Modulated Two-Dimensional Patterns in Reaction

Modulated Two-Dimensional Patterns in Reaction-Diusion Systems R. Kuskey Department of Mathematics Tufts University

P. Milewskiz Department of Mathematics University of Wisconsin, Madison Abstract

New modulation equations for hexagonal patterns in reaction-diusion systems are derived for parameter regimes corresponding to the onset of patterns. These systems include additional nonlinearities which are not present in Rayleigh-Benard convection or Swift-Hohenberg type models. The dynamics of hexagonal and roll patterns are studied using a combination of analytical and computational approaches which exploit the hexagonal structure of the modulation equations. The investigation demonstrates instabilities and new phenomena not found in other systems, and is applied to patterns of ame fronts in a certain model of burner stabilized ames.

1 Introduction Instabilities in many physical or chemical systems often lead to spatio-temporal patterns. Typical systems in which these patterns occur include convection [1]-[2], chemical reactions [3]-[5], and nonlinear optics [6]. These patterns usually occur as the superposition of periodic waves of some component of the system, such as temperature or concentration. Borrowing the terminology from convection studies, a single such wave is usually called a roll. Near the onset of instabilities, the dynamics of the periodic waves are well described by amplitude equations for their envelopes. These amplitude equations are used to study the modulations and instabilities of the patterns. Also, the computation of pattern dynamics is often more straightforward using modulation equations rather than the full system, as, for example, in the study of patterns on a reaction front in [7]. Perfect hexagonal patterns appear as the superposition of three roll solutions with wave vectors at 120. Modulation equations for hexagonal patterns in general reaction-diusion models typically include additional nonlinear terms, which describe instabilities and new phenomena not studied in Rayleigh-Benard convection models [8] or Swift-Hohenberg type models [9]. In this paper we derive new evolution equations for modulated hexagonal patterns in general reaction-diusion systems.  y z

Supported in part by a NSF Mathematical Sciences Postdoctoral Research Fellowship. Present address: School of Mathematics, University of Minnesota, Minneapolis, MN. Supported in part by NSF Grant DMS 9401405.

1

These equations are written in a form in which the hexagonal symmetries are apparent. That is, the new coordinates are X1 , X2 , and X3,

p

p

X1 = X; X2 = , 21 X + 23 Y; X3 = , 12 X , 23 Y; (1.1) where (X; Y ) = ,1 (x; y) are the usual two-dimensional Cartesian coordinates. Then the coupled evolution equations for the amplitudes Aj (X1; X2; X3), j = 1; 2; 3 of the three rolls are Aj  = aAj + cAk Al + dAj   + bAj jAj j2 + fAj (jAk j2 + jAl j2) +ih1 (Ak Al ) + ih2[(Ak ) Al + (Al ) Ak ]; j j

j

k

l

(1.2)

for permutations of j; k; l = 1; 2; 3: The subscripts j for j = 1; 2; 3 denote dierentiation along the direction Xj , that is, in the direction of the wave vector for roll j , @ = @ ,1 @ ,1 @ : (1.3) @j @Xj 2 @Xk 2 @Xl For some of the phenomena studied, we will also consider modulations to roll j in the direction orthogonal to Xj (that is, modulations along the roll). For this, we introduce the derivatives in the direction j , orthogonal to j with

p ! @ = 3 @ , @ : (1.4) @j 2 @Xk @Xl Including those modulations, 1.2 become  2 Aj  = aAj + cAk Al + d @ + 2ik @2 Aj + bAj jAj j2 + fAj (jAk j2 + jAlj2 ) c +ih1 (Ak Al ) + ih2 [(Ak ) Al + (Al ) Ak ]: (1.5) j

j

j

k

l

The dierence between these modulation equations and those studied in [9]-[10] are the terms with coecients h1, h2, referred to as nongradient terms [2] or non-potential terms, as discussed further at the end of Section 2 and in [11], [12], and [13]. We shall employ a combination of analytical and computational approaches which exploit the hexagonal structure of the modulation equations to determine the stability and dynamics of patterns. We use a pseudo{spectral numerical algorithm on a hexagonal grid that allows the removal of unwanted boundary eects that arise from a rectangular mesh, as used in [10]. The analysis and computations demonstrate the stability dierences between a general reaction diusion system and previously studied evolution equations for certain convection problems [2], [9]-[10]. In the reaction-diusion systems we consider, there are additional instabilities of both roll and perfect hexagons, which are also observed in Marangoni convection [12]-[14]. There are also other stable modulated two dimensional patterns which are a combination of rolls and hexagons, sometimes referred to as general hexagonal patterns. The computational results validate and complement the stability criteria obtained analytically. 2

In the linear stability analysis we perturb the three rolls forming the hexagonal pattern with the same small wave number. We call such perturbations \symmetric", in contrast to the more general \asymmetric" case of dierent wave number perturbations for each roll. Although such asymmetric perturbations are analytically less tractable, we observe numerically that in the case of h1 6= 0 and h2 6= 0, they lead to observable instabilities. These instabilities are important since they may occur in parameter ranges where the pattern is stable to symmetric perturbations. The application of these methods is demonstrated for a general reaction-diusion system and in the speci c problem of a burner-stabilized ame front [7] which has the important features of the general system.

2 The modulation equations We consider the general reaction-diusion system, as motivated by various models such as [4], [15], and [16], ! @ u~ = D @ 2 u~ + @ 2 u~ + f (~u; ); t > 0; ,1 < x; y < 1; (2.6) @t @x2 @y2

u~ bounded as x; y ! 1; where t and x; y are the temporal and spatial variables, respectively, and u~(x; y; t; ) and f (~u; ) are vector functions u~ = (~u1; u~2; : : : ; u~m); f (~u; ) = (f1 ; f2; : : : ; fm): (2.7) The matrix D is diagonal with nonnegative elements on the diagonal. The parameter  is real. We assume that m  2. We assume that f (0) = 0, so that there is a basic uniform state of u~ = 0. Then one can consider a perturbation u~ = u 6= 0 to this basic state, such as the deviation from a chemical reaction in which the species are kept constant and uniform in space and time, as in the Brusselator model [16]. In the study of a pattern on a reaction front, one can consider u as the deviation from a planar front [7]. The parameter  is a control parameter; for values of  above a critical value c, the basic state u~  0 loses stability to a periodic solution. To determine conditions for the instability of the basic solution and the types of solutions which appear as a result of the instability, we linearize (2.6) about the basic state, that is, for the perturbation u  1. This yields ! @u = D @ 2 u + @ 2 u +  u; (2.8)  @t @x2 @y2 where  is the Jacobian matrix such that ! @f u ;  ) i (~ j u  (2.9) @ u~j u~=0 u ; (i = 1; : : : ; m): where superscripts indicate components of the vector u, and repeated superscripts indicate summation over j = 1; : : : ; m. We substitute u = et+ik1 x+ik2y v; k12 + k22  jkj2; (2.10) 3

into (2.8) to obtain

,jkj2Dv + v = v:

(2.11) Here (k1; k2) and  are the wave vector and growth rate, respectively, of the perturbation u. Equation (2.11) is an eigenvalue problem for the growth rate (k1; k2; ), with corresponding eigenvector v . Without loss of generality, we assume that the critical value is c = 0, and that the following conditions hold for : (1) For  < 0 all eigenvalues  have negative real parts. (2) For  = 0 and jkj = kc 6= 0, there exists an eigenvalue  = 0, and all other eigenvalues  have negative real parts. (3) For a range of  > 0 there is a range of wave numbers k, < jkj < k+ such that for each jkj in this range there exist an eigenvalue (k1; k2; ) > 0 and  < 0 for jkj outside that range. In addition, the real parts of all other eigenvalues are negative. These assumptions indicate that the basic solution, which is stable for  < 0, loses its stability when  passes through the critical value c = 0 via a steady bifurcation. There exists a continuous band of wave numbers of perturbations for the instability region  > 0. The curves k form the neutral stability curve which has a minimum at jkj = kc, which corresponds to the rst unstable mode as  is increased above c. In the following analysis we consider systems near critical, that is  , c = O(p) where   1, p > 0. In studying the patterns in this regime, one can consider any superposition of modes as a solution for u. We are interested in patterns for which the interaction between the modes are strongest, the simplest of these being hexagonal patterns. Evolution equations for the slowly varying modulations or rolls have been used to study the stability of two-dimensional patterns and the propagation of defects in patterns in convection [2] and in reaction diusion kinetics [3]. Here, we derive these amplitude equations for the general setting of (2.6). At the end of this section we compare the form of these amplitude equations with related work. Near criticality (  1), we de ne a small parameter   1  = 2 2 + : : : : (2.12) Also, the solution u is small and we expand as u  u1 + 2 u2 + 3 u3 + : : : : (2.13) We consider the interaction of three modes, by writing u1 = R1e1 v0 + R2e2 v0 + R3e3 v0 + complex conjugates : (2.14) where v0 is the eigenvector in (2.11). The wave vectors of the modes ej are separated by angles of =3, so that p

p

e1 = eik x; e2 = eik (,x=2+ 3y=2) ; e3 = eik (,x=2, 3y=2) : (2.15) We assume the amplitudes Rj are slowly varying in space and time, so that Rj = Rj (X; Y; T;  ) where X = x; Y = y; T = t;  = 2 t: (2.16) c

c

4

c

Typically only one slow time scale is introduced in convection or Swift-Hohenberg-type models. Here the introduction of two time scales is necessary for the systematic derivation of the amplitude equations, as seen below. We expand the nonlinear function f (~u; ) about the basic solution u~ = 0, that is, for small perturbations u

f (~u; )  u + (u; u) + (u; u; u) + (u; u; u; u) + : : : ; where u is de ned in (2.9) and



(

)

2 (u; u)  2!1 @@fu~ij(~@uu~; k ) uj uk u~=0 ) ( 3 1 @ f (~ u ;  )

(u; u; u)  3! @ u~j @i u~k @ u~l uj uk ul ; u~=0

(2.17) (2.18) (2.19)

where repeated superscripts indicate summation. Since  , , and  are in general functions of , we expand as

  0 + 1 + : : :

(2.20)

@  1 = @ ; =0

(2.21)

where 0 = j=0,

and similarly for and . Substituting (2.12),(2.13),(2.17),(2.20) in (2.6), and collecting the coecients of like powers of  yields

"

where

# 2 2 ! @ @ @ O() : L0 u1  @t , D @x2 + @y2 , 0 u1 = 0; O(j ) : L0 uj = rj : j = 2; 3; : : : ;

(2.22)

  r2 = 0(u1; u1) + 2D u1Xx + u1Y y , u1T r3 = 21u1 + 2 0 (u1; u2) + 0(u1; u1; u1) +2D u2Xx + u2Y y + D (u1XX + u1Y Y ) , u2T , u1 :

(2.23)

It is possible to solve for uj , j = 2; 3; : : :, if the right hand side of the equation for uj in (2.22) is orthogonal to the homogeneous solutions (v0e1 , v0e2, v0e3) of the adjoint equation L0 u = 0. This solvability condition is applied with an appropriately de ned inner product. In this case, it is

Z 2=k1 Z 2=k2 0

0

(rj
el v0)dydx  hrj ; el v0i = 0; 5

(2.24)

where k1; k2 are chosen in order to include a period of the integrand in (2.24). We use the normalization hek v0 ; el v0i = kl . These solvability conditions yield equations for the slowly varying amplitudes Rj . We outline this process in some detail to illustrate a particular feature: that the appropriate terms in the amplitude equations are obtained at two dierent orders in the perturbation expansion. (For the sake of clarity, the reader is reminded that when k is used as a subscript it denotes indexing and not wave number). First, we consider the equation obtained at O(2). Applying the solvability condition yields hr2; v0el i = RlT , c0RmRn = 0; (2.25) with (2.26) c0 = h 0(em v0; env0 ); el v0i; where l; m; n = 1; 2; 3 and cyclic permutations give us the three amplitude equations. Of course, these are not the full evolution equations for Rj , j = 1; 2; 3. As is well known from symmetry considerations (see, for example, [1]) in the case when the modulations are temporal only, the correct amplitude equations include cubic terms. In the context of deriving the evolution equations from a perturbation expansion, including cubic terms corresponds to including higher order terms, that is, those obtained from the solvability conditions (2.25) and from the next order [7]. From a dynamical point of view, the cubic terms play the role of stabilizing the hexagonal pattern which bifurcates subcritically from the basic state. This complication is not encountered in the derivation of evolution equations in the context of Rayleigh-Benard convection. In that application, the symmetries are such that the coecient c0 = 0 in (2.25). If the coecient c0 = 0, then it is appropriate to consider modulations on the time scale  only, so that (Rj )T = 0. Then the solvability conditions (2.25) are automatically satis ed, and the evolution equations will be given by the solvability conditions at O(3). This is the procedure used in [2]. In many applications c0 6= 0 ([3], [7]), so that the solvability condition at O(2) is combined with the O(3) solvability condition to obtain the correct evolution equations, as we discuss below. Regardless of the value of c0 , we determine u2 which is used in the higher order equations. We shall do this for the case c0 6= 0, since this is the general case for reaction-diusion equations. The rest of the derivation is straightforward, and is outlined in Appendix A. After solving for u2 (A.3), we substitute in r3 and apply the solvability condition (2.24). This yields solvability conditions of the form

Sj T + Rj  = aRj + c0(S k Rl + Rk S l ) + dRj   + bRj jRj j2 + fRj (jRk j2 + jRl j2) +ih1 (Rk Rl ) + ih2 [Rk Rl + Rl Rk ]; (2.27) j j

j

k

l

and j; k; l = 1; 2; 3 with cyclic permutations to obtain the three solvability conditions. Since Rj , j = 1; 2; 3 are the amplitudes corresponding to the homogeneous part of u1 and Sj is the amplitude for the homogeneous part of u2, we combine the solvability condition from O(2) and O(3 ) in an equation for Aj = Rj + Sj for j = 1; 2; 3,

Aj T + Aj  = aAj + (c0 + c1 )Ak Al + dAj   + bAj jAj j2 + fAj (jAk j2 + jAl j2) +ih1 (Ak Al ) + ih2 [(Ak ) Al + (Al ) Ak ] + O(2): (2.28) j j

j

k

6

l

Truncating at O() and writing c0 + c1 = c,@^ = @T + @ , ^b = b, and similarly for a, f; d; h1; and h2 , we obtain (1.2) (after dropping the hats). These equations are written in a form in which the hexagonal symmetries are apparent, using (1.3). These equations describe the dynamics of modulated hexagonal patterns, as discussed further in Section 3.1. In Section 3.2 we use (1.2) to determine the stability of rolls to hexagonal perturbations. To describe instabilities to other perturbations, such as the zigzag instability to perturbations transverse to the rolls, it is necessary to include higher order derivative terms, corresponding to modulations along the rolls. In this case, the term dAj j j in (1.2) is replaced by 2  d @j + 2ik c @2j Aj (see 1.5). This is discussed further in the following section. We comment on the fact that the terms in (2.28) come from O(2) solvability conditions and the O(3) solvability condition, so that there is a factor of  in front of some of the terms. As mentioned above, the cubic terms such as bAj jAj j2 and fAj (jAk j2 + jAl j2) are crucial in describing the stability of hexagons, since these nonlinear interactions stabilize the hexagonal patterns which bifurcate subcritically from the uniform state. These terms are obtained from the solvability conditions at O(3), that is, at the next higher order from (2.25). Therefore, for consistency, we include all terms obtained from the solvability conditions at that order. In the case of c0 6= 0 in (2.25), we obtain the terms ih1 (Ak Al )j and ih2 [(Ak )k Al + (Al )l Ak ] in the amplitude equations. As shown in the following stability analysis and computations, these terms also have an eect on the stability of the patterns. Since there is a coecient of  in the equations (2.28) we discuss how these terms may be balanced. The coecient c0 can take small values, so that all terms may be of the same order for certain physical parameter values. In this case, as in the case when c0 = 0 in (2.25), one can set c0 = c, and allow A to be a function of the slow time  only, not of T . In general for c0 = O(1), one applies the method of [15] to combine contributions at dierent orders, as shown above. In [15] this approach was used to study mean eld eects described by evolution equations which include higher order terms. In the case when the additional higher order terms describe instabilities or stability not determined from the lower order equations, these terms must be included. This is the case in this paper, as well as in [15]. The main dierence between (1.2) and the evolution equations studied in [3] and [10] are the nonlinear terms with coecient hj , the nongradient terms. Envelope equations with nongradient terms have also been derived for Rayleigh-Benard convection models. In the context of nonequilteral hexagonal patterns h1 = 2h2 [2], and in the context of nonBoussinesq eects h1 = 0 [18]. The nongradient terms with coecient h1 have been discussed in [11] in the context of generalized Swift-Hohenberg models with nonlinear gradient terms. Equations of the form (1.2) for general h1 and h2 are obtained in the context of Marangoni convection, as shown in [12] and references therein, and by [13]. In the stability analysis below we consider symmetric perturbations to perfect hexagons and rolls for arbitrary coecients in (1.2). Stability of these patterns to other types of perturbations and the stability of nonequlilateral hexagons has been studied in [14]. In [12] the stability was considered for coecients consistent with Marangoni convection. In the case when h1 = h2 = 0, as in [3] and [10], it is useful to consider Aj = rj eij . With the assumption that the amplitude evolves adiabatically with the phase it is possible to derive phase equations for j describing the hexagonal patterns, thus reducing the complexity of the 7

stability analysis [9]-[10], [17]. When h1 6= 0 and h2 6= 0, there are additional complications in these coupled phase equations so that it is not necessarily advantageous to pursue such an approach.

3 Stability In this section we consider the stability analysis of simple nite amplitude solutions to (1.2). We rst consider the stability of perfect hexagons to long wave perturbations, and then the stability of a single roll to general perturbations. In both cases we show that when h1; h2 are nonzero, there are new stability constraints. The analysis uses (1.2), without transformations which normalize some of the coecients. This allows a comparison with previous analyses, such as [7] and [10], to understand the eects of the additional quadratic terms. 3.1

Stability of hexagons

First we consider the stability of symmetric hexagonal modulations,

Aj = r0 eiqX ;

(3.29)

j

as solutions of (1.2). In the following we nd it useful to de ne the quantity

B = c , h1 q + 2h2q;

(3.30)

The stationary solutions (3.29) exist for r0 and q satisfying

a + Br0 + (b + 2f )r02 = dq2 > 0: That is,

r0 =

q

,B  B 2 , 4(a , dq2)(b + 2f ) 2(b + 2f )

(3.31)

;

(3.32)

where r0 is real for 2 dq2 < a , 4(b B+ 2f ) :

(3.33)

Then one can determine the stability of such solutions for

Aj = (r0 + p^j )eiqX +i ^ : (3.34) Substituting in (1.2) and linearizing about p^j = 0 and ^ j = 0 yields coupled linear equations for p^j and ^ j . With the ansatz that j

j

p^j = pj et+i 1 X +i 2 Y ;

^ j = j et+i 1 X +i 2 Y ; 8

(3.35)

we obtain stability conditions for the regular hexagonal patterns in terms of the eigenvalues  of the matrix given in Appendix B. That is, the regular hexagons are stable to small perturbations for Re() < 0. The perturbations in (3.35) are symmetric in the sense that the three rolls are modi ed by the same wave number ( 1, 2). This corresponds to perturbing the hexagonal pattern as a whole with a long wave modulation. We consider the limit of long wave perturbations, i.e. 1  1 and 2  1. To leading order ( 1 = 2 = 0) the eigenvalues of the matrix are 1;2 3 4 5;6

= = = =

0

(multiplicity 2)

,3Br0

Br0 + 2br02 + 4fr02 ,2Br0 + 2br02 , 2fr02

(multiplicity 2) :

(3.36)

For q = h1 = h2 = 0, the stability conditions are the same as those obtained for perfect hexagons in [7]. The conditions j < 0, j = 3; 4; 5 are shown in the a-q plane in Figure 1. For the case of h1 6= 0 and h2 6= 0, when the additional nongradient nonlinearities play a role in the dynamics, there is a region of instability which is not found for the case of h1 = h2 = 0. 4.5 4 3.5

a

3 2.5

(

2

H)

1.5 1 0.5 0 −3

−2

−1

0

q

1

2

3

Figure 1: Existence and stability of perfect hexagonal patterns (3.29) for c = d = 1, b = ,1, f = ,2, h1 = ,1, h2 = 1. Perfect hexagonal patterns exist above the curve (H). The shaded region corresponds to stability to zero mode perturbations ( 1 = 2 = 0 in (3.35)). For these values of q and a the eigenvalues j < 0, j = 3; 4; 5. To determine all stability conditions, one must determine the O( 12) and O( 22) corrections to the eigenvalues  = 0. Scaling  = O( 12) or  = O( 22) we determine the higher order corrections. For 0 < 1 = 2  1, the conditions for  < 0 are 2 2 2 2 d + 4d q , (h1 + h2) r0 > 0; 5 9

45d + 4d2q2(2br02 , Br0 ) + ((2b , 4f )r02 , 3B )(h21 + h22)r02 + 4dq5(h1 , h2 )r0 , 2h1h2 r02(4br02 , 2fr02 , 3Br0 ) > 0: (3.37) These conditions are shown in Figure 2. In Figure 2a, the stability region is shown for h1 = h2 = 0, the case of Rayleigh-Benard convection or Swift-Hohenberg-type models. In Figure 2b, the stability region is shown when nongradient nonlinearities in uence the pattern dynamics.

4.5 2.5 4 3.5

2

a

a

1.5

H)

(

1

3 2.5

(

2

H)

1.5 1

0.5

0.5 0 −2

−1.5

−1

−0.5

0

0.5

q

1

0 −3

1.5

a

−2

−1

0

q

1

2

3

b

( )

( )

Figure 2: Eect of non-gradient terms on the stability of perfect hexagonal patterns for c = d = 1, b = ,1, f = ,2. Perfect hexagonal patterns exist above the curve (H). a) For values of q and a in the shaded region and h1 = h2 = 0, as in [10], perfect hexagonal patterns are stable to long wave perturbations ( 1  2  1 in (3.35)). b) For values of q and a in the shaded region and h1 = ,1 and h2 = 1, perfect hexagonal patterns are stable to long wave perturbations ( 1  2  1 in (3.35)). 3.2

Stability of rolls

Some of the numerical computations demonstrate that hexagons may lose their stability to rolls. We therefore consider the stability of rolls in the context of (1.2), that is, the stability of rolls to hexagonal perturbations. Proceeding as in the previous analysis, we consider perturbations on a roll A1 = r0 eiqX1 , A2 = A3 = 0, with solutions of the form A1 = (r0 + p^1)eiqX1 10

A2 = p^2 eiqX2 A3 = p^3 eiqX3 :

(3.38)

The roll solutions satisfy a , q2d + br02 = 0, and exist for

a , q2 d > 0: ,b

(3.39)

Writing the perturbations as

p^j = pj et+i 1 X +i 2 Y ; we determine the stability for all 1 and 2 using the notation,

p

(3.40) (3.41)

p

(3.42) ,1 = 1; ,2 = , 1 1 + 3 2; ,3 = , 1 1 , 3 2: 2 2 2 2 The conditions are a > q2d; (Eckhaus stability criterion) 3 ,2A2 + d(,22 + ,23)  2qd(,3 , ,2) > 0; (h2,3 + h1 ,2  B )(h1,3 + h2,2 )  B )r02 + ((A2 , ,23d) + 2qd,3)((A2 , ,22 d) , 2qd,2) > 0; (3.43) where

A2 = ,2(b , f )r02:

(3.44)

Figure 3 shows the region in the a-q plane for which rolls are stable to hexagonal perturbations. We note that most of this region corresponds to q < 0. However, for q < 0 rolls are unstable to transverse perturbations, the so{called zigzag instability. As discussed in Section 2, this instability can be observed using the equations (1.5) instead of (1.2).

4 Burner-stabilized Flames In this section we consider a three{dimensional model of burner-stabilized ames, in which the ame is treated as a two-dimensional surface. Then, two{dimensional patterns of this surface (see 4.59 below) can be studied with the methods discussed in Section 2. The model we employ is a straightforward extension to three dimensions of the two{dimensional models used in [19] and [20], and the details are given in [7]. As discussed in [7] and [20], an analysis based on this model has no restriction on the wavelength of the pattern, in contrast to the analyses of [21] and [22] which consider only long wavelength instabilities. We mention a few of the aspects of this model which are pertinent to the following discussion. The burner is modeled as a heat sink [23], with assumptions of weak thermal 11

a

2.5

2.5

2

2

E)

a

(

1.5

R

( )

1

R

( )

1

0.5

0.5

0 −2

E)

(

1.5

0 −1.5

−1

−0.5

0

0.5

1

1.5

−2

−1.5

−1

−0.5

q

0

0.5

1

1.5

q

a

b

( )

( )

Figure 3: Stability of rolls to hexagonal perturbations. a) Existence and stability of rolls for c = d = 1, b = ,1, f = ,2, h1 = ,1, h2 = 1. Rolls exist above the curve (R). The Eckhaus stability boundary (stability to longitudinal perturbations) is given by (E). In a) the shaded region corresponds to stability to zero mode perturbations ( 1 = 2 = 0 in (3.40)). In b) the shaded region corresponds to stability to hexagonal perturbations ( 1 = 2 = :3 in (3.40)).

12

expansion of the gas and large activation energy. Then the reaction zone is a narrow region of width O(1=Z ), where Z is the (nondimensional) Zeldovich number, which is proportional to the activation energy and typically large in combustion models (Z > 10). For large Z , As Z ! 1, this zone can be approximated by a front whose location is given by

x3 = (t; x1 ; x2 ):

(4.45)

We employ nondimensional cartesian coordinates (x1 ; x2; x3 ), with the burner located at x3 = 0. We also assume that there is a single de cient component of the reaction so that only its mass fraction evolves, while the mass fractions of all other components are so abundant that they can be considered to be constant. The reaction-diusion model is based on perturbations expansions in powers of Z ,1 of the nondimensional temperature T and the mass fraction Y of the de cient component. The Lewis number L, which is the ratio of thermal to mass diusivities, is assumed to be close to unity and the coecient H describing heat loss to the burner is assumed to be small. Then L and H are scaled as (4.46) L = 1 + Z ; H = K Z: Employing a coordinate system which moves with the ame front

z = x3 , (t; x1 ; x2 ); x = x1 ; y = x2 ;

(4.47)

and seeking solutions of the form

  Y = Y0 + Z1 Y1 + o Z1   T =  + Z1 T1 + o Z1 S  T1 + Y1 ;

(4.48)

the model is given by  + Y0  1 (4.49) 8 9 1 = 2 ! ! ! @  + m , @  @  =
 + 0; @S ! 0 as z ! +1 @z  ! 0; S ! 0 as z ! ,1; jj < 1; jS j < 1 as x; y ! 1: 13

(4.51)

Also, assuming that the reaction goes to completion at the ame front, we set Y  0 behind the ame (for z > 0). In (4.50) the parameter m is the ow rate of the fuel, and  is the Dirac delta function. The operator
is the Laplacian in the moving coordinate system. A steady solution of (4.50)-(4.51), termed the basic solution, is given by ( ; z>0 0(z) = 1emz (4.52) ; z < 0; 8 > z>0 < B; ,h < z < 0 S0(z) = > B , mzemz ; : Bem(z+h) , mzemz ; z < ,h; 0 = h; (4.53) where  ,K  1 B = 2 ln(m); h = m ln Bm : (4.54) The basic solution (4.52)-(4.53) represents a stationary planar ame located at the stando distance x3 = h. Since, Y1 = 0 behind the ame, B in (4.54) is the O(1=Z ) correction to the ame temperature. As shown in [7], the stando distance h in (4.54) is a U -shaped function of m, whose minimum approaches zero as K approaches 2=e. We note that the basic solution becomes unbounded as m ! 0. Hence only values of m bounded away from zero are considered. In addition we consider values of the heat loss parameter K > 2=e [19], in order to avoid ashback (h < 0) for certain values of m. The behavior of this basic solution is discussed further in [19]. To determine the linear stability of the basic solution we introduce perturbations of the form  = ,h 0 ; w =  , 0(z) ,  ddz 0 v = S , S0(z) ,  dS (4.55) dz : The linear problem for these perturbations is obtained by substituting (4.55) into (4.50)(4.51) and linearizing about  = w = v = 0, ! " # ! 2 2 2 ! 1 0 ! w ! @ @ @ @ @ w w L v  @t + m @z v , @z2 + @x2 + @y2  1 v = 0;(4.56) for z 6= 0; ,h, with jump conditions which are linear in w, v, and , (see Appendix C) which we denote 0 1  J B@ w CA = 0 (4.57) v and boundary conditions @v ! 0 as z ! +1 w  0 for z > 0; @z w ! 0; v ! 0 as z ! ,1; jwj < 1; jvj < 1 as x; y ! 1: (4.58) 14

In this model we consider ows in the z-direction only, and allow the domain to extend to in nity in both the x and y directions. Thus we expect to obtain two dimensional patterns which do not have a preferred orientation in the x , y plane. We nd that (4.56)-(4.58) has solutions of the form

0 1 0 1 w W B@ v CA = Re!t+ik1 x+ik2y B@ V CA + c:c:;  [W ]0

(4.59)

where k12 + k22 = k2 , R is an arbitrary complex constant, and c.c. denotes complex conjugate. The expressions for W (z) and V (z) and the dispersion relation are in Appendix C. For 0) the basic solution is stable (unstable). The neutral stability boundary corresponding to ! = 0 is shown for selected values of m and K = 1 in [7]. This boundary, given explicitly by 2 ,h  = ,2 , 8 mk 2 + m2K(m e, ) ;

2

= m2 + 4k 2 ;

(4.60)

delineates the region of cellular instability of the basic solution. For admissible values of the ow rate m, the neutral stability curve has a local maximum at (c; jkj = kc), where kc 6= 0 corresponds to the most rapidly growing mode of the perturbation (4.59). We perform a local analysis in a neighborhood of the point (c; kc) seeking stationary non-planar solutions which represent time-independent two-dimensional cellular

ames which bifurcate from the basic solution (4.52)-(4.53). We employ an expansion in terms of , as described in the previous section,

  c(1 + 2 + : : :) w  w1 + 2w2 + 3 w3 + : : :

(4.61) (4.62)

and similarly for v and . Accordingly, we employ the scaled temporal and spatial variables as in (2.16), T = t,  = 2 t,  = x and  = y. The expansion of  implies that the basic solution S0(z) is expanded in  as well. Substituting (4.61)-(4.62) in (4.50)- (4.51) and equating like powers of  we obtain the sequence of equations

L wvjj with jump conditions

!

= rrj1 j2

!

z 6= 0; ,h;

0 0 0 1 BBB j0 j J B@ wj CA = BBBB jj12 vj B@ j3 j4 15

1 CC CC CC CC A

(4.63)

(4.64)

and boundary conditions (4.58). To leading order, (O()) this yields the linear problem (4.56)-(4.58) with r11 = r12 = 10 = 11 = 12 = 13 = 14 = 0. Then

0 1 0 1 p w1 W (z) k c B C B C (4.65) @ v1 A = (R1 e1 + R2 e2 + R3 e3 + c:c:) @ V (z) A ; k1 = , 2 ; k2 = 32kc 1 1=m where W (z) and V (z) are given in (C.8)-(C.9) with , k, and ! replaced by c, kc, and zero, respectively, and e1, e2 and e3 are as in (2.15). Since we perform our analysis in the neighborhood of the local maximum (c; kc) of the neutral stability curve, all other modes decay exponentially in time. At this order the complex coecients Rn(T; ; Y; Z ) (n = 1; 2; 3)

are undetermined. We proceed as in Section 2 to determine the amplitude equations for Rn. The nonzero inhomogeneous terms rj1, rj2, j0, j1, j2, j3, j4 for j = 1; 2; 3 are given in [7] for the case of no spatial modulations (Rj is a function of  only). Here, there are additional terms from the spatial derivatives. The solvability condition is applied, as in (2.24). In this case it is

*

! !+ rj1 ; w = 0; rj 2 v

(4.66)

where the vector (w; v)T is the steady solution to the problem adjoint to (4.56)-(4.58) with  = 0. This adjoint solution is given in [7]. At O(2), we obtain the solvability condition (2.25) with

3 , 8m2 + 3m2  Ke,h Ke,hp (,m2 + 2 ) !  5  c0 =  , m + : (4.67) 4m2 4m2 c The solution for j = 2 is determined as described in Appendix A, where the coecients Bn are 0 1 W2n(z) Bn = B (4.68) @ V2n(z) CA [W2n]0 =m

where W2n and V2n for n = 1; 2; 3; 4 are given in [7] and @V ; ; V = W25 = @@W 25 (kc2 ) @ (kc2 )

(4.69)

For j = 3 we apply the solvability condition to obtain the amplitude equations (2.28). The graphs of the coecients for K = 1 are given in Figures 4a and 4b. Accordingly, the stability region for perfect hexagonal patterns on the reaction front is given in Figure 5 for these parameter values. We note that it is necessary to take c = c0= to account for the fact that the terms in the amplitude equation are obtained from two solvability conditions at dierent orders, as in (2.28). 16

K=1

K=1

0.1

2

0

1.5

−0.1

1

−0.2

0.5

−0.3

0

−0.4

−0.5

−0.5

−1

−0.6

−1.5

−0.7 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−2

1

0.4

0.5

0.6

0.7

0.8

0.9

m

m

a (b) Figure 4: a) The coecients c (dot-dashed), b (dotted), f (dashed), and h2 (solid) for K = 1 in Section 4. b) The coecients d (solid) and h1 (dotted) for K = 1 in Section 4. ( )

17

1

4.5 4 3.5

a

3 2.5

(

2

H)

1.5 1 0.5 0 −3

−2

−1

0

q

1

2

3

Figure 5: Stability region of perfect hexagonal patterns as predicted by the analysis of Section 3 for the model described in Section 4 with m = :7 and K = 1, so that c = ,:287079 (c = c0=,  = :1), b = ,:410856, d = :994014 f = ,:625134, h1 = 1:50953 and h2 = ,:245025. Perfect hexagonal patterns exist above the dotted curve. For values of q and a in the shaded region perfect hexagonal patterns are stable to long wave perturbations ( 1  2  1 in (3.35)).

5 Numerical experiments In this section we present numerical computations of (1.2) and (1.5). These experiments demonstrate the new behavior described by the evolution equations (1.2) in the general model of Section 2 and for the application of Section 4. The method used to solve the amplitude equations is described in detail in Section 6. In the gures we present the actual pattern u1 de ned in (2.14). In doing so, we must choose the critical wavenumber kc discussed in Section 2. For a particular application of the form (2.6) one has to calculate this wavenumber. For our purposes we choose an appropriate kc for the general model and calculate kc for the model of burner stabilized ames. Since the modulations equations (1.2) and (1.5) govern slow modulations of the rolls, kc  qmax where qmax is the highest Fourier mode resolved by our numerical scheme. This is consistent with the form of (2.14), since Rj gives a slowly varying modulation to the pattern with wave number kc. The gures shown are approximately the size of the computational domain in the X; Y plane (recall that the computational domain is hexagonal). The computations validate the stability analysis of Section 4 and demonstrate additional instabilities to asymmetric perturbations occurring for h1 ; h2 6= 0 and parameter ranges for which the pattern is stable to symmetric perturbations. Figure 6 shows the evolution of (1.2) for a case in which the initial rolls would be Eckhaus{ stable if h1 = h2 = 0. According to Figures 3a, ref g3b, these rolls are unstable. This is con rmed by the computation, in which the nal state consists of asymmetric hexagons, that is, hexagons of the form (2.14), for which the modulations Rj are not all equal. In the notation of (6.5) the nal state has most of the energy in b12;0 , b24;0 , b33;1 . Since the fast 18

variation in the gure is due to kc, this asymmetry is not immediately apparent. Figure 7 shows the evolution of (1.2) for a case where the initial hexagons are unstable according to Figure 1. For h1 = h2 = 0 such hexagons would be stable. Here the solution does not seem to reach a uniform nal state, but a state in which a patch of irregularities coexists with a region hexagons. Figure 8 corresponds to the evolution of (1.5) for a roll which is stable according to the calculations performed in Section 3, but with  6= 0 is zig{zag unstable in (1.5). The zig{zag instability is not apparent since there are other asymmetric perturbations that grow faster. If the run is repeated with h1 = h2 = 0 then the zig{zag instability is observed. The nal pattern is composed of perfect hexagons. We have not performed a systematic study of the appearance of asymmetric instabilities as h1 and h2 are varied. For certain values of h1 and h2 the zig-zag instability is still observed. Figure 9 shows the evolution of (1.2) for the model of burner-stabilized ames discussed in Section 4. The coecients in the evolution equations are determined for m = :7 and K = 1. This choice of parameters is representative of a range of values of m and K for which the analytical stability results are similar. In this case a perfect hexagonal pattern, which is stable under symmetric perturbations, is perturbed asymmetrically. The pattern is unstable to this perturbation, and Figure 9 shows the evolution to a stable asymmetric hexagonal pattern.

6 Numerical method The equations (1.2) and (1.5) were solved numerically using spectral dierences in space and nite dierences in time. The method used diers from previous methods in two important ways, which improve accuracy and speed. Since these improvements are applicable to a large class of problems, we explain them in detail here. Consider the equations (1.2). They are usually written in the form @Rj = L (@ ; @ )R + N (R ; R ; R ); (6.1) j X Y j j j k l @ where permutations of j; k; l yield the three amplitude equations, and repeated indices no longer denote summation. For numerical purposes, the linear operators Lj may contain any combination of X; Y derivatives, and the nonlinear term N may also contain terms where Rj ; Rk ; Rl are dierentiated. In integrating (6.1), one is interested in the dynamics of one or a few adjoining hexagons (recall that these equations govern long modulations of some hexagonal solution with wavenumber kc in (2.15)). To simplify the computations, the solution is usually assumed periodic in (X; Y ). Then, to minimize the eects of a rectangular domain on solutions with hexagonal structure, one must make the rectangular computational domain large enough so that the central hexagons are unaected by the boundaries. To overcome this problem we choose to solve the equations with the hexagonal symmetry apparent (1.2,1.5), and assume that the solutions are periodic in (X1; X2; X3). Those

19

(a)

(b)

(c)

(d)

Figure 6: Snapshots of the pattern from a computation of (1.2) with parameters a = 1:5, c = 1:0, b = ,1:0, d = 1:0 f = ,2:0, h1 = ,1:0 and h2 = 1:0. The initial data is a roll q = 0:5 with a small random perturbation. From the analysis in Section 3, the roll is unstable (see Figures 3a,3b). The gures show the contours of u1 in (2.14). For the purposes of the gure we choose kc = 8:0. The nal state consists of asymmetric hexagons. The solution is shown at (a) t=0, (b) t=2.25, (c) t=30, (d) t=90. 20

(a)

(b)

(c)

(d)

Figure 7: Snapshots of the pattern from a computation of (1.2) with parameters a = 2:0, c = 1:0, b = ,1:0, d = 1:0, f = ,2:0, h1 = ,1:0 and h2 = 1:0. The initial data is a perfect hexagon (q = ,0:25) with a small random perturbation. From the analysis in Section 3, the hexagon is unstable (see Figure 2). The gures show the contours of u in (2.14). For the purposes of the gure we choose kc = 6:0. The solution is shown at (a) t=0, (b) t=6, (c) t=10, (d) t=300. 21

(a)

(b)

(c)

(d)

(e)

(f )

Figure 8: Snapshots of the pattern from a computation of (1.5) with parameters a = 2:0, c = 1:0, b = ,1:0, d = 1:0,  = 1:0, f = ,2:0, h1 = ,1:0 and h2 = 1:0. The initial data is a zig{zag unstable roll q = ,0:5 with a small random perturbation. The shading variation in the gures shows the variation of u1 in (2.14). For the purposes of the gure we choose kc = 6:0. The solution is shown at (a) t=0, (b) t=3.1, (c) t=6, (d) t=11.45, (e) t=11.6, (f) t=12.

22

(a)

(b)

(c)

Figure 9: Snapshots of the pattern from a computation of (1.5) for the combustion model of Section 4 with parameter values m = :7 and K = 1, so that kc  3:9. The initial pattern, corresponding to a = 1 and q = ,:5, is stable to hexagonal perturbations, as shown in Figure 5. For suciently large asymmetric perturbations, the pattern destabilizes and evolves to an asymmetric hexagonal pattern. The snapshots are at times t = 0, t = 35, and t = 250. The shading variation corresponds to the variation of u1 in (2.14).

23

equations have the general form @Rj = L(@ ; @ )R + N (R ; R ; R ): (6.2) j j j j k l @ Considering (X1; X2; X3 ) as independent variables, we can solve (6.2) with periodic boundary conditions on (X1 ; X2; X3) (that is, solving the equations on the 3-torus). This increases the dimension for the problem by one, but it is easy to show that it does not increase the computational work for a given resolution in the X; Y plane. The method does, nevertheless preserve the hexagonal symmetry of the problem, and is equivalent to integrating the problem on a hexagonal grid in space. Once a solution in terms of a triple Fourier series for (X1; X2; X3 ) is computed, we map it back to a double Fourier series in (j ; j ), to graph and interpret the results. Recall that (j ; j ) are coordinates parallel and perpendicular to the wave{vectors in (2.15), and thus have clear signi cance as modulations parallel or perpendicular to the rolls. For mode j , and even permutations of j; k; l, we have

p

p

(6.3) Xj = j ; Xk = , 21 j + 23 j ; Xl = , 12 j , 23 j : Thus, from a solution in terms of the Fourier decomposition of Rj in (X1; X2; X3) of the form N X N N X X Rj (X1; X2 ; X3;  ) = ajlmn( )ei(lX1 +mX2 +nX3) ; (6.4) l=,N m=,N n=,N

we can reconstruct the solution in (j ; j ) as

Rj (j ; j ;  ) =

4N 2N X X

p=,4N q=,2N

 p  i 21 pj + 23 qj

bjpq ( )e

:

(6.5)

The coecients bjpq ( ) are found in terms of ajlmn( ) by the relation

bjpq =

N X l=,N

ajl;l+ 21 (q,p);l, 21 (q+p) ;

(6.6)

for q + p even and ,N  l  12 (q  p)  N , and bpq = 0 for q + p odd. We now brie y describe the method used to compute triply periodic solutions to (6.2). We use a method which consists in computing directly the solution of the Fourier transform of (6.2), and which computes the linear part of (6.2) exactly [24]. For nonlinear problems with sti linear parts this spectrally accurate method has the advantage of circumventing the stiness by treating the linear part analytically. By denoting F the Fourier transform operation from (X1; X2 ; X3) to the Fourier variables K = (K1; K2; K3), F ,1 its inverse, and R^j (K;  ) the Fourier transform of Rj , the transform of (6.1) is then dR^j = L^ R^ + F fNg : (6.7) j j d 24

Here, L^j (K) is the Fourier symbol of L in (6.2), that is, for even permutations of j; k; l,

p



!!

 L^j (K) = L i Kj , 21 (Kk + Kl ) ; i 23 (Kk , Kl )

:

(6.8)

The nonlinear term N in (6.7) is computed in real space. If N contains terms where Rj ; Rk ; Rl are dierentiated, these derivatives are rst computed in the Fourier domain before computing N . We now use an integrating factor to eliminate the linear part of (6.7). De ning R^j = S^j (K;  )eL^j  , equation (6.7) becomes

dS^j = e,L^  F fNg : (6.9) d If one considers the initial value problem on a periodic domain (3-torus), and discretizes (X1; X2; X3 ) with N  N  N points, then equation (6.9) becomes a set of N 3 rst order, ordinary dierential equations for the amplitudes of the Fourier modes. Thus, (6.9) can now be integrated in time with a standard nite dierence method such as Runge{Kutta. The greatest advantage of using (6.9), is that the linear part is integrated exactly (clearly, if N = 0, we recover the exact solution to the partial dierential equation (6.1)). Thus, this method has no stability restrictions arising from the linear operators in (6.1). j

7 Conclusion We have derived new evolution equations for hexagonal patterns in general reaction-diusion models (1.2). These equations are obtained using a perturbation expansion in which the new non-gradient terms appear at the same order as the usual cubic terms which stabilize the bifurcating hexagonal pattern. We study the new instabilities described by these nonlinearities using both analytical and computational methods. A stability analysis gives the stability criteria for perfect hexagons and rolls, which are con rmed by the computations. Furthermore, we demonstrate computationally that there are other stable patterns composed of both rolls and hexagons. The numerical method is based on the hexagonal structure of the equations, thus removing boundary eects that can arise from a rectangular grid. We have computed these patterns for both a general reaction-diusion system and a speci c model of burner-stabilized ames. These results show a rich variety of modulated two dimensional patterns described by these evolution equations.

Acknowledgement The authors wish to thank Prof. Bernard J. Matkowsky for bringing this problem to their attention.

25

A Appendix A The solvability condition obtained at O(2) is

hr2; v0el i = RlT , c0RmRn = 0;

(A.1)

Assuming that (A.1) is satis ed, u2 is determined from

Lu2 = u1T , 0 (u1; u1) , 2D(u1Xx + u1Y y ) ,

3 X

l=1

hr2; el v0iel v0 :

(A.2)

The solvability condition obtained at O(2) has been used strategically to add zero to the equation for u2. It is then straightforward to solve for u2 from (A.2). Substituting (2.14) in (2.6), we nd that

u2 = B1[R12 e21 + R22e22 + R32e23 + c:c:] + B2 [R1 R1 + R2 R2 + R2R2 ] + B3[R1 R2 e1e2 + R1 R3e1 e3 + R3R2 e3e2 + c:c:] + B4[R1 R2e1e2 + R1R3 e1e3 + R3 R2e3 e2 + c:c:] +2ikcB5 [R11 e1 + R2 2 e2 + R3 3 e3 + c:c:] +S1e1 v0 + S2 e2v0 + S3 e3v0 + c:c:: (A.3) where we have used the derivative notation of (1.1). The coecients Bj can be obtained explicitly from the equation (A.2) and we perform this computation in Section 4 for the combustion application. The expression for u2 is used in r3 to obtain the solvability condition (2.27).

B Appendix B The stability of hexagonal modulations (3.29) is obtained from considering the system

 = M; (B.1) where  = [p1 ; p2; p3; 1 ; 2; 3 ]T and M is 2 m (, ) m12 m12 m14 (,1) m15 (,1 ; ,2) m15 (,1; ,3) 3 11 1 66 m12 m11 (,2) m12 m15 (,2; ,1) m14 (,2 ) m15 (,2; ,3) 777 66 m m12 m11 (,3 ) m15 (,3; ,1) m15 (,3 ; ,2) m14 (,3) 77 : 12 M = 66 m45 m45 777 66 m41 (,1) m42 (,1; ,2) m42 (,1 ; ,3) m44 (,1) 4 m42 (,2; ,1) m41 (,2) m42 (,2 ; ,3) m45 m44 (,2 ) m45 5 m42 (,3; ,1) m42 (,3; ,2) m41 (,3 ) m45 m45 m44 (,3) (B.2) Here we use p p 1 3 1 (B.3) ,1 = 1; ,2 = , 1 + 2; ,3 = , 1 , 3 2; 2 2 2 2 26

and the entries of M are given by

m11 (,) m12 m14 (,) m15 (,; ) m41 (,) m42 (,; ) m44 (,) m45

= = = = = = = =

a + 3br02 + 2fr02 , d,2 , q2; Br0 + 2fr02; ,2ir0 dq,; ih2 r02 + ih1 r02,; 2idq,=r0; ih2  + ih1 ,; ,Br0 , d,2; ,Br0 :

(B.4)

For h1 = h2 = 0 this yields the same stability results as in [10]. For hj 6= 0, determining the stability results in terms of the coecients is considerably more tedious than in [10], since the matrix M is full.

C Appendix C Denoting [f ]a the jump in f across z = a, [f ]a = f (t; a+; x; y) , f (t; a,; x; y); the jump conditions at z = 0; ,h, written generally in (4.57), are speci cally  = m1 [w]0 [v]0 + [w]0 = 0; " # @w , m[w] + m v(t; 0+; x; y) = 0; 0 @z 0 " # " 2# @v + @w + m[w] = 0; 0 @z 0 @z 0 [v],h = 0; " # @v , Kw(t; ,h; x; y) = 0: @z ,h

(C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7)

The basic mode of (4.50)-(4.51), for which we study the evolution has components

W =

(

0;

,epz ;

z>0 z z>0 < A2 elz ; pz pz V = > Ce + D1 e + D2ze ; ,h < z < 0 : Eepz + D2 zepz ; z < ,h 27

(C.8)

Where,

o n p p = 12 m + m2 + 4! + 4k2 ; l = m , p; 2l ; C = 2l + (l2 , k2) A2 = m m (p , l)2 2l , C; D = (p2 , k2 ) ; E = D + Ceh(p,l): D1 =  + m 2 1 p,l The solution (4.59) is nontrivial if and only if the dispersion relation 2l(l , p)2 + m(l2 , k2) + Km(l , p)e,h(p,l) = 0 is satis ed.

28

(C.9)

(C.10) (C.11)

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[17] R.B. Hoyle, \Long wavelength instabilities of square patterns", Physica D 67 198223,1993. [18] H.R. Brand, \Envelope equations near the onset of hexagonal patterns", Prog. Theor. Phys. Suppl. 99 (1989), 442-449. [19] B.J. Matkowsky and D. Olagunju, \Pulsations in a burner stabilized premixed plane

ame", SIAM J. Appl. Math. 40, 551-562, (1981). [20] D. Olagunju and B.J. Matkowsky, \Burner stabilized cellular ames", SIAM J. Appl. Math. 48 (1990) 645-664. [21] J.D. Buckmaster, \Polyhedral ames - an exercise in bimodal bifurcation analysis," SIAM J. Appl. Math. 44 (1984), 40-55. [22] L. Shtilman and G. Sivashinsky, \On the hexagonal structure of cellular ames," Canadian J. Phys. 68 (1990), 768-771. [23] G.F. Carrier, F.E. Fendell, and W.B. Bush, \Stoichiometry and ameholder eects on a one-dimensional ame", Comb. Sci. Tech. 18, 33-46, (1978). [24] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods in Fluid Mechanics, New York, Springer-Verlag, 1988.

30