Landau equation
in our case does not depend on the value of the angle
between mode wavevectors and is given by c 3, 4 =
−2/b; the self-interaction coefficient c 3, 5 = −1/b.
Two out of three diffusional coefficients in the second
amplitude equation are not zeros:
c 3, 1, 1 = − ia σ ;
c 4a 2
3, 1, 2 = −
(1 + σ)(1 + σ − iΩ) .
(79)
Finally the amplitude equation may be written in
the form:
σ
∂ 2 a 1 = −
b(1 + σ) (2|a 2| 2 + |a 1 | 2 )a 1
−
−
4a 2
(1 + σ)(1 + σ − iΩ) (k 1∇ 1 ) 2 a 1
ia
1+σ (∇ 1∇ 1 )a 1 + σR 2
(1 + σ) a 1 . (80)
5. Amplitude Equations for
Convective Problems
5.1. Convective instabilities
The convective problems are characterized by the
presence of the additional convective term in
Eq. (52) and can be written in the following form
[Rovinsky & Menzinger, 1992, 1993]:
∂
u(r, t)=D(∇·∇)u(r,t)+εV(n·∇)u(r,t)
∂t
+f(u(r,t),R), (81)
where n denotes a constant vector determining the
direction of flow and the matrix V sets the velocities
values; it is assumed that the velocities are
of the order ε. Then for the Turing instability
one must call the function BifurcationTheory as
follows:
BifurcationTheory[
DD.(Nabla[r].Nabla[r])**u + f[u,R] +
eps V.({1,0}.Nabla[r])**u -
Nabla[t]**u == 0,
u, R, t, {r,2},{r,1},{U,Ut},
{{a1,0}, {a2, alpha}}, c, eps, 2, k, 0]
Here the velocity direction is chosen parallel to the
x-axis. Note that the small parameter of expansion
eps in this case appears explicitly in the operator
equation describing the problem. The result produced
may be written as:
∂ 1 a 1 = c 2, 1, 1
c 2, 2
(ik 1 ∇ 1 )a 1 + c 2, 3
c 2, 2
a 1 + c 2, 3(R 1 )
c 2, 2
a 1 ,
∂ 2 a 1 = c 3, 4
c 3, 3
|a 2 | 2 a 1 + c 3, 5
c 3, 3
|a 1 | 2 a 1 + c 3, 6
c 3, 3
a 1
+ c 3, 6(R 1 , R 2 )
c 3, 3
a 1 + c 3, 1, 1
c 3, 3
(∇ 1 ∇ 1 )a 1
+ c 3, 1, 2
c 3, 3
(ik 1 ∇ 1 ) 2 a 1 + c 3, 2, 1
c 3, 3
(ik 1 ∇ 1 )a 1
(82)
+ c 3, 2, 1(R 1 )
c 3, 3
(ik 1 ∇ 1 )a 1 + c 3, 2, 2
c 3, 3
(n∇ 1 )a 1 .
It can be noted that the set of equations is very
similar to Eq. (65) with the addition of some terms.
Here we write the expressions for the coefficients
which are changed or added comparing with coefficients
of (65). The additional linear term in the
first equation is due to the velocity matrix V and
its coefficient is given by:
c 2, 3 = ikU † VU .
Similarly, there are two linear terms in the second
equation, their coefficients now depend on V:
c 3, 6 = k 2 U † VR(f u − k 2 D)VU − k 2 (U † VR(f u − k 2 D)VU)(U † VU)/(U † U) ,
c 3, 6 (R 1 , R 2 )=(U † R(f u −k 2 D)f uR UR 1 )(U † f uR UR 1 )/(U † U)+U † f uR UR 2
− U † f uR R(f u − k 2 D)f uR UR 1 R 1 + U † f uRR UR 1 R 1 /2
(83)
− ikU † VR(f u − k 2 D)f uR UR 1 + ik(U † VR(f u − k 2 D)f uR UR 1 )(U † VU)/(U † U)
+ ik(U † VR(f u − k 2 D)VU)(U † f uR UR 1 )/(U † U) − ikU † f uR R(f u − k 2 D)VUR 1 .