ñòð. 63 |

êîâàíå ðiâíÿííÿ).

The key idea making it possible to solve a linear heat equation

ut = uxx

by the method of separation of variables is reduction of it to two ordinary differential

equations (ODE) with the help of the special ansatz (see, e.g. [1, 2])

u(t, x) = R(t, x)?1 (?1 (t, x)?2 (?2 (t, x)).

Unfortunately, the method of separation of variables can not be applied to nonli-

near second-order partial differential equations. However, some progress is possible

if we apply the anti-reduction procedure. The main idea is the same with the one of

the method of separation of variables. Namely, we look for a solution of a nonlinear

differential equation using the special ansatz reducing it to several equations which

have a less number of independent variables [3]. With application to the nonlinear

heat equation

(1)

ut = [a(u)ux ]x + F (u)

it means that its solution is searched for (1) in the form

(2)

G(t, x, u, ?0 (?), . . . , ?N (?)) = 0,

where ? = ?(t, x) is a new independent variable, ?0 (?), ?1 (?), . . . , ?N (?) are smooth

functions satisfying some system of ODE.

The principal difficulty of the anti-reduction procedure is the proper choice of

the ansatz (2). In the present paper we construct a number of ansatzes reducing

nonlinear heat equations of the form (1) to ODE. These ansatzes are obtained by

using Q-conditional invariance of the equation under study with respect to some Lie–

B?cklund vector field (the definition of Q-conditional invariance with respect to the

a

Lie vector field was suggested in [4]).

Definition. We say that Eq. (1) is Q-conditionally invariant under the Lie–B?cklunda

vector field

? ? ?

+ ··· (3)

Q=? + (Dt ?) + (Dx ?)

?u ?ut ?ux

Äîïîâiäi ÍÀÍ Óêðà¿íè, 1994, ¹ 5, P. 40–43.

252 W.I. Fushchych, R.Z. Zhdanov

if there exist such a finite-order differential operator

X = R0 + R1 Dt + R2 Dx + R3 Dt + · · ·

2

and the function R that the equality

Q(ut ? auxx ? au2 ? F ) = X(ut ? auxx ? au2 ? F ) + R? (4)

?x ?x

holds.

In the above formulae (3), (4) Dt , Dx are total differentiation operators and ?, R,

R0 , R1 , . . ., are functions on t, x, u, ux , uxx , . . ..

Roughly speaking, Eq. (1) is Q-conditionally invariant with respect to the vector

field (3) if the system

Eq. (1),

?(t, x, u, ux , uxx , . . .) = 0

is invariant under the vector field (3) in a usual sense. That is why, to study Q-

conditional invariance of Eq. (1) one can apply the standard infinitesimal algorithm [5].

But the system of determining equations for ? is nonlinear (let us remind that in the

classical Lie approach determining equations are always linear).

We look for conditional symmetry operator (3) with

g(u) ? C 2 (R1 , R1 ).

2

(5)

? = Dxg (u),

Lemma. Eq. (1) is Q-conditionally invariant with respect to the Lie–B?cklund vector

a

field (3), (5) if the functions a(u), F (u), g(u) are given by one of the following for-

mulae:

F (u) = (?1 + ?2 ?(u))(?(u))?1 ,

? ? ? ? (6a)

1) a(u) = ?(u)?(u), g(u) = ?(u);

F (u) = (?1 + ?2 ?(u))(?(u))?1 ,

? ? ? (6b)

2) a(u) = u?(u), g(u) = ?(u);

F (u) = (?1 + ?2 ?(u))(?(u))?1 ,

? ? ? (6c)

3) a(u) = ?(u), g(u) = ?(u).

In (6) ?1 , ?2 are arbitrary constants, ? ? C 3 (R1 , R1 ) is an arbitrary function.

The proof of the lemma is rather tedious, therefore it is omitted. We restrict

ourselves by proving that Eq. (1) with a(u), F (u) from (6a)

ut = [?(u)?(u)ux ]x + (?1 + ?2 ?(u))(?(u))?1

? ? ? (7)

is Q-conditionally invariant with respect to the Lie–B?cklund vector field (3) under

a

...

? 2

? = ?(u)uxx + ? (u)ux .

Consider an over-determined system

ut = (??ux )x + (?1 + ?2 ?)??1 ,

? ??

...

?

? ? ?uxx + ? u2 x = 0.

?

Introducing a new independent variable v = ?(u), we get

? 2

vt = ?(u)?(u)vxx + vvx + ?1 + ?2 v,

vxx = 0

Conditional symmetry and anti-reduction of nonlinear heat equation 253

or, equivalently,

2

vt = vvx + ?1 + ?2 v,

(8)

vxx = 0.

...

?

The Lie–B?cklund vector field Q = (?uxx + ? u2 ) ?u + · · · takes the form

?

a x

? ? ?

? + ··· (9)

Q = vxx + vtxx + vxxx

?v ?vt ?vx

Acting by the operator (9) on the first equation from (8) we get

?

Q(vt ? vvx ? ?1 ? ?2 v) = Dx (vt ? vvx ? ?1 ? ?2 v) + (4vx + 2vvxx )vxx .

2 2 2 2

Hence, it follows that system (9) is Q-conditionally invariant under the Lie–

B?cklund vector field (9).

a

To construct solution invariant under the Lie–B?cklund vector field (3), (5) one

a

has to solve an equation ? ? Dx g(u) = 0. General solution of the above equation

2

reads

(10)

g(u) = ?0 (t) + x?1 (t),

where ?0 , ?1 are arbitrary smooth functions. Replacing in (10) g(u) by ?(u) we get

ansatz for Eq. (7) invariant with respect to the Lie–B?cklund vector field (3) with

a

? (11)

?(u) = ?0 (t) + x?1 (t).

Substitution of (11) into Eq.(7) yields the system of two ODE for ?0 (t), ?1 (t)

?0 = (?2 + ?2 )?0 + ?1 , ?1 = (?2 + ?2 )?1 ,

? ?

1 1

which general solution has the form

?1 ?1

+ (e2?2 t ? 1)?1/2 arctg(e?2?2 t ? 1)1/2 ,

?0 = ?

?2 ?2 (12)

?1 = ?2 e?2 t (1 ? e2?2 t )?1/2 .

1/2

Substituting the obtained formulae into (11) we get the exact solution of the

nonlinear heat equation (7). Since the maximal in Lie’s sense invariance group of

Eq. (7) is the two-parameter group of translations with respect to t, x, solution (11),

(12) can not be obtained by the symmetry reduction procedure. Consequently, it is

essentially new.

In the same way we construct Q-conditionally invariant ansatzes for Eqs. (5), (6b)

and (5), (6c). They are of the form

(13)

?(u) = ?0 (t) + ?1 (t)x.

Substituting (13) into the corresponding nonlinear equations we get the following

systems of ODE:

?0 = ?2 ?0 + ?2 + ?1 ,

? ?1 = ? 2 ?1

?

1

and

?0 = ?2 ?0 + ?1 ,

? ?1 = ? 2 ?1 .

?

254 W.I. Fushchych, R.Z. Zhdanov

Provided the functions a(u), F (u) take a more specific form, it is possible to

construct ansatzes reducing Eq. (1) to three, four and even five ODE. Corresponding

results are listed below

a(u) = ?uk , F (u) = ?1 u + ?2 u1?k , uk = ?0 (t) + ?1 (t)x + ?2 (t)x2 ;

1)

F (u) = ?1 + ?2 e?u , eu = ?0 (t) + ?1 (t)x + ?2 (t)x2 ;

a(u) = ?eu ,

2)

a(u) = ?u?3/2 , F (u) = ?1 u + ?2 u5/2 ,

3)

(14)

u?3/2 = ?0 (t) + ?1 (t)x + ?2 (t)x2 + ?3 (t)x3 ;

a(u) = ?u?4/3 , F (u) = ?1 u + ?2 u7/3 ,

4)

u?4/3 = ?0 (t) + ?1 (t)x + ?2 (t)x2 + ?3 (t)x3 + ?4 (t)x4 .

(the formulae 4 from the above list were obtained by Galaktionov [6]).

Here ?, ?1 , ?2 are arbitrary real constants; ?0 (t), ?1 (t), . . ., ?4 (t) are arbitrary

smooth functions.

It is interesting to note that the cases 1, 2, 4 exhaust all possible nonlinearities

a(u) such that invariance group of Eq. (1) is wider than two-parameter translation

group [7].

Besides the above mentioned cases, we established that Eq. (1) with a(u) = 1,

F (u) = 2 C3 + C2 u ? 1 C1 u3 , Ci ? R1 is Q-conditionally invariant under the Lie–

2

3 3

B?cklund vector field (3) with ? = uxx ? C1 uux ? 1 C3 + C2 u ? 1 C1 u3 . This fact

2

a 3 3

can also be used for antyreduction of the corresponding nonlinear heat equation.

In conclusion let us mention another important point. It is well-known that Eq. (1)

admits the Lie–B?cklund vector field only if it is equivalent to the linear heat equation

a

or to the Burgers equation [8]. Consequently, the conception of conditional invariance

widens essentially our possibilities to use a non-Lie symmetry to solve nonlinear

partial differential equations.

1. Miller W., Symmetry and separation of variables, Massachusetst, Addison–Wesley, 1977, 288 p.

2. Fushchych W.I., Zhdanov R.Z., Revenko I.V., On the new approach to variable separation in the

wave equation with potential, Äîêë. AH Óêðàèíû, 1993, ¹ 1, 27–32.

3. Fushchych W.I., Serov M.I., Repeta V.K., Conditional symmetry, reduction and exact solutions of

nonlinear wave equation, Äîêë. AH Óêðàèíû, 1991, ¹ 5, 29–34.

4. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, D. Reidel Publ.,

1987, 212 p.

5. Ovsyannikov L.V., Group analysis of differential equations, M., Hayêa, 1978, 400 p.

6. Galaktionov V.A., Invariant subspaces and new explicit solutions to evolution equations with

quadratic nonlinearities, Report ¹ AM–91–11, School of Math. University of Bristol, 1991.

7. Dorodnitsyn V.A., Knyazeua I.V., Svirshchevsky S.R., Group properties of the heat equation with

source in the two-and three-dimensional cases, Äèôôåðåíö. ypaâíåíèÿ, 1983, 19, ¹ 9, 1215–1223.

8. Ibragimov N.Kh., Transformation groups in mathematical physics, M., Hayêa, 1983, 280 p.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 255–266.

On linear and non-linear representations

?

of the generalized Poincare groups

in the class of Lie vector fields

W.I. FUSHCHYCH, R.Z. ZHDANOV, V.I. LAHNO

We study representations of the generalized Poincar? group and its extensions in

e

the class of Lie vector fields acting in a space of n + m independent and one

dependent variables. We prove that an arbitrary representation of the group P (n, m)

with max{n, m} ? 3 is equivalent to the standard one, while the conformal group

C(n, m) has non-trivial nonlinear representations. Besides that, we investigate in

detail representations of the Poincar? group P (2, 2), extended Poincar? groups P (1, 2),

e e

P (2, 2), and conformal groups C(1, 2), C(2, 2) and obtain their linear and nonlinear

representations.

1 Introduction

The central problem to be solved within the framework of the classical Lie approach

to investigation of the partial differential equation (PDE)

F (x, u, u, u, . . . , u) = 0, (1)

r

12

where symbol u denotes a set of k-th order derivatives of the function u = u(x),

k

is to compute its maximal symmetry group. Sophus Lie developed the universal

infinitesimal algorithm which reduced the above problem to solving some linear over-

determined system of PDE (see, e.g. [1–3]). The said method enables us to solve

the inverse problem of symmetry analysis of differential equations — description

of equations invariant under given transformation group. This problem is of great

importance of mathematical and theoretical physics. For example, in relativistic field

theory motion equations have to obey the Lorentz–Poincar?–Einstein relativity prin-

e

ciple. It means that equations considered should be invariant under the Poincar? e

group P (1, 3). That is why, there exists a deep connection between the theory of

relativistically-invariant wave equations and representations of the Poincar? group

e

[4–6].

There exists a vast literature on representations of the generalized Poincar? group

e

P (n, m) [6], n, m ? N but only a few papers are devoted to a study of nonlinear

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