On the variational form of evolution equations

Physica 17D (1985) 215-226 North-Holland, Amsterdam

ON THE VARIATIONAL

FORM OF EVOLUTION

EQUATIONS

Eugene LITINSKY Eliahu Nahayassi

3/5,

Bat-Yam,

Israel

Received 14 February 1985

In this paper a new definition is given for representability in the form of a variational derivative; a particular case of this definition yields the classical expression for the variational derivative. A close connection will be demonstrated between the variational derivative and the exterior differentiation operator d. For the case of functions of one variable one obtains effective computational criteria which solve the “hidden variability” problem. The rigorous mathematical theory is illustrated as applied to the study of a few nonlinear equations of mathematical physics.

1. Introduction There

is no need to explain the importance of studying evolution equations of the type u, = Q
(1.1)

for any function h = h(x) in a suitable function space E over an interval [a, b]. This functional, which is linear in h, depends on u as a parameter, u E E. It can be interpreted as a differential l-form on the infinite-dimensional manifold E. In this case we can define the exterior differentiation operator d on differential forms. (We recall that in the finite-dimensional case

d-f(x)=?

$J,(X)dxi,

(1.2)

I

where f(x)

is a function of x E R”, or an O-form; further,

d(~~(r)dxi)=,~i(~-~)dxiAdxi, i

dxiAdxi= I

I

0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

-dxjAdxi,

(1.2a)

216

E. LAinsky/

On the uariationd form of evolution equurions

where &Wi(x)dxi is a l-form, x E R”.) The main result of this paper runs as follows. Provided Q satisfies certain natural smoothness conditions, there exists a functional J such that Q is the variational derivative of J if and only if d F, (Q) = 0 (identity relative to u in E). In the case of a single space variable, effective formulas will be obtained for the computation of d F,( Q) from which it will follow that the equality dF,, (Q) = 0 (in a suitable space E) is equivalent to the identity

ii

ahK-$o -f$ [h’k’(x)g(x)-dk’(+Wl

dx) =0

for any h(x) and any g(x) in E. Conversely, if d F, (Q) = 0, a functional J(U) such that Q = SJ/h

(I .3)

can be computed

from the formula

J(u)= Jo’UL(Qb)d~. 2. Rigorous statement and proof of the main results Let E be a Banach space of functions on an interval [a, b], with norm IIu II = C;,+ max, E ta, hjJuck)( x)1 (such space is sometimes denoted by D”). Let Q = Q(x,, x1,. . . , x,) be a real-valued function of n + 1 real variables. If Q is continuous (this is the case in the physical applications investigated below), then, for fixed u E E, the formula Fu(Q)h=~aQ(u(x), defines a continuous

(2.1)

u(‘)(x), U(‘)(x),...,u(‘@(x))h(x)dx linear functional

on E, i.e. F,(Q) E E *, where E*

is the dual space of E,

II =

su~,,,,,,-~lF,(QVl. Thus, F,(Q) is an element of E * which depends on u as a parameter in a certain domain U c E, u E U c E; in other words, the map u + F,(Q) generates a map U --*E *. Throughout the sequel, when no confusion can arise, we shall write F,, for F,(Q). If u E E, we denote ir( x) = { u(x), tP( x), u@)(x) ,‘.., u’“‘(x)) E R”+l, and Q( ir)( x) = Q( u( x), u(‘)(x), u(*)(x), . . . , u(“)(x)). A bold face italic letter denotes a vector in R”+‘. As usual, C*(R”+‘) denotes the class of real-valued functions ‘on R”+’ having continuous second partial derivatives. We begin the exposition of the main part of this paper with two remarks. IIF,(Q)

Remark a. Given a differentiable map J: E --) R’ (i.e., a O-form), the derivative map (see section 4) J,’ may be written as J,’ = d J, where d is the exterior differentiation operator (see introduction and [l]). b. The expression for J,’ obtained in lemma 5 of section 3 (below) will be transformed integration by parts: Remark

via

E. LAinsky/ On the vuriationalform of evolution equations

217

We carry over all our reasoning to a subspace EL c E such that the terms outside the integral vanish for all u, h E E,$ For example, E,$ might be the space E, of functions in E that vanish at the endpoints a, b of the interval together with all their derivatives, up to order n - 1 inclusive. On E, we have (2.3) The kernel of this functional is an expression which is customarily called the variational derivative of the functional J(u) = /,“P( G)(x) dx, and denoted by 6J/Su. (This observation is the essential point of remark b.) Combining remarks a and b, we obtain the basis for the new definition presented below. It will be assumed that the map F,: E, --, E;*, u E E,, is given by the formula

4th) = j”Q@)(x)hb)dx

(2.4)

a

for any h E E,, where E, is some subspace of E (determined in practice by imposing boundary conditions). Definition. The kernel Q(a)(x) of the functional F, will be called the variational derivative of J throughout E,, if there exists a map J: E, + R’ such that (d J)(u) = F, for any 1(E E,. In that case we write SJ/Gu = Q(a)(x). If no such J exists, we say that Q is not a variational derivative. (Our new definition is invariant, since this is true of the exterior differentiation invariance property is an advantage.)

operator

d; this

Remark. Of course, the role of the space E, in the above definition may be filled by the space of functions of many variables, u = u(x,, x2,. . . , x,,). However, in this paper (which is essentially an illustration of the use of the definition) we shall confine ourselves to the case n = 1. The following problem is fundamental for use of the concept of variational derivative: Given an expression Q(u(x), u”‘(x), u@(x),..., u’“)(x))= Q(fi)(x), does there exist a map J(u): E, + R’, u E E,, such that Q(~)(X) = SJ/Gu? (Usually, Q(c)(x) is the right-hand side of some evolution equation, u, = Q( fi).) Sometimes it is a simple matter to select a map J such that Q( ir) = SJ/&u. But if this cannot be done, the following problem suggests itself: is it possible that Q( fi) is indeed a variational derivative, but that we simply cannot find a suitable J; in other words, is there a possibility of “hidden variability”? The theorem presented below gives computationally effective necessary and sufficient conditions, given Q( ii)(x), for the existence of a map J such that Q(i() = &J/au. The most important aspect of these conditions is the fact that they are necessary and sufficient, for this precludes the very idea of “hidden variability” for a given Q(n). Theorem A. Let F,: E, --*ET, u E E,, and assume that F,(h) = /,6e(ir)(x)h(x)dx, h E El. Then Q( ii)( x) is a variational derivative if and only if d F, = 0.

where Q E C*(R”+‘),

Proof. Let dFu = 0. Then theorem 1 of section 4 implies that Poincare’s theorem (see [l]) is applicable, and we can thus assert the existence of a map J: El --+R’ such that F, = (d J)(u), u E El, i.e., Q(fi) = &J/au.

218

E. Litinsky/On

the vuriational form of evolution equations

Conversely, let Q(i() be a variational derivative. This means that there exists J: E, --) R’ such that F, = dJ(u), and then dF, = d2J( u) = 0. This completes the proof of theorem A. It will operator It was Hence it

be shown in the remark after lemma 3 of section 4 that F,,!= 52,, where the structure of the G?,,will be specified in lemma 3, (4.9). proved in [l], chapter 3, section 2, that for any g, h E E,, (dF,(h))g = (F:(h))g - (F,(g))h. follows that (d F,( h))g = (i2,h)g - (O,g)h, and

(dF&r))g=j’(

i a

aQ;;($))

[h’*‘(x)g(x)-g’*)(x)h(x)]}dx.

k-0

0)

Note that in the definition and thereafter we used a certain subspace E, of E. Putting E, = E, (where E, is defined in remark b), we obtain the classical definition of the variational derivative, as noted in remark b. The same situation is obtained if E, is the space of periodic functions on [a, b]. On the other hand, if we let E, be, for example, the entire space E, then the formula for what we have called the variational derivative is modified; it is no longer

(2.5) with J(U) = j,bFD(fi)(~)dx, but a different expression. It follows immediately from formula (I) above that d F,, = 0 (identically in U) if and only if, for any u, g, h E E,,

= j(ab

“Q;;/;x)

[h’“‘(x)g(x)-g”‘(x)h(x)])dx=O.

(11)

Thus the validity of (II) for any u, g, h in E, is an equivalent criterion for Q(G) to be a variational derivative. On the other hand, if dF,, = 0, then the formula (III) defines a map J: E, + R’ such that (d J)(u) = Fu. For the proof see [l], chapter 3, section 2. Throughout the above theory we adopted a global approach: the point of interest was whether the kernel Q(ti) is or is not a variational derivative in the entire space E,. However, the theory carries over without change to the case u E UE E,, where U is any open set in E, in which Poincare’s theorem is valid (see [l]).

3. Examples and applications

E. Litinsky/

On the variational form of evolution equations

219

1) Let Q( 6) = u”, n = 0,1,2,. . . . Then w = nu”-‘, and for any g, h E Ea. j’(wg)(x)h(x)dx=j~(wh)(x)g(x)dx.

(3.1)

a

a

Thus, by (II), Q( ii) is a variational derivative. Using (III), we get

J(u)=~*~;+;‘;) dx, Q(G)=& 2) Q(io= U(~)(X),

(3.2)

1,2,3,. . . . Then o = (d/dx)&. conditions in E,, the equality

/

abyh(x)dx

k =

= j”g(x)F

dx

(3.3)

a

viewed as an identity in g and h, is valid only for

/

It is readily seen that, because of the boundary

yhyh(x)dx

= - lbg(x)y

k = 2p,

p =

1,2,3,.

Indeed, if

dx,

k =

2p - 1 we have

(3.4)

and we leave it to the reader to prove that the equality

/ bg(x)$-$&x=f+$dx

-

(3.5)

a

cannot hold in E, as an identity in g and h (elements of E,,). Consequently, if k = 2p - 1, Q(a) is not a variational derivative. Using (III), one sees that when k = 2p the appropriate J( 24) is J(u) = (- l)p/2/,b[u’P’(x)]2dx. 3) Q( ir) = cu(x)u(‘)(x), where c is a constant. Then o = c1((‘) + cu(x) d/d#x, it is convenient to put wi = c@)(x), w2 = cu(x) d/dx. Then obviously, for any g, h E E,,,

j*b,d(x)h(x)dx =jbb,h)(xh+)dx. a u

(3.6)

But

jhb&)+)dx+jbbQ%+$+ix a

identically in g(x), that case

h(x) E E,, as can be verified by taking h = u(x), g= u(x)zP(x),

jb(W2g)(x)h(x)dx-jh(W2h)(x)g(x)dx= a

(3.7)

a

a

-3jhU2(x)[U”‘(x)]*dx<0,

u(x) 3 const. In

(3.7a)

(I

and since We have adopted a global point of view, it suffices to find one pair g, h and one II in E, such that [?w,g)(x)h(x)dx#

j%+h)(x)g(x)dx a

(3.8)

220

E. Litinsky/

On the vuriationalform

of evolution equations

in order to state that Q(u) is not a variational derivative. In so doing one must choose u with an (n + 1)th derivative, so that g = u(x)u(‘)(x) should be n times differentiable; the boundary conditions for g are satisfied. However, it is always possible to choose u in E,, not a constant, n + 1 times differentiable, since such functions are dense in E,. In this case, then Q(ii) is not a variational derivative. 4) Q( ii) = 6u(‘) - uc3); Q(u) is the right-hand side of the Korteweg-de-Vries equation. Then w = 6~“) + 6ud/dx + d3/dx3. An elementary calculation using integration by parts and the boundary conditions in E, yields, for example, A~f~(wg)(x)h(x)dx-lh(wh)(x)g(x)dx=6jhu”’(x)g(x)h(x)dx (1 a +12/Qu(x)g’1’(x)h(x)dx+2/hg’2’(x)h’1)(x)dx. (1 D

(3.9)

In order to prove that Q(i() is not a variational derivative, it will suffice (as in example 3) to find g, h, u E E, such that A f; 0. To this end, we let u be an n + 3 times differentiable function, whose n + 2 first derivatives vanish at the endpoints of [a, b]. Such functions are dense in E,. Next, we put g = u, h = - 2~‘~’ + 18~ - u(l); it is readily seen that h E E,. Putting g = u in the formula for A, we obtain A=~b{18u(x)u’1’(x)-2u’3’(x)}/t(x)dx, a *hence, putting h = 2ut3)(x) + 18u(x)u(‘)(x), A =~h{18u(x)u(1)(x) a

(3.10) we obtain (3.11)

-2u(3)(x)}2dx.

We can set u = (x - a)“(x - b)“p(x), where p(x) is an arbitrary polynomial; then 18u(x)u(‘)(x) 2u())(x) f 0 on [a, b] and A > 0. We have thus proved that Q(i() is not a variational derivative. We note the well-known fact that Q,(a) = 3u2 - u,, is a variational derivative (of the functional J(u) = /,“{ u3( x) + h[ u(i)( x)12} dx). Thus, Q( ic) = (d/dx)( SJ/Su). The theory presented above shows that there is no way to reduce the right-hand side of the Korteweg-de Vries equation to variational form, i.e., to eliminate the operator d/dx acting on SJ/Gu. It is known from the theory of this equation that the very possibility of representing the free term in the form (d/dx)(aJ/&u) is essentially equivalent to the statement that the flow generated by the equation is Hamiltonian: in this context the fact that d/dx cannot be eliminated is quite self-evident. 5) Q( ic) = - uc4)- u(‘) - [ u(‘)12; this is the right-hand side of the Kuramoto-Sivashinsky [7, 81 equation. It is easy to see that if J(u) = /,“{[u(‘)(x)]~/~ - [u(2)(x)]2/2} dx, then SJ/Su = - d2)- d4). Consider Ql(ir)= -[u”‘(x)]~, oi = -2u(‘)(x)d/dx. Al~f~b(~lg)(x)h(X)dx-~b(~1JI)(x)g(X)dx=4~~~n)(x)gn)h(x)dx +2

hu’2’(x)h(x)g(x)dx. I (1

(3.12)

Let u(x) f const, g(x) = 1 and h = u. Then A, = 2iab[u(‘)(x)]* dx < 0. Thus, Q,( ic), and together with it Q( ir), is not a variational derivative, 6)

Q(~)=u”‘(~)+~((1-[u(~)(x)]~)u(~)(x))+au(~).

(3.13)

E. Litinsky/ On the vuriutionalform of evolution equutions

221

This equation describes Benard convection in a nearly isolated liquid layer (see [3, 5, 61). Reworking the expression for Q(C), we get (3.14)

Q(fi)=u(4)(~)+u(2)(~)+a~(~)-3[u(1)(~)]2u(2)(~).

Let

(3.15) then uC4)+ u(‘)+ CYU = 6J,/Su. Let Q,(h) = - ~[u(‘)(x)]~u(~)(x); then

Wl

=

-~u(‘)(x)u~~)(x)-&

~[u”‘(x)]~$

(3.16)

Next,

A, =~b(~lg)(x)~(x)dx-~b(~lh)(x)g(X)dx u

=

(I

-6/bU’“(x)u’2’(x)g”)(x)/t(x)dxU

3/b[u”‘(x)]2g”‘(x)h(x)dx (1

+6/hU(1)(x),(2)(x)h(1)(x)g(x)dx U

(3.17)

+ 3jb[u”‘(x)]‘g(x)h’“‘(x)dx, (1

where we have taken into consideration that -6jb~cl(x),c2’(x)g”‘(x)h(x)dx =;lb[#‘(x)]2gC2)(x)h(x)dx+

(3.18)

3~b[u”‘(x)]2g”‘(x)h”‘(x)dx,

6jhu(“(x)u’“(x)h”‘(x)g(x)dx : -3~“[#‘(x)]‘h”‘(x)g(x)dx-

Hence A, = 0. This Q,(i() is a variational +j,h[rr”)(x)14dx. Thus Q(r)= 6J/i3u, where J(u)=

s”{;u’(x)-

(3.19)

3[b[U”‘(x)]2g’“(x)h(l)(x)dx, derivative. The appropriate

[u(l)(x)]2+ [u(2)(x)]2+ [uyx)14 2

2

J1( u), evaluated

by (III), is

(3.20)

A few words should be said here about the possible use of the fact that the free right term of the evolution equation u, = Q(fi) is a variational derivative, u = u(r, x), t a time parameter. In the case Q( 2) = SJ/Gu, where J(u)(t) = /,6P(fi)(x)dx, we have the equality

Q(a)=

2 (-&)“$-$ k-0

(3.21)

E. Litinsky/On the variutionol form

222

if the space in question is

E,.

ofevolutionequutions

Furthermore,

auqic) a@)‘\~_.

(3.22) This inequality means that J(u) is a suitable basis for construction of a Lyapunov function, provided only that it satisfies certain additional conditions. However, we can at least state that with increasing t the solution u(t, x) goes from one set ux = {u: u E E,, J(u) = X } (in nonrigorous language, this is a “manifold of lower dimension”) to another u+ X, > h. Knowing some additional, purely geometrical properties of the family { ux} (e.g., the system {a,,} is isomorphic in a suitable sense to a system of nested spheres), one can draw far-reaching conclusions about the stability and asymptotic behavior of u( t, x) as a function of time. Qualitatively speaking, representability of Q( fi) as a variational derivative implies that the behavior of u as a function of time is in a certain sense regular. In this context, the non-variation property of the free term in the Kuramoto-Sivashinsky equation, proved above, is-at least-an indirect indication that this equation may be connected with turbulence.

4. Some lemmas To justify the use of Poincare’s theorem in the proof of theorem A in section 1, we must verify that F, is differentiable as a map from E into E*, u E E. This will be done in the present section. By definition (see [l]), the map F, is differentiable at a point u if the following two conditions hold: (i) F,, is continuous at u as a map U + E *, where U is a neighborhood of u in E. (ii) There exists a linear map 52: E * E * such that ” F,+, gE

-F,--Og~l=o(llgll),

E.

Throughout sup, E lJ IIx II < cc.

the sequel, “bounded

set” will mean a set U in a Banach

space such that

Lemma

1. Let Q E C2(R”+i) and let IIII o be a norm in R”+‘. Then, for any bounded set W, WC R”+‘, there exists a constant r( W, Q), depending on W, Q and II II ,,, such that for any u, f~ W:

Q(u+f)-Q(u)=

2 q,.,, i==o

(4.1)

’

where 1~1
o= {uo, ui,...,~,},

f= {fo, fi,...,f,).

To prove this it will suffice to consider the Taylor expansion (4.2)

E. L&sky/

On the vuriotional

form

223

of evolution equutions

Let D denote the matrix with elements (dk

,)

fof) au,au* ,

a'Qb+

=

k,m=0,1,2

,...,

n.

(4.3)

Then rzf *(Of, j), where (*, * ) is the euclidean scaler product in R”+l; the vector and linear operator norm (*, *) will be denoted by II* IIi. We have 1~1< 4 IID II1 llfll f. Any norm inequality II * II I I cIo II * II o, where ci,, is a fixed constant. We now put

Yk.m

def =

a’Qb+

sup

kd

; UE w, fE

au, au,

II * II ,-,

[OJ]

w, t,E

in R”+’ satisfies an

(4.4)

It is clear that 0 s yk,,, < cc. It is readily shown that there exists a constant I’, for which II D II 1 5 To, provided the elements d,,, of D. satisfy the inequality Id,, ml S yk, m where yk, m are constants. Thus 1~1 5

$I-&,‘,

llfll

g.

Putting r( W, Q) z iT,,c&, we complete the proof of the lemma 1.

Lemma

If Q E Cz(Rfl+‘), then F, is a continuous

j,bQ(fi)(x)g(x)dx.

function of U, u E E. Let U be any bounded domain in E, u E U, h E U. Denote W(U) ef { u E R ‘+I: YUE U and

Proof.

k E [a, 61 such that ii(x) = u}. It is easy to see that W(U) is a bounded set if U is bounded; indeed, if UE W(U), then k lUil = t i=O

]U(i)(XJ) I IIu II

I

y

(4.5)

i-o

if any function u E U satisfies the inequality II u Consider an arbitrary g E E, II g II = 1. Then

l(fL+h - F,)(dl

ii+ s JblQ( (I s

1” i 0

k)(x)

II s y.

- Q@)(x)1

“e,‘;!Lx) hck’(x) +

. Idx)l E

dx

dx.

(4.6)

k-0

Note that yk < 00, where

UE

Recall that W(U)

l(K+, -

W(U),

u=

{uo,ul

,...)

u,}

.

(4.7)

is a bounded set, u(x) E W(U) and h(x) E W(U) for any x E [a, b]. We then obtain

F,)(g)1

s(b-a)llhll

i

yk+r(w(u),

Q)llhl12(b-a),

k-0

whence it follows that I;;+,, + I;: as II h

II -

0, proving the lemma 2.

(4.8)

224

E. Litinsky / On the vuriotionul form of evolution equutions

Lemma 3. Let F”E E*, with F”(g)= /,bQ(G)(x)g(x)dx for any g E E, Q E C*(R”+‘). Let Q, be the linear operator from E to E *, i.e., S2,,hE E* for any h E E, defined by the formula

(4.9) for any h, g E E. Then IIF,,,, - F, - S2,h II=u( IIh

II).

Proof. As before, the proof utilizes the Taylor expansion

=

Q@+&)(x)-Q(h)(x)-

2

(4.10)

aQ;;~;x)h’k’(x)

k-0

Again using lemma 1 with

II 1) II o = c~,oluil,

obtain

we

I&+,, - F. - S2,h)gj 5 r( W(U), Q)(a - 6)

(II g II =

1): (4.11)

II h II 2,

where u E E, h E U, with U an arbitrary but foxed bounded neighborhood of u in E and W(V) the set defined in the proof of lemma 2. This inequality completes the proof of lemma 3. Remark. The linear operator 52, is commonly called the derivative F; of the map F,: E --$E* at the point u. As observed previously, it follows from lemma 3 that, for fixed u, 0, is a continuous linear operator from E to E*. Lemma 4. Under the assumptions of lemma 3, &I,,,as a function of u, is a continuous map from E to L( E, E*), where L(E, E*) is the Banach space of linear operators from E to E* with the norm hfzE,

IIL?II=SU~{IIS~~II:

l~h~~=l}=sup{)(JZh)(g)l:

h,geE,

llhll=llgll=l}.

Proof. Let W be a bounded set in R”+l, f, u E W. Proceeding as in the proof of lemma 1, we obtain

aQb+f)

--e

“k

aQb) avk

i a*Q(u+tof) avkav,

m-0

f In’

OSl,ll

(4.12)

and then it follows that there is a constant r( W, Q) for which

aQ(u+f)

--

aQb)

Q)

sr(w

i

avk

If

I

m

9

avk

9

(4.13)

m-0

where f- {fo,fi,...,fn>. Let U be an arbitrary bounded neighborhood of u in E. Let W(U) be the corresponding set in R”+‘, as

On the variational form of evolution equations

E.

225

defined in the proof of lemma 2. Let E being arbitrary and fixed belongs to U, E E U, then, noting that aQ(ir+s)

= aQ(fi+g)

a( dk) + E(k))

(4.14)

cwk

we obtain

G(W(U),Q)(b-U)II~II

Finally, putting

.ilgll

II h II = II g II =

II 52u+e - 51, II s T( W( U),

.IIEII.

(4.15)

1, we see that Q)( b - a) II E II .

(4.16)

This completes the proof. Combining the conclusions of lemmas 2, 3 and 4, we have the following Theorem 1. Let Fu be a map E + E *, u E E. Let F, be defined by a formula F,(h) = j,he(ii)(x)h(x)dx. If Q E C*(R”+*), then F, is a continuously differentiable map or, in other words, the differential l-form F, belongs to the smoothness class C’. Finally, we need Lemma 5. Let J be a map of E into R’, such that J(u)=

j%(P)(x)d u

x

UEE,P=B(X~,X~

,..., x,,)

a real-valued function of n + 1 real variables. Let P E C*(R”+‘). Then the derivative map J,’ is given by the formula

J,I( h) = /“( (1

i

aPa’@;;Jx) hck)( x)) dx

(4.17)

k-0

for any h E E. For the proof we again appeal to the expansion (4.1) of lemma 1:

J(IlJp@+])(x)-P(ii)(x)-

k ap;f;&oxJ _ftk)(x)} dxl

s J.4el dx

k-0

and then, reasoning as in the proof of lemma 3, we reach the desired conciusion.

(4.18)

226

E. Litinsky/On the variutionulform of evolution equations

Acknowledgements

We are deeply indebted to Professor G.I. Sivashinsky for his interest in this paper.

References (11 H. Cartan, Calcul Differentielles. Formes Differentielles (Hermann, Paris, 1967). [2] Y. Pomeau and S. Zaleski, Le Journal de Physique 42 (1981) 515-528. [3] C.J. Chapman and M.R.E. Proctor, J. Fluid Mech. 101 (1980) 755. [4] A. Novick-Cohen and L.A. Segel, Physica 10D (1984) 277-298. [5] M.C. Depassier and E.A. Spiegel, Astron. J. 86 (1981) 496. [6) V.L. Gertsberg and G.I. Sivashinsky, Progr. Theor. Phys. (1981) 1219. [7] Y. Kurarnoto, Progr. Theor. Phys. Suppl. 64 (1978) 346. [8] G.I. Sivashinsky, Acta Astronautica 4 (1977) 1177.