Pattern propagation in nonlinear dissipative systems

Physica 14D (1985) 348-364 North-Holland, Amsterdam

PATYERN P R O P A G A T I O N IN NONLINEAR DISSIPATIVE S Y S T E M S E. BEN-JACOB, H. BRAND, G. DEE, L. K R A M E R and J.S. L A N G E R Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA Received 5 April 1984 We discuss the problem of pattern selection in situations where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. Our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known; and we point out that these known properties are consistent with a marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. We then describe a more general interpretation of the marginal-stability hypothesis and, finally, present numerical evidence for its validity from three different pattern-forming models,

1. Introduction

Pattern formation occurs in a wide variety of dissipative systems driven beyond the limits of stability of their spatially homogeneous states. In general, it is not well understood how such systems restabilize to form the often complex structures seen in nature. We are thinking, for example, of the patterns formed in Rayleigh-Brnard convection [1-4], the Taylor instability [4], crystal growth [5], chemical reactions with diffusion [6-8], etc., where the restabilized structures are periodic and are characterized by their wavelengths. Much research in this area has been aimed at finding some stability principle which would select a naturally preferred wavelength from a family of mathematically allowed states of a system [9-15]. No general form of such a principle has yet been found, however, and it seems probable that none exists. In this paper we shall take a different approach by looking, not necessarily at the ultimately restabilized pattern, but at the way in which that pattern might form as a propagating disturbance produced following an abrupt change of the system parameters [16, 17]. We shall consider an initially structureless system which is "quenched" suddenly so that it becomes unstable against pattern-forming deformations. A perturbation which at first is confined to a small region will grow

locally into a well-developed pattern-convective rolls, spatially varying chemical concentrations, e t c . - and this pattern will spread out to cover the rest of the space. As we shall see, the pattern may spread by propagating at a well-defined velocity. We then ask the following questions: How fast does the front of the pattern move into the undisturbed, unstable region? What wavelength of pattern is generated by the moving front? Part of our motive for studying this problem is the idea that patterns observed in nature may sometimes originate in a sequence of events such as that described above. In fact, pattern propagation in a Taylor-Couette column recently has been observed by Ahlers and Cannell [18]. Another major reason for our interest is that this propagation problem has some similarity to the problem of dendritic crystal growth. In a way which has yet to be understood, the front of a propagating pattern behaves like the tip of a dendrite, generating growing sidebranches behind it as it moves [19]. The most striking similarity is that, in all the cases we have studied, the propagation speed and wavelength of the pattern are selected by the same marginal-stability principle that has been successful - but unproven - in predicting dendritic growth rates. The role of instabilities in pattern selection will be a central theme throughout this paper. The scheme of this paper is as follows. In section 2, we start by examining a relatively simple

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E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipative systems

nonlinear diffusion equation of the form:

8,u = O~u + f ( u ) ,

(1.1)

wlaere f ( u ) is chosen so that (1.1) has a stable, uniform, stationary state at, say, u = 1 and an unstable state at u = 0. This equation has a long history in the mathematical and biological literature [20-23]. It is well-known to possess a family of solutions in which the state u---0 is converted to u = 1 at moving fronts. Strictly speaking, this is not a pattern forming model because the state which emerges is uniform, u --- 1, rather than oscillatory. But other crucial elements of the selection problem are present, and the model is simple enough to permit rigorous analysis. In particular, the selection mechanism is equivalent to a marginal-stability principle. A further reason for studying (1.1) is that it is related to the "amplitude equation": 8 , v = a~v + v - Ivl2v.

(1.2)

Here, the complex function v can be interpreted as the envelope of an oscillating wave form arising, in the near-threshold limit, from any one of a large class of true pattern-forming models. In section 3 we show that (1.2), like (1.1), possesses a family of propagating solutions; and we argue that (1.2) is close enough mathematically to (1.1) that the same selection mechanism must be operative. This suggests the possibility that a marginal-stability principle may be valid also away from threshold for the pattern-forming models. A mathematical interpretation of this conjecture is described in section 4. Supporting numerical evidence for several cases is presented in section 5. The simplest of these cases is the Hohenberg-Swift-Pomeau-Manneville (HSPM) equation [9, 14, 24] OtW=

[E--( 0 2 + 1)2]w - w 3,

(1.3)

which can be shown to reduce to (1.2) in the limit of small e. We also present analogous results for coupled reaction-diffusion equations and for a third model [19] of interest in dendrite theory.

349

The results of our various analyses and numerical studies can be summarized by several conjectures. Each of these conjectures pertains to equations of the form (1.1)-(1.3) where (0), i.e. u, v, or w = 0, is a linearly unstable stationary solution, and where there exist linearly stable stationary solutions (S) with u, v, or w ~ 0. We consider the evolution of such a system in which the unstable state (0) is perturbed initially (t = 0) by a small function which vanishes, or perhaps decays sufficiently rapidly, for x > 0. Then:

Conjecture 1. There exists a natural propagation speed c* such that, if an observer moves with velocity c, the subsequent evolution is seen to approach (0) as t ~ oo if c > c*, but not to do so if c < c*. Conjecture 2. There exists a naturally selected pattern (S*), which is one of the stable stationary solutions (S), and which will be seen as t ~ oo by an observer who is not moving. (This observer, however, must be located at large enough x so that the pattern he sees is not affected by transients that may persist near the initial perturbation.) Conjecture 3. The naturally propagating mode (c*, S *) is marginally stable against localized perturbations, that is, against perturbations which decay sufficiently fast as x --, + o0. In the case of eq. (1.1), this marginal stability conjecture is equivalent to the statement that c* is the slowest stable propagation speed. The principle of slowest stable speed is also valid for eqs. (1.2) and (1.3) provided that one excludes from consideration certain slowly propagating solutions which are, for topological reasons, inaccessible from the stated initial conditions. 2. The nonlinear diffusion equation The mathematical properties of equations of the form (1.1) have been completely and carefully described in the literature, perhaps most definitively by Aronson and Weinberger [22]. In this section we shall restate some of those properties in

E. Ben-Jacob et a L / Pattern propagation in nonlinear dissipative systems

350

with the boundary conditions lim -¢-.._..~o

r~ ~

x ~

u

(a)

(b}

Fig. 1. a) A schematic form for the function f ( u ) in eq. (1.1); b) the corresponding potential q~(u).

terms which are more physically intuitive than they are rigorous, and then present a linear stability analysis which we believe to be new. The linear analysis is not so powerful as the fully nonlinear methods that have been used for the study of this particular equation; but it has the advantage of being generalizable to more complex situations. The appendix to this paper contains a summary of the strategy used by Aronson and Weinberger in obtaining their results. We include this because we think it is important to understand both the power and the limitations of these methods while thinking about possible generalizations. The function f ( u ) o n the right-hand side of (1.1) can conveniently be chosen to have the form shown in fig. la. Here, f is smooth and positive between the stationary points at u = 0 and 1. In our figure, another stable stationary point occurs at u = - b . With no loss of generality, we can assume that the variables have been scaled so that d f / d u = 1 at u = 0. The kinds of propagating solutions that are of interest to us are shown schematically in fig. 2. These are shape-preserving solutions which move with constant speed c; that is, in a frame of reference moving at speed c in the + x-direction (x ---}x - ct), they are solutions of (2.1)

0 = a2uc+ c O x u c + f ( u c ) ,

(o1

(b)

(C)

Fig. 2. The schematic form of the propagating wave fronts for velocities a) c > c*; b) c _< c* (one "overshoot"); and c) c < c*.

uc(x)=l;

--~

lim u c ( x ) = 0 .

(2.2)

x ~ - - ~

There exists not just one, but a range of values of c for which (2.1) has solutions of this kind. To see this, note that (2.1) can be interpreted as the mechanical equation of motion for a particle of unit mass whose "displacement" uc is a ftinction of " t i m e " x. The particle is undergoing damped motion with damping constant c in a potential • (u,.) such that d~ f ( u c ) = d u C,

(2.3)

as shown in fig. lb. The trajectories shown in fig. 2 are those in which the particle starts at u = 1 with zero kinetic energy, falls to the left, and finally comes to rest at 0. Any damping constant c which is large enough to prevent the particle from escaping over the barrier at - b is a mathematically possible steady-state speed for the diffusion front. In the next paragraphs, we show that these steady-state solutions are linearly stable at all apeeds c greater than some c*, stability being defined in a technical sense which we shall discuss shortly. Thus, we are faced with a nontrivial selection problem: which of these modes of propagation would we observe in a real system? This selection problem for eq. (1.1) has been solved completely by Aronson and Weinberger. Their result, which is described in more detail in the appendix, is that the natural propagation speed is the smallest c for which the function uc(x) remains everywhere non-negative. The term "natural" was defined somewhat formally in the introduction. At a more intuitive level, it may be understood as referring to the mode of propagation which is achieved by essentially all physically attainable initial states of the system. Specifically, we may consider perturbations which initiaUy are restricted to the left-hand side of the system (x < 0) and ask how they propagate to the right. If all such initial states eventually produce

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipative systems

the same mode of propagation, we call that the " n a t u r a l " mode. The existence of such a mode, itself, is a nontrivial result of the exact analysis. In the language of the mechanical analog suggested by figs. 1 and 2, the natural mode corresponds to the smallest damping constant c such that the representative particle approaches its rest point u = 0 without overshooting. The main result of the stability analysis which follows is that this direct solution of the selection problem by Aronson and Weinberger turns out to be precisely consistent, for reasons which we do not fully understand, with the marginal-stability hypothesis. The natural speed turns out to be the marginally stable c*. The concept of stability is being used here in the same way that it has been used in the dendrite theory, which is a slightly different interpretation than is usual. The states of interest to us are ones in which the system, as observed in the laboratory frame of reference, is unstable over a semi-infinite part of the x-axis. Ordinarily one thinks of stability as being an absolute property of a system, unaffected by the frame of reference in which the system is being observed; and in that sense these states are always unstabl e . The interpretation that is useful here, on the other hand, pertains to observations made in the frame of reference which moves at the speed of the propagating front. An initially localized perturbation, observed at a fixed point in that frame, will either grow or decay. A perturbation which decays is considered stable even if it generates a growing disturbance, like a dendritic sidebranch, which moves away from the point of observation. Now let us see how this interpretation of stability works out in detail for the nonlinear diffusion equation. Consider a specific propagating state u c, and introduce a small deformation ul:

u(x, t) = uc(x ) + ul(x )e -~t,

(2.4)

where the variable x denotes position in the frame moving at speed c. The linearized equation for u~ is = -

+

+ re(x)]

Ul,

(2.5)

351

with

Fc(x ) -f'(uc).

(2.6)

(The prime means differentiation with respect to the argument u.) Because of the way we have selected the steady-state solutions u c and our normalization of f , we have lim X---~ + OO

F~(x)

= f ' ( 0 ) = 1;

(2.7)

I'c(x)

=f'(1) = -y.

(2.8)

and lim X~

--00

Some care must be given to the choice of boundary conditions for u v For the limited purposes of this discussion, we can consider only localized perturbations; thus u~(x) must vanish in the limits x ---, +oo. More than this, we e x c l u d e - o r , at least, consider separately-perturbations which simply shift the system from one to another of the otherwise stable propagating states. This exclusion turns out to be equivalent to a more stringent definition of localization than that stated above. In the limit x ~ +oo, the steady-state solution of (2.1) must have the form Uc(X

) =

A ( c ) e -qlx "~ 9(17) e -q2x,

X ----~ -bOO,

(2.9) where

c(

ql = g -

1] lj2 (2.10)

q2 = ~-+

-

and the coefficients A and B are functions of c which must be determined by the nonlinear terms in f. (Of course, the appearance of B in (2.9) is mathematically meaningful only when A = 0.) A function u 1 which decays less rapidly than the slow term in (2.9), that is, which decays like e x p ( - q ' x ) with q'< qx, would be less localized than the steady-state solution u c itself. In fact, the Aronson-Weinberger analysis tells us that any amount of such a perturbation ultimately leads to a state with speed c ' > c such that q~(c')= q'. In what follows, we adopt the convention that al-

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipative systems

352

lowable perturbations u 1 must decay at least as fast as exp ( - q~x) in the forward direction. A large part of the stability analysis can be accomplished simply by using (2.7) and (2.8) in connection with (2.5) to determine the possible behaviors of the functions ux in the limits x--+ + ~ . First consider x--+ - m and suppose u~ is proportional to exp(Q(-)x]. Eq. (2.5) tells us that

Q ( - ) = Q {+_-) =

C[¢2 ]1/2. ~ +_ --~ + v - ~o

(2.11)

Localization requires Re Q ( - ) > 0, thus we must choose the plus sign in (2.11), and we can write u x ( x ) ~ exp [Q{+-)x],

x ~ - m.

=

Q(++) c [c2 _ = --~+_[~--1-~o

]1/2.

(2.13)

The localization criterion, described in the last paragraph, is R e Q ( 2 ) < Re - ~ - +

-~---1

.

(2.14)

Here there is the possibility that both Q~+÷) and Q{÷) satisfy (2.14). When that happens, we know that an acceptable Ul(X ) can be formed because all solutions of (2.5), including the one which behaves like (2.12), have the asymptotic form

ux ( x ) = a exp [ Q~++)x ] + b exp [ Q(~ )x ] , x -", + ~ ,

u I = e-'X/2q~ ( x ) ,

(2.15)

where a and b are w-dependent quantities whose magnitudes are fixed by the choice of unit amplitude in (2.12). Accordingly, for c > 2, we can have only stable solutions of (2.5) with Re ~ > 0. For c < 2, on the other hand, the bottom of the range of allowable ~'s moves down to ~ = c 2 / 4 1 < 0. Thus the states with c < 2 are unstable. For present purposes, we need not touch on the question of whether the solutions of (2.5) form a com-

(2.16)

so that (2.5) takes the form of a Schrrdinger equation:

(2.12)

Next, for x ~ + ~ , let ux be proportional to exp[Q(+)x]. We find:

Q(+)

plete set of eigenstates of a linear operator; our problem is simply to identify the onset of localized instabilities. We shall, however, discuss a related eigenvalue problem in section 4. The only possibility for instability of states with c > 2 is that the amplitude a(~o) in (2.15) might vanish for some discrete set of negative values of oa. To investigate this possibility, it is useful to write

+

(2.17)

Note, however, that the boundary conditions ~b at x ~ + ~ are not (yet) well posed by the condition that u 1 be localized. In this version of the problem, it is clear that oa = c 2 / 4 - Fc(m ) = c 2 / 4 - 1 locates the bottom of a continuum of solutions. Instability of u c for c > 2 is equivalent to the existence of one or more bound-state (~0 < 0) solutions of (2.17). By differentiating (2.1), we see that (2.17) is exactly satisfied for oa = 0 by the function

+o(X) = e

d,,. dx

(2.18)

If uc(x ) is a monotonically decreasing function, that is, if the fictitious particle moving in the potential • in fig. lb does not overshoot u c = 0 in its approach to equilibrium, then ~o(x) has no nodes and, by the standard analysis, we know we cannot reduce ~0 and find a state which decays exponentially in both limits x--+ + ~ . On the other hand, if u c ( x ) overshoots the region as shown schematically in fig. 2b, then ~k0 does have a node, and it must be possible by choosing 0a < 0 to move that node to infinity and thus satisfy the conditions for a bound state. The condition that uc(x ) be a monotonically decreasing function, like that shown in fig. 2a, is the general criterion for stability of the propagating front for this particular equation. In the lan-

E. Ben-Jacob et aL/ Pattern propagation in nonlinear dissipative svstems

guage of the mechanical analogy, we can distinguish two ways in which instability may occur as we reduce the "damping constant" c [25]. If the trajectory u~(x) does not change sign until c is reduced to its "critical-damping" value, c = c* = 2, then we refer to this situation as "case I". This class of situations is particularly important because the determination of stability requires only an examination of the linearized eq. (2.5) well away from the front. In "case II", overshoot occurs at values of c greater than critical damping. In this situation, one must examine the fully nonlinear behavior of the system in order to determine c*. In both cases, the Aronson-Weinberger result is precisely equivalent to marginal stability; that is, c* is the slowest c for which the front remains stable. To make this classification more clear, we discuss briefly a simple example [26] which shows both cases I and II depending upon the value of a parameter. In accord with the conventions for f l u ) defined at the beginning of this section, we choose =

u(1-u)(b+u),

(2.19)

with b > 0. This cubic form for f permits us to find explicit steady-state solutions u~(x) for certain, special, b-dependent values of the speed c. In particular, for

J

o

f/~"

> b

I

Fig. 3. The velocity of propagating fronts as a function of the parameter b in eq. (2.19). The curve labeled cs is the velocity of the slowly moving "domain wall" solution connecting the stable states u = 1 and u = - b . The other unbroken curve is the velocity c* of the front which connects the stable state u = 1 with the unstable state u = 0. The dashed curve is the velocity c b for b > 1/2.

decaying mode, e x p ( - q x ) , as x --, + ~ . Thus, for b > ½, (2.21) must be the special solution in which the fast mode is changing sign as c decreases through Cb; that is, B(cb) = 0 in (2.9). Because the behavior of uc(x) at large x is dominated by the slow mode, uc itself remains monotonically decreasing, and we have case I with c* = 2. On the other hand, when 0 < b < ½, it is the slow mode that is changing sign; A ( c b ) = 0 in (2.9). Thus Uc(X) can no longer remain monotonic for c < cb, and we have case II with c* = c b. This situation is summarized in fig. 3. Also shown in fig. 3 the velocity of the unique domain wall us(x ) which joins the stable states u = 1 and u = - b . Specifically, Us(X)

c =- c b = (2b) 1/2 + (2b)-1/2,

I/2

353

--

- ~ )

tanla T

(2.23)

(2.20) and

it is easy to check that u ~ ( x ) = ½(1 - t a n h - ~ ) ,

1-b c s = ( )-2b-1/2.

where ql, q = ~q2,

f o r b > ½, forO
(2.24)

(2.21)

(2.22)

and qx and q2 are given by (2.10). The important feature of (2.21) is that it contains only a single

Note that c s is negative for b > 1, which is consistent with the fact that the state u = - b then has the lowest "energy", and therefore must be the one which grows. For b < 1, c s is the smallest speed for which there exists a bounded propagating solution joining u = 1 and u = 0. In the mechanical analog, trajectories with smaller "damping" c would start from rest at u = 1 and have enough

E. Ben-Jacob et a L / Pattern propagation in nonlinear dissipative systems

354

energy to pass over the barrier at u-- - b , thus becoming unbounded and never returning to u = 0. T o explore further the dynamics of cases I and II, we have solved the fully time-dependent eq. (1.1) numerically using f(u) in (2.19). In these numerical experiments we perturbed steady-state solutions uc(x ) for c = c* by adding a small but finite undershoot in which u became negative near the propagating front. For case I situations, b > ½, such a perturbation always initiated an evolution toward a state containing a domain wall connecting u = l and u = - b and a propagating front connecting u = - b and u = 0. In accord with the speeds shown in fig. 3, the front always moved out ahead of the wall in the + x-direction. Thus, case I solutions seem to be unstable against arbitrarily small undershoots. In contrast, case II solutions for b < ½ were found to be stable against such sign-changing perturbations so long as they were sufficiently small and localized.

The first step in our analysis is a demonstration that propagating solutions of (1.2) exist for any speed c and some range of nonvanishing values of k. We are not able to carry out a complete stability analysis for these states; but a "case I" assumption makes it seem likely that the only unstable states with small k are the special ones found previously for which k = 0 (exactly) and 0 < c < 2. Despite the apparent existence of a class of propagating solutions which is even larger than before, we shall argue that it is extremely unlikely that any state other than c = 2, k = 0 can be generated by propagation. The propagating solutions of (1.2) cannot be states in which v(x) is stationary in a moving frame of reference because the x-dependent waveform which is generated must be fixed in the laboratory frame. The states which do propagate are found by transforming to polar coordinates: v = p e i~°

3. The amplitude

(3.3)

and defining the x-dependent winding number

equation

~(x),

A logical first generalization of the nonlinear diffusion eq. (1.1) is the complex amplitude eq. (1.2). As mentioned in the Introduction, (1.2) can be interpreted as the equation of motion for the envelope of near-threshold (0 < e << 1) oscillations governed by the HSPM eq. (1.3). Specifically, one can write

x = axCp,

(3.4)

where o, cp, and K are real functions of x and t. The amplitude equation, in the laboratory frame, becomes the pair of equations o , o = a~p + (1 - ~ 2 ) o - p3

(3.5)

/ ~ \1/2

w(x,t)=[-~)

v(2el/2x, e t ) e i X + c . c . + . . .

and (3.1)

(For details, see ref. 14). Thus (1.2) has some features of a true patternforming system. In particular, it admits periodic, stationary solutions of the form

vk(x ) = (1 - k2) z/2 e 'kx.

(3.2)

Values of k unequal to zero would correspond to wave numbers different from unity in (3.1); and such wave numbers might, in principle, be selected by some mechanism.

Ox( Ox +oxK) 2/¢

(3.6)

It is p and r, but not cp, which can propagate uniformly; that is, we can find solutions of (3.5) and (3.6) for which p and r are functions only of x - ct. As before, we shift notation so that x denotes position in the moving frame, x ~ x - ct. The equations for Pc and Kc are

02p¢+ cOxpc + ( 1 -

K c2 )

p c __ p3 _~ 0

(3.7)

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipative systems

and

+

CK c

= ck0,

(3.8)

.where k o is a constant which appears because we have performed a first integration of (3.6) in deriving (3.8). Looking at (3.8) in the limit x --, - oo, we see that k0 must be the wave number of the state which is being formed by propagation. That is, lim ~c(x) = ko;

X ~ --OO

lim O~(x) = 00,

X ~ --00

(3.9)

and, from (3.7), p~ = 1 - k02. Our problem now is to see whether (3.7) and (3.8) have solutions satisfying (3.9) such that p~(x) vanishes in the limit x ~ + oo. To do this, we write (3.7) and (3.8) in the form of three coupled first-order equations:

3xpc = -q~p~,

(3.10)

Oxq ~ = - c q ~ + 1 - Kz~ + q~ - 0~,

(3.11)

Eq. (3.10) defines a new variable q~(x); (3.11) and (3.12) are direct transcriptions of (3.7) and (3.8) respectively. As already noted, these equations have a fixed point, to be denoted by a superscript (0), at p~o) = Po, ~ o ) = ko ' and q~O)___0. There is also a pair of fixed points, denoted by superscripts ( + ) and ( - ), at 0~ ± ) = 0 and /-!(£-1) x~ ±)-- + ~ 2~ 4 ] 1/2] 1/2

+ c2k2l

)

;

(3.13)

] 1/2~ 1/2

Both of the fixed points ( ± ) are consistent with the required behaviour of o(x, t) in the limit x --, + oo; thus existence of a propagating solution is

355

equivalent to the existence of a connection between (0) and either ( + ) or ( - ) . This connection can be established by examining the flows determined by (3.10)-(3.12) in the neighborhoods of these fixed points in the three-dimensional space of the variables Oc, re, and qc. Linearization of (3.10)-(3.12) at a fixed point yields a system of equations of the form:

Ox =A ,

(3.15)

where ~ is the vector made up of components Pc - P~*), rc - x~*), and qc - q~*), in that order; the superscript (*) denotes ( + ) or (0); and A (*) is the matrix

A(*) =

- q(*),

0,

-p(*)

0,

2q~(*) - c,

2x(~*)

- 2p~(*),

- 2K~('),

2q~(*) - c

(3.16)

At the fixed point (0), detA (°) = 2 c ( 1 - ko2) > 0 for ko2 < 1, thus at least one eigenvalue of A (°) must be real and positive, and at least one trajectory must flow out from this point. A more detailed analysis indicates that there is only one such outward flowing trajectory under most conditions; that is, the unstable manifold at this point is one dimensional. At ( ± ) , the eigenvalues of A ( ± ) are - q c ~±) and a complex conjugate pair for which the real part is 2 q ~ ± ) - c . The fixed point ( + ) is therefore completely stable in the sense that all trajectories in its neighborhood spiral into it. We know from the work of Aronson and Weinberger that a connection between (0) and ( + ) exists in the case k o = 0, c > 2; and the above analysis tells us that the attractive nature of this fixed point persists as we move to nonvanishing k 0. Thus there must exist a continuous family of propagating solutions with finite winding numbers go(x). (We do not know whether such solutions exist for all/,o2 < 1.) The linear stability of these solutions can be tested by an asymptotic analysis analogous to that described in section 2, eqs. (2.11)-(2.15), but we

356

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipative systems

have not yet been able to find an analog of the node-counting argument that might allow us to eliminate the possibility of discrete, "case II" instabilities. The linearized forms of (3.5) and (3.6), in the moving frame, are __ 6 0 0 1

=

2 8xPl + COxP1 q- ( 1

-- K c2 --

302)pl

and then let both Pl and "1 be proportional to exp [Q(+)x]. The resulting quadratic equation for Q(+) is [(Q(+)

C,2

[_(+)__¢

+ 4(,{.+))2[Q(+)

-- 2G."c"1,

2

(3.22)

+ q ~ . + ) ] 2 = O,

(3.17) which has solutions

- oa. t =

0~. 1 + cO.,., 1

2

Q(~)

2,,,

+o, 7.(o,.o,.),1 +770,.0,

2G,.O

,

]

,o,.)ol . (3.18)

In the limit x ---) - oc, both Pl and "1 behave like exp[Q(-)x]. The four possible values of Q(-) at fixed oJ are determined by --c2 +~o-202o]

×

Q( )+~

--~+oa +4k~Q ( ) - = 0 . (3.19)

For our purposes, it will be sufficient to look at small values of k 0, in which case the solutions of (3.19) are determined to a zeroth approximation by the vanishing of one or the other of the quantities in square brackets on the left-hand side. Inspection of the approximate solutions to order k02 indicates that there can be at most two roots Q ( - ) which satisfy the localization requirement, Re Q ( - ) > 0. This implies that there must be at least three decaying solutions of (3.17) and (3.18) in the limit x--* + ~ in order to guarantee the existence of a solution at all x. Analysis of (3.17) and (3.18) at x--) + ~ is slightly complicated by the fact that pc(x) introduces an x-dependence which persists in this limit. Specifically,

&.(x)=Acexp[-q~+)x],

x~ +~.

(3.20)

To eliminate these exponential terms, we write 1

•l(x) = ~ T e x p [ +q(+)x]~l(x),

(3.21)

C

2

-~+ i (+)

c

2-

i "n.(c+ ) ±!

a (+)

[(2

- - a(+) -*c "I-

1/2

x(,*)

i"~+))2 - -

6)

(3.23) The localization condition, Re Q(+)< -q~+), can be satisfied for more than two of these solutions only if Re ~0 > 0. Thus, there appear to be no instabilities. Note that in none of the above discussion were we restricted in our choice of the speed c. There are, as we saw previously, unstable propagating states for c < 2 and k 0 = 0; but these states are not obtained by taking the limit k 0 --* 0 in the present analysis. Specifically, (3.13) implies that, for c < 2, ,~+) --* (1 C2/4) 1/2 as k 0 --* 0. This is a state for which the complex propagating function G(x, t) is constant behind the moving front but has a rotating phase within the front and in the region of exponential decay ahead of it. Apparently these states are stable, whereas the purely real solutions u c obtained previously for c < 2 are distinctly different and are unstable. (For c > 2, the limit k o --* 0 does recover the real, nonscillatory states found previously.) The obvious question is whether any of these complex propagating states can be generated from physically achievable initial conditions. By "physically achievable", let us agree to mean that the initial state is strictly localized, that is, that v(x, t = 0) vanishes outside of some finite region on the x-axis. It seems highly unlikely that such a function can evolve into a complex propagating state with finite (or even vanishingly small) k 0. To -

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipatioe s.vstems

357

see this, consider the number of phase rotations, N:

4. "Case I" interpretation

N = 2 ~- -r

The general feature of case I pattern selection is that the natural propagating mode is determined solely by the linear dynamics of the unstable state into which the pattern is moving. This is obviously a very special situation; in section 2 we already have seen a case II example where this hypothesis fails. But such a selection criterion would be extremely useful in dealing with complex systems where nonlinear analysis is impractical; thus we should like to have some understanding about when such a criterion might be valid and how it might be applied most simply. In the next paragraphs, we shall present an equivalent interpretation of case I marginal stability which provides a convenient computational procedure and which also sheds a little light on the general question of applicability. As will be seen, we are far from having a complete answer to the latter question. As introduced in section 2, the crucial concept is stability in the moving frame. Consider the following interpretation. Suppose we start with a small, localized perturbation of a uniform, unstable state of a model system such as the HSPM equation, (1.3). As illustrated schematically in fig. 4a, this perturbation will grow in magnitude and extent in accord with the underlying instability. (The functions denoted W shown in fig. 4 may be visualized either as the actual waveforms or as the envelopes of oscillating patterns.) Figs. 4b and 4c show the same sequence of functions in frames of reference moving at speeds c > c* and c = c* respectively. If c > c*, and if we observe the system at a point x = x o + ct which is stationary in the moving frame, then W ( x o + c t , t ) will be a decreasing function of t. That is, we can achieve a certain kind of stability by moving fast enough to outrun the perturbation. The case I marginal stability hypothesis states that the natural c* is the speed of the moving frame of reference in which the leading edge of a perturbation neither grows or decays. This is illustrated in fig. 4c, where the dashed line indicates the additional guess that the

Kd x .

(3.24)

Strictly speaking, x is not defined where v = 0. However, because of the parabolic nature of the equation of motion (1.2), v must be smoothly varying everywhere after an arbitrarily short time interval, and x must be defined everywhere except possibly at a discrete set of "phase-slip" points where v passes through zero. This diffusive smoothing of the initial o cannot generate an infinite number of such points. Thus it seems safe to assume that the initial N is finite. Moreover, dN = l_~~ j 0x(_2Kq+0xK)dx; dt 21r _ ~ q -= - 0 x In O;

(3.25)

so that N can change only during phase-slips where q becomes undefined. The propagating states v,,, on the other hand, have indefinitely large values of N because x ~ x~+):~ 0 as x ---, + ~ . If we insert a propagating state into (3.25), we find dN c 1 ~+)~+)_ dt = - ~ - x c qc - -

1 (x~+,_k0)" 2---~

(3.26)

If k o is positive (negative), then the right-hand side of (3.26) is negative (positive). Thus, propagation is associated with a steady reduction of fN I. On the other hand, propagation requires the existence of an indefinitely large number of phase loops ahead of the front which are steadily unwound at x ~ + ~ as the front moves forward. It is hard to imagine how such an array of loops can be generated during the formation of the front by any initial state with finite N, especially if the steady-state motion is such that N is decreasing. Our conclusion is that the natural propagating solution of the complex amplitude eq. (1.2) is the same as that of the nonlinear diffusion eq. (1.1) with f = u - u 3. The selected propagating state v~ will be identical to u c with c = c* = 2.

marginal

stability:

an

equivalent

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipative systems

358

{o)

(b~

(c)

Fig. 4. Schematic time evolution of a perturbation in a) the laboratory frame of reference; b) a frame of reference moving at velocity c < c*; and c) the frame of reference moving at velocity c = c*.

leading edge of a spreading perturbation may coincide in shape with the leading edge of the naturally propagating pattern. A case II situation would be one in which the front formed in this way would itself be unstable. This consideration leads us to believe that the following calculation produces at least a lower bound to the natural propagation speed. To compute c* explicitly according to the above prescription, start with a Fourier representation of the spreading disturbance in the laboratory frame:

W(x,t)= f dkA(k)exp[ikx-~%(k)t],

(4.1)

where w0(k ) is the linear amplification rate for a perturbation of wave number k, and A(k) is an amplitude which must be assumed to be very slowly varying as a function of k; that is, the initial perturbation is sharply localized so that all Fourier modes are present. Then let x = ct, let t become very large, and write

W(ct, t) .=A(k) exp [ -

w(fc)t],

d ¢°__o= dk ic;

(4.3)

and (4.4) condition,

(4.6)

C*

Note that the interior wave number k* is not necessarily the same as Rek(c*), which is the wave number of the small, growing oscillations ahead of the front. At this point, some examples are in order. Consider first the nonlinear diffusion model (1.1) with the normalization adopted in section 2. The dispersion relation is w0(k) = k 2 -

1;

(4.7)

therefore ~c = ic/2, 0~(k) = c2/4 - 1, and c* = 2. The wave number k* vanishes in accord with the fact that no pattern is formed. These results are correct for case I conditions as described in section 2. The HSPM equation, (1.3), provides a much more interesting example. For this case, ~,o ( k ) = - e + ( k 2 - 1) 2.

which

de-

(4.8)

Let ~:1 and ~ca denote the real and imaginary parts of k respectively. Equations (4.3) and (4.5), after a little algebra, reduce to ~:2---1[-1

o~(k) - ~o(~C) - i~cc.

Re ~0(~:) = O.

k*= - llmw(c*).

(4.2)

where the complex wave number f¢(c) is determined by a saddle-point condition:

The marginal-stability termines c*, is simply

In pattern-forming situations, w must turn out to be complex, because -Im0~ must be the frequency at which nodes in the waveform are passing the moving observation point. As long as nodes are not being created or destroyed at the front (in the language of the last section, no phase slips are occurring there), this flux of nodes is conserved as it moves back into the fully developed pattern. Because the pattern moves at speed c* in the moving frame, its fundamental wave number must be

+ ( 1 + 6 e ) 1/2] e

k? = I + 3~:z2-- 1 +

...,

c* -- 8k2(1 + 4Xc~) = 4~ 1/z + . . . ,

(4.9) (4.10)

(4.11)

- I m w ( c * ) -- 8X:2(1 + 3~c~)3/2= 4ex/z + , . . . (4.5)

(4.12)

E. Ben-Jacob et al. / Pattern propagation in nonlinear dissipatioe systems

The small-e limits shown here are the results that would be obtained from the amplitude eq. (1.2) via the transformation (3.1). The exact results, for finite e, will be used in connection with the numerical experiments described in section 5. The relation between these-calculations and the more straightforward analysis of the kind performed in sections 2 and 3 suffers from the same uncertainty that accompanies the distinction between case I and case II. The quantities Qt+) occurring in (2.12), (3.22), etc. are playing essentially the same role as the wave numbers ik in this section. In sections 2 and 3, stability of the propagating solutions was tested by looking primarily at the linearized equations in the limit x - , + oo, far ahead of the front; and very little attention had to be paid to the region x ~ - o o where the stable state had formed. It is intuitively satisfactory, then, that the calculation defined by eqs. (4.3)-(4.5) is equivalent to solving a linear eigenvalue equation for perturbations of a uniform unstable state, in a moving frame, with boundary conditions fixed at, say, x = 0 and x = L - - , o o in that frame. The unsatisfactory aspect of the argument is that it is not clear how the boundary conditions are related to the linear stability problem. In particular, it is not clear how the boundary conditions at x = 0 are supposed to be related to the smooth continuation of perturbations into the region x -o - oo. In some sense, the moving front in a case I situation has only the effect of a moving boundary; but just when and how this happens is difficult to determine. The general structure of the eigenvalue problem in the moving frame is t o W = 3 2 o W - c O~,W,

35 9

which looks like (4.4) but is intended to be used in a different way. Specifically, (4.14) is supposed to be solved for a set of wave numbers kl(w) (l = 1, 2, 3, 4 in the case of the HSPM equation), and the eigenvalues o~ are to be determined by insisting that there exist coefficients A~ such that the function W ( x ) = )-~At ex p 0 k , x )

(4.15)

l

fits the boundary conditions at x --- 0 and L. For a fourth-order operator fgo with two homogeneous boundary conditions imposed at each end of the interval (0, L --* oo), it has been shown in ref. 19 that the condition for an eigenvalue is that the roots kt(to ) arrange themselves so that two of them have identical imaginary parts. The minimum value of Re w occurs when the real parts of this pair of roots are equal. Thus, the most unstable eigenvalue occurs when two roots of (4.14) coalesce, which is equivalent to the saddle point condition (4.3). The marginality condition is trivially the same as (4.5).

5. Numerical results

We turn now to some numerical studies. These have been designed to test the validity of case I marginal stability in strongly nonlinear situations well beyond the onset of pattern-forming instabilities. That is, our principal interest in what follows is to check the hypotheses of section 4 in circumstances which are outside the range of validity of an amplitude equation.

(4.13) 5.1. The H S P M model

where I20 is the linear operator with constant coefficients whose Fourier spectrum is ~00(k) and where we are now specifying boundary conditions on W at x = 0 and x = L. The eigenfunctions W must have the form exp(ikx), thus (4.13) produces a relation between to and k: to = too(k) - ick,

(4.14)

Our first example is the HSPM model described by eq. (1.3), for which the marginal-stability predictions are summarized in eqs. (4.9) through (4.12). The control parameter E has been introduced in (1.3) in such a way that the state w = 0 becomes unstable for e > 0. The restabilized stationary solutions of (1.3), for any fixed e in the

E. Ben-Jacobet al./ Patternpropagation in nonlineardissipative systems

360

range 0 _< r < 1, are periodic functions with fundam e n t a l wave n u m b e r s k occurring in bands of finite width in the neighborhood of k - - 1 . (For > 1, there are uniform, stable, stationary solutions w = ___(e- 1) ~/2 which are not particularly interesting for present purposes.) The H S P M e q u a t i o n has the additional feature that there exists a L y a p u n o v function:

F{w}=fdx[~(a~w)Z-(axW) + ½(1 - e)w 2 + ¼w'],

T

r

r~

280

290 X

300

310

W

0

2

270

(5.1)

I hO

Fig. 5. A typical wave front solution for the HSPM model [eq. (1.3)] for e = 0.9.

such that (1.3) is equivalent to O,w

=

-

(5.2)

aF/aw

a n d d F / d t < O. Thus it is interesting to see whether the wavelength selected b y pattern-propagation is the same as the one which minimizes F. In fact, the two wavelengths turn out to be different. W e have integrated (1.3) in an interval 0 < x < L, using an implicit finite-difference scheme based on R i c h t m y e r ' s method, and with b o u n d a r y conditions Oxw = 03w = 0 at x = 0 and x = L. In most of o u r c o m p u t a t i o n s we have used an L of order 600 and mesh spacings Ax of order 10 -t. Our initial conditions have generally been of the simple form w ( x , t = 0)

=ore

_x2 ,

(5.3)

with a << 1; but we also have checked that our results are n o t sensitive to this choice. Starting with a small perturbation like (5.3), we observe a p a t t e r n of oscillations to emerge first near x = 0 a n d then to spread in a propagatory m a n n e r as described in previous parts of this paper. A typical front of the propagating pattern at e = 0.9 is shown in fig. 5. N o t e that the oscillatory part of the pattern on the left is stationary in the l a b o r a t o r y frame, and that new oscillations arise as the envelope of the pattern moves to the right. Fig. 6 shows the local wave n u m b e r k as a function of x for the entire system whose front is s h o w n in fig. 5. N o t e that the wave n u m b e r in the

b o d y of the pattern has settled down accurately to the predicted value k* = 1.076 as determined by eqs. (4.8)-(4.12) and eq. (4.6) for e = 0.9. Ahead of the front, where the amplitude of the wave form is decreasing rapidly, k passes through values of order k 1 = ReTc = 1.176. Also shown in this figure is kmj n --- 0.998, which is the wave n u m b e r of the stationary periodic state wk for which F{ wk) in (5.1) is a minimum. Clearly this m i n i m u m is not selected by pattern propagation. O u r results for the propagation speed c*(e) (actually, c*/4¢~-) are shown in fig. 7. The solid curve is obtained theoretically from (4.11). Accurate measurements of the propagation speed require long c o m p u t e r runs largely because it is necessary to allow enough time for the front of the pattern to b e c o m e well separated from a transient I

]

I

k 1.2

/

i/

I.I

k~

LC kuiN-'~

/

J 1 I00

I 200 x

3OO

400

Fig. 6. The local wave number k as a function of x for the entire system whose front is shown in fig. 5.

361

E. Ben-Jacob et al./ Pattern propagation in nonlinear dissipative systems 1.15

I.IC

C* 4v~" h05

I.O0

I

0.5 IE

I.O

Fig. 7. Comparison of the numericallyobserved velocity(open circle) and the prediction of the marginal stability [eq. (4.11)] for the HSPM model.

which remains near the site of the initial perturbation. In fig. 6, this transient appears near x = 0 as a region of small k which is weakening (k ~ k * ) and spreading diffusively. As shown in fig. 7, our velocity c* at e = 0.9 is consistent with the marginal-stability estimate from (4.11). Similar but somewhat less extensive numerical experiments were performed at six other values of e in the range 0.1 < e < 0.8. The results in all cases were consistent, within our limits of accuracy, with the theoretical predictions for both c* and k*. The case e = 0.9 has been studied and described in greatest detail because it is the most stringent test of the marginal-stability hypothesis.

5.2. R e a c t i o n - d i f f u s i o n equations

=

Dx % ,2U 1 +

a1(1 - U?)U1- bU2,

=

. b

(5.5)

-D2 k2 - a 2

Eq. (4.14) then becomes

( which is a quartic equation for k ( w ) . The condition that two of the four roots of (5.6) coalesce at a point where Re w = 0 is sufficient to determine c* and, via (4.6), k*. The parameters used in preparb 1.3

1.2 I

2

l.I I

1.0

C*

The second model of pattern propagation that we have studied is described by the pair of coupled reaction-diffusion equations: 0,Ua

For values of [bl less than some critical b c which depends on the parameters D t, a t, etc., eqs. (5.4) have a continuous family of linearly stable, periodic solutions [15] which exhibit propagatory behavior. In particular, we have studied propagation of oscillatory patterns into the unstable state U1 = U2 = 0 for two supercritical values of the parameter e = ( b c - [ b l ) / b c as shown in fig. 8. Graphs of U 1 or U2 as functions of x look qualitatively similar to the propagation front for the HSPM model shown in fig. 5. The solid curve in fig. 8 is the marginal-stability prediction for c * ( e ) determined according to procedures described in section 4. That is, we linearize around the unstable state U 1 = U2 = 0 to obtain an analog of (4.13) in which I20 is the 2 × 2 matrix:

(5.4)

8 , U 2 = D 2 0 U2 - a 2 ( 1 + U? ) U2 + bUt.

So far as we know, this system of equations is not derivable from any stationarity principle, and there exists no L y a p u n o v function analogous to (5.1).

I

0

0

I

I 0.1

I

I O.Z

Ibl b~

E -= q:-

Fig. 8. Comparison of the numerically observed velocity (circles) and the prediction of marginal stability [eq. (5.6)] for reaction-diffusion equations [eq. (5.4)].

362

E. Ben-Jacob et al./ Pattern propagation in nonlinear dissipative systems

ing fig. 8 w e r e : D 2 / D 1 = 4, a 1 = 1,a z = 1.2, for which bc = 1.3. As in the HSPM case, agreement with the numerical simulations is well within our numerical accuracy for both c* and k*. Moreover, the wave numbers k* selected by propagation appear to be different from those obtained in ref. 15 by looking at static solutions in a model where e slowly becomes subcritical at the boundaries of the system. The latter selection mechanism coincides with energy-minimization in cases where such a variational principle exists; thus we have another indication of the intrinsically dynamical nature of the propagatory mechanism. We have also studied the reaction-diffusion equation (5.4) in situations where propagation occurs into an unstable oscillatory state. That is, there exist stationary oscillatory solutions of (5.4) which are linearly unstable against Eckhaus-like deformations. An initially localized perturbation of one of these states generally produces a new stable state which spreads by propagation throughout the system. The moving interface which separates the stable and unstable states has interesting dynamical properties which we hope to describe in a separate publication.

8 x V = a 3 V = O at the points x = t and x = t - L , L ~ oo. The idea was that the point x = t looks like the tip of a dendrite growing in the + x-direction at unit speed; and the oscillatory pattern which is generated in the region x < t looks like emerging sidebranches. The selection problem was to predict the value of V at x = t (a rough analog of the radius of curvature of the tip of the dendrite) and the spacing of the "sidebranches." The version of this problem which is analogous to our previous examples is the following. Let V(x,t) be defined over the whole x-axis, let the initial state be V(x,O)= Vo > 0 everywhere except for a localized perturbation near x = 0, and ask: What pattern propagates into the system? At what speed does it propagate? The marginal-stability analysis outlined in section 4 can be applied directly; in fact, the linearization of (5.7) about V0 can be scaled onto the linearized HSPM equation with e = 1. The results are

5.3. Dendrite caricature

k* = (x/ff + 3)(5 - 7~-) 1/2 V~/2 ~ 0.7657Vol/2.

A third model that we should like to mention is one that has some features of interest in the theory of dendritic crystal growth [19]. The model is described by the fourth order nonlinear equation a t V = - V O 2 V ( 1 +aO~V)-a4x V.

(5.7)

The quantity a is required to be positive in order to assure that solutions of (5.7) remain bounded; but a is not the relevant control parameter in this problem. In fact, the mathematical structure of (5.7) is interestingly different from the two previous examples. Note that any V = Vo = constant is a stationary solution of (5.7), and that states with V0 > 0 are unstable. It is V0 which plays the role of a control parameter in this problem. In ref. 19, eq. (5.7) was solved in the region x < t subject to the moving boundary condition:

c* = 2 ( ¢ ~ - - 1)1/2(v0- + 2) Vo3/2= 1.622V03/2

3¢g (5.8) and

(5.9) We have checked these predictions numerically, although not yet with the accuracy that we were able to achieve for the HSPM model, and have found apparent agreement. We also remark that, in ref. 19 where the propagation speed was fixed at c * = l , the value of V at the " t i p " x = t was found to undergo small oscillations in the neighborhood of V0 -- (1.622)-2/3 = 0.724 in agreement with (5.8).

Acknowledgements The authors would like to acknowledge the valuable contributions made to this work by R. Ball,

E. Ben-Jacob et aL/ Pattern propagation in nonlinear dissipative systems

J.P. Eckmann, Y. Gefen, N. Goldenfield, P. Hohenberg, and G. SchiSn. This material is based upon research supported in part by the National Science Foundation under Grant No. PHY7727084, supplemented by funds from the National Aeronautics and Space Administration, and in part by the Department of Energy, Contract No. DEAM03-76F00034.

Appendix A strategy for analysis of the nonlinear diffusion equation Consider the nonlinear diffusion equation, (1.1), in a frame moving at speed c in the +x-direction: d,u =

+ c Oxu + f ( u ) .

(A.1)

As an illustration of the comparison method used by Aronson and Weinberger [22], we shall show that, if c > c*, no initially localized solution of (A.1) can remain nonzero at any fixed position in this frame of reference. More specifically, if 0 < u ( x , 0 ) < l for x_<0 and u ( x , 0 ) = 0 for x > 0 , then u(x, t) must vanish in the limit t --, oo at any fixed x. Therefore, the natural propagation speed cannot be larger than c*. Because fronts with c < c* are linearly unstable, it seems reasonable to conclude that the natural speed must be c*. Aronson and Weinberger complete a rigorous demonstration of this fact by showing that the natural speed also cannot be less than c*. Interested readers should consult the original papers for complete details. The comparison method is based on a wellknown special property of the linear parabolic equation

OtW= 02xW + c OxW + etW,

(A.2)

where a is any constant. If W is everywhere positive at t = 0, it can never become negative anywhere at any later time. This follows simply by noting that a minimum in W can never cross the

363

x-axis because, at such a crossing, W and axW would vanish and a t W = OzW would be positive. The comparison theorem follows directly from this positivity property. Let ua(x, t) and ub(x, t) both be solutions of (A.1), and suppose u,(x, O) > ub(x,O ) for all x. Then u,(x, t ) > ub(x, t) always. To see this, subtract the u b equation from the u~ equation. The result has the form (A.2), with W = u a - u b, near any point where u, approaches Ub; and positivity of W is equivalent to the stated result. As a simple but important application of this theorem, consider the initial condition 0 < u(x,O) < 1. The functions u = 0 and u = 1 are solutions of (A.1); thus, by comparison, u(x, t) must remain in the strip 0 < u(x, t) < 1. Now consider an initial u(x,0) which, as described in the paragraph following (A.1), is nonzero in this strip for x _< 0 and vanished for x > 0. A very useful comparison function is ua(x, t) for which the initial state is:

Ua(X'0)=

1, Uc(X),

x <0, X>0,

(A.3)

where tic(x) is the stationary solution of (A.1) that satisfies h~(x) ~ B e x p ( - q 2 x ) ,

x ~ + oo,

(A.4)

and q2 = c / 2 + ( c 2 / / 4 - 1) 1/2 as in (2.10). Note that ~¢ is asymptotic to the rapidly decaying part of the function u c in (2.9). By suitable choice of B, equivalent to shifting ~c along the x-axis, we can arrange that tic(0)= 1. The situation is illustrated in fig. 9. The fact that 0 ~ c < 0 at x = 0 for c > c* can be verified by a phase-plane analysis or, equivalently, by returning to the mechanical analog; the particle can approach the origin purely in the fast mode if it has a finite speed as it passes u~ = 1. The function ua(x,t ) is nonincreasing as a function of t. Initially, it is stationary everywhere except at x -- 0, where it has a discontinuity in its derivative. In the neighborhood of this point, (A.1) behaves like a diffusion equation; the kink smoothes downward and, after any small time 8,

364

E. Ben-Jacobet al./ Patternpropagation in nonlineardissipativesystems References

Fig. 9, A typical localized perturbation and the appropriate comparison function as is explained in the appendix.

comparison theorem with u~(x, 8 + t) = Ub(X, t) tells us that u~(x, t) m u s t Ua(X , 8)<~ Ua(.X,0 ). The

decrease m o n o t o n i c a l l y toward some stationary s o l u t i o n of (A.1). (A clever a n d powerful result!) I n fact, u~ m u s t decrease to zero, The only n o n zero s t a t i o n a r y solution of (A.1) is u,.(x) as described in (2.9), which, because c > c*, must have a positive, slowly decreasing (in x ) c o m p o n e n t a s y m p t o t i c to A e x p ( - q l x ) . But u~(x, t) is everywhere b o u n d e d above by u~(x,O), which has n o such slowly decreasing exponential tail. It follows that u~ c a n never a p p r o a c h u c a n d therefore must v a n i s h as t b e c o m e s large. T h e last step in the a r g u m e n t should be obvious. If o u r original f u n c t i o n u(x,O) is b o u n d e d above b y u~(x,O), a n d u~(x, t) decreases to zero, then u(x, t) also m u s t decrease to zero. Clearly what is h a p p e n i n g is that the n a t u r a l diffusion front generated b y u(x,O) is slower t h a n c a n d appears to be m o v i n g b a c k w a r d s in the chosen frame of reference. T h e way in which this analysis reproduces o u r earlier definitions of c* should also be obvious. T h e f u n c t i o n u~(x, t) can settle into a n o n zero s t a t i o n a r y state when uc(x ) ceases to have a n e x p o n e n t i a l tail which is long in c o m p a r i s o n with that of ~c(x). This occurs when either qx = q2 at c = c* = 2 (case I) or the a m p l i t u d e A(c) in (2.9) v a n i s h e s at some c* > 2 (case II).

[1] F.H. Busse, in: Hydrodynamic Instabilities and the Transition to Turbulence (Springer, Berlin, 1981). [2] L. Koschmieder, Adv. Chem. Phys, 32 (1975) 109. [3] J.P. Gollub, in: Nonlinear Dynamics and Turbulence, D.D. Joseph and G. Iooss, eds., (Pitman, 1983). [4] H.L. Swinney, in: Hydrodynamic Instabilities and the Transition to Turbulence (Springer, New York, 1981). [5] J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. [6] M. Flicker and J. Ross, J. Chem. Phys. 60 (1974) 3458. [7] D. Feinn, P. Ortoleva, W. Scalf, S. Schmidt and M. Wolff, J. Chem. Phys. 69 (1978) 27. [8] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems (Springer, New York, 1979). [9] Y. Pomeau and P. Manneville,J. Phys. Lett. 40 (1979) 609. [I0] M.C. Cross, P.G. Daniels, P.C. Hohenberg and E.D. Siggia, Phys. Rev, Lett. 45 (1980) 898. [11] Y. Pomeau and P. ManneviUe,J. Phys. 42 (1981) 1067. [12] M.C. Cross and A.C. Newell, preprint. [13] A.C. Newell, in: Proceedings of the U.S.-Japan Conference on Partial Differential Equations, Tokyo, July 1982 (North-Holland, Amsterdam, 1982). [14] M.C. Cross, P.C. Hohenberg, P.G. Daniels and E.D. Siggia, J. Fluid Mech. 127 (1983) 155. [15] L. Kramer, E. Ben-Jacob, H. Brand and M.C. Cross, Phys. Rev. Lett. 49 (1982) 1981. [16] G. Dee and J.S. Langer, Phys. Rev. Lett. 50 (1983) 383. [17] E. Ben-Jacob, H.R. Brand and L. Kramer, unpublished. [18] G. Ahlers and D. Cannell, Phys. Rev. Lett. 50 (1983) 1583. [19] J.S. Langer and H. Mi~ller-Krumbhaar,Phys. Rev. A 27 (1983) 499. [20] R.A. Fisher, Ann. Eugenics 7 (1937) 355. For a more general discussion and other references, see J.D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977). [21] A. Kolmagorov, I. Petrovsky and N. Piscounov, Bull. Univ. Moscow, Ser. Internat., Sec. A 1 (1937) 1. [22] D.G. Aronson and H.F. Weinberger,Adv. Math. 30 (1978) 33. [23] P.S. Hagan, Studies in Appl. Math. 64 (1981) 57. [24] J. Swift and P.C. Hohenberg, Phys. Rev. A 15 (1977) 319. [25] A.N. Stokes, Mathematical Biosciences31 (1976) 307. [26] K.P. Hadeler and F. Rothe, J. Mathematical Biology 2 (1975) 251. E. Magyari, J. Phys. A 15 (1982) L139.