+ A2 (s) X* (s, ~) + X* (s, ~) A3 ~s) + A~ (s) + 8F (s, X* (s, 8)) ds. Thus, letting m ~ ~ in (6'), we obtain the identity t

X* (t, 8) -- X* (0, ~) = I X* (s, ~) A~ (s) X* (s, ~) + A= (s)-X" (s, ~) +'X* (s, 8) A~ (s) + A~ (s) + ~F (s, X* (s, ~)) ds, 0

i.e., sinceTis arbitrary, X*(t, e) satisfies the identity (16) for any t E R. Furthermore, X*(t) is differentiable in t, since the expression on the right of the identity is an integral of a continuous function and is differentiable with respect to its upper limit. Differentiating (16), we obtain

dX* ~, @7~ = X* if, 8) A 1 (0 X* (t, @ + A~ (t) X* ~, ~ + X* (t, ~ A 3 (t) + A~ ~) + 8F (t, X* (t, ~). i.e., X*(t, e) satisfies Eq. (i) as required. The uniqueness of X*(t, s) is obvious. Indeed, if Y*(t, ~), X*(t, e) were solutions of Eq. (i) such that llY*(t, e)ll < i, IIX*(t, e)ll < I, one could show by the arguments proving (15') that [[ X* if, ~ - - Y* (t, 8)II ~< ~ 11x * ~, 8) - - r (t, ~)II. (17)

But s i n c e ~ < 1, i t would f o l l o w f r o m ( 1 7 ) t h a t of the theorem.

X*(t,

~) ~ Y * ( t ,

~),

completing the proof

LITERATURE CITED i.

2. 3.

4.

Yu. A. Mitropol'skii and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973). Yu. A. Mitropol'skii, "On the investigation of an integral manifold for nonlinear equations with variable coefficients," Ukr. Mat. Zh., i, No. 3, 270-279 (1958). A. M. Samoilenko and V. L. Kulik, "On the question of the existence of a Green's function of the problem of an invariant torus," Ukr. Mat. Zh., 27, No. 3, 348-359 (1975). Ya. S. Baris, "Integral manifolds and solutions of the Riccati matrix equation," Dokl. Akad. Nauk UkrSSR, No. I, 7-10 (1986).

INVERSE PROBLEM METHOD FOR BURGERS' EQUATION UDC 517.9

O. Yu. Dinariev and A. B. Mosolov

In this article we prove that Burgers' equation

u, + uux--u=

= 0

(i)

has a representation in the form of a compatibility condition for two linear equations which can be used to solve Eq. (i) by the method of the inverse scattering problem. Let u = u (t, ~6CZ(R2), be a function such that for a fixed t it tends to zero as Ixl + +~ and (t,.) 6 LI(R)! for all t. Define matrices

U=

,

V=

TUx-

0

o

Tu

i~- T u 0

where I is an arbitrary real parameter. Consider a system of linear equations for a vector function ~ = ~ ( t , ~6C2(R2), with values in R2: ~ = U~ :

(2)

V~.

(3)

A compatibility condition for this system is the following equation: U , - - V x + [U, V] = O.

(4)

All-Union Scientific-Research Institute of Natural Gases, Moscow. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 7, pp. 969-971, July, 1989. Original article submitted October 13,'1986. 0041-5995/89/4107-0827512.50

9 1990 Plenum Publishing Corporation

827

The left-hand side of Eq. (4) is identically equal to matrix

:)

(u~ + u u ~ - u=)

i

-

"0

so Eq. (4) is equivalent to Eq. (i). A representation of Eq. (i) in the form of a compatibility condition allows us to use the method of the inverse scattering problem [1-3] when solving Eq. (i). Let us consider the direct scattering problem for Eq. (2). Eq. (2) has the following asymptotic behavior as x § •

, - , - = , (t) o Je

+t~•

Clearly, any solution to

(5)

9

Here

=+

0]{=_],

where scattering coefficients a and b depend on t and X. To determine the scattering coefficients we rewrite Eq. (2) in a simpler form. Carrying out a transformation

,=e

9.

Equation (2) becomes

~;_--

2

/-- e-'~"\2X 0 )*'

=

and asymptote (5) becomes ~-~• With this representation,

calculation of the scattering coefficients is straightforward: a = exp

b= ~

exp

u (t, x) dx

(++; ) ( ; u (t, x) dx

u (t, x) exp

- - -~-

,

)

u (l, y) dy - - i~x dx.

We use the scattering coefficients a = a(t) and b = b(t, ),) t o o b t a i n x): u (t, x) =

U (t, x)

+~

a-~(t)++

,

the

function

u = u(t,

ix +[-~m (t-----7"~ b (t, ;9 ea"d~.

U (t, x) =

I U(t,y)dy

--

We now determine the time dependency of the scattering coefficients. Substituting (5) into (3) we obtain =• (1)~- e-~'tcz• (0), ~• (l)----~• (0), from which we conclude a(1) = a(0), b(l, %) = e-~% (0,~).

Since a does not depend on time, quantity ]0-----S udx

is an integral of Eq. (i).

Func-

tion F(%) ----b(l, %)/b* (t, ~) -~ b(O, l)Tb*(0 , ~) is a generating function for a countable number of other integrals of Eq. (i). We cite two examples of such integrals defined for functions u = u(t, x) which decrease sufficiently quickly as Ix[ ~ +~: I1 =

828

xU (t, x) dx,

U (t, x) = u (t, x) exp

- - -~- ~

,

Thus, Burgers' equation (I) can be solved by the method of the inverse scattering problem: first, a(0) and b(0, ~) are calculated using the Cauchy condition u(0, x) = u0(x), and then u(t, x) is calculated from b(t, k). This method is formally equivalent to first using the C o l e - H o p f substitution [4], which reduces (i) to a linear thermal conductivity equation, and then solving the latter by the Fourier transform method. The fact that there is a U - V pair for Burgers' equation is of interest, since it is a counterexample to the comanonly held view that dissipative partial differential equations cannot be solved by the method of the inverse scattering problem. LITERATURE CITED V. E. Zakharov and A. B. Shabat, "Integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem. I," Funkts. Anal. Prilozh.,

i.

.

.

4.

8, No. 3, 43-53 (1974). V. E. Zakharov and A. B. Shabat, " I n t e g r a t i o n o f n o n l i n e a r e q u a t i o n s of m a t h e m a t i c a l physics by the method of the inverse scattering problem. II," Funkts. Anal. Prilozh., 13, No. 3, 13-22 (1979). V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Problem Method [in Russian], Nauka, Moscow (1980). G. B. Whitham, Linear and Nonlinear Waves [Russian translation], Mir, Moscow (1977).

DYNAMICS OF SYSTEMS WITH A COOPERATIVE EFFECT UDC 517.9

V. A. Dobrynskii

Let us assume that we are studying a dynamical system consisting of a sufficiently large number N of identical components xi, i = i, N and suppose that it is impossible to follow the interaction of the elements it is composed of. It is only known that the dynamics of the N

cooperative variable

x=~

x i is given by the differential equation =

(I )

and E(0) = 0. We have to find out the structure of the interaction of the components x i. It is obvious that this problem is not soluble if additional considerations concerning the nature of the interaction among the components are not used. However, if one assumes a priori that the interaction among the components x i is linear and homogeneous at all levels (that is, when two, three, four or more elements take part simultaneously), then it is not hard to reconstruct the indicated structure, in the process, it is necessary to extend the class of bilinear dynamical systems of Molchanov [i] to include polylinear systems. Definition.

A dynamical system N

dx,/ t = =1 (xi) +

N

N

~=i

/=I

a, (x,, x,) + . . . + /=1

(x,,

. . . . .

xk)

(2)

is called polylinear if the functions al (xi),a2 (xi,x~)..... aM (x~,x~, ...,x~) are linear in all their arguments and are the same for any collection of variables entering into them. Definition. A polylinear function am (x~, x~..... xh) is called typical if the fact that a coefficient of a p-th (I ~ p ~< m) order of linearity term implies that the sum of the coefficients of the terms of this order of linearity is nonzero. Definition. A dynamical system, the right-hand side of which contains only typical polylinear functions is called a typical polylinear system. Institute of Hydrobiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 7, pp. 971-973, July, 1989. Original article submitted November 14, 1986.

0041-5995/89/4107-0829512.50

9 1990 Plenum Publishing Corporation

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