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?2 = 0;

3. 22 u = ?3 u, (8)

?3 = 0;

4. 22 u = 0. (9)

Here ?1 , ?2 , ?3 , k are arbitrary constants.

Theorem 2 The symmetry of the equations (6)–(9) is described as follows:

1. (a) The maximal invariance group of the equation (6) when k = (n+5)/(n?3),

k = 0, 1 is the extended Poincar? group P (1, n) generated by the operators (2) and

e

? 4 ?

(10)

D = xµ + u.

1 ? k ?u

?xµ

(b) The maximal invariance group of the equation (6) when k = (n + 5)/(n ? 3),

n = 3 is the conformal group C(1, n) generated by the operators (2) and operators

3?n ?

?

D(1) = xµ + u,

?xµ 2 ?u

(11)

?

= 2xµ D(1) ? (x? x? )

(1)

Kµ .

?xµ

2. (a) The maximal invariance group of the equation (7) when n = 3 is the

extended Poincar? group P (1, n) generated by the operators (2) and

e

? ?

?4 .

D(2) = xµ (12)

?xµ ?u

(b) The maximal invariance group of the equation (7) when n = 3 is the

conformal group C(1, n) generated by the operators (2) and operators

?

Kµ = 2xµ D(2) ? (x? x? )

(2)

(13)

.

?xµ

3. The maximal invariance group of the equation (8) is generated by the opera-

tors (2) and

? ?

Q = h(x) , I=u ,

?u ?u

where h(x) is an arbitrary solution of the equation (8).

Symmetry reduction and exact solutions of nonlinear biwave equations 405

4. The maximal invariance group of the equation (9) is generated by the opera-

tors (2), (11) and

? ?

Q = q(x) , I=u ,

?u ?u

where q(x) is an arbitrary solution of the equation (9).

The proof of the Lemma 1 and the Theorems 1, 2 is carried out by means of the

infinitesimal algorithm of S. Lie [2, 5]. Since it requires very cumbersome computa-

tions, we adduce a general scheme of the proof only.

Within the framework of the Lie’s approach an infinitesimal operator of the equa-

tion (5) invariance group is looked for in the form

? ?

X = ? µ (x, u) (14)

+ ?(x, u) .

?xµ ?u

The criterion of invariance of the equation (5) with respect to a group generated

by the operator (14) reads

X (2 u ? F (u))

2

(15)

= 0,

22 u=F (u)

4

where X is the 4-th prolongation of the operator X.

4

Splitting the equation (15) with respect to the independent variables, we come to

the system of partial differential equations for functions ? µ (x, u) and ?(x, u):

µ

?u = 0, ?uu = 0, µ = 0, n,

j

?j = ??i ,

i 0 i

?0 = ?i , i = j, i, j = 1, n,

(16)

?0 = ?1 = · · · = ?n ,

0 1 n

2??u = (3 ? n)?00 , ? = 0, n,

?

22 ? ? ?F (u) + F (u)(?u ? 4?0 ) = 0.

0

(17)

Besides, when n = 1, there are additional equations

(18)

?00u = 0, ?01u = 0,

that do not follow from the equations (16) and (17).

In the above formulae we use the notations ?? = ?? µ /?x? , ?µ = ??/?xµ and so

µ

on.

System (16) is one of the Killing equations in the Minkowski space-time. Its

general solution is well-known and can be represented in the following form:

? ? = 2x? xµ cµ ? xµ xµ c? + b?µ xµ + dx? + a? ,

(19)

? = ((3 ? n)cµ xµ + p)u + ?(x),

where cµ , b?µ = ?bµ? , d, a? , p are arbitrary constants, ?(x) is an arbitrary smooth

function.

Substituting the expression (19) into the classifying equation (17) and splitting it

with respect to u we arrive at the statements of the Lemma 1 and the Theorems 1, 2

according to the form of F (u).

406 W.I. Fushchych, O.V. Roman, R.Z. Zhdanov

It follows from the assertions proved that the equation of type (4) is invariant

under the extended Poincar? group P (1, n) if and only if it is equivalent to one of the

e

equations (6), (7) or (9). Let us note that an analogous result was obtained for the

wave equation (1) in [3].

The following statement is also a consequence of the Theorems 1, 2 but because

of its importance we adduce it as a theorem.

Theorem 3 Equation (5) admits the conformal group C(1, n) if and only if it is

equivalent to the following:

1. 22 u = ?1 u(n+5)/(n?3) , (20)

n = 3;

2.22 u = ?2 eu , (21)

n = 3.

Let us note that conformal invariance of the equation (20) has been first ascertai-

ned in [12] and that of equation (21) – in [2] by means of the Baker–Campbell–

Hausdorff formula. It is also worth noting that conformal invariance of the nonlinear

polyharmonic equations has been studied in [13], which enables constructing some

their exact solutions.

In conclusion of the Section let us emphasize an important property of the linear

biwave equation (9) with n = 3, which is a consequence of the Theorems 2, 3.

Corollary. There exist two inequivalent representations of the Lie algebra of the

conformal group C(1, 3) on the solution set of the equation (9) [2, 6, 8]:

(1) (1)

1. Pµ = P µ , Jµ? = Jµ? ,

? ?

Kµ = 2xµ D(1) ? (x? x? )

D(1) = xµ (1)

, ;

?xµ ?xµ

(2) (2)

2. Pµ = Pµ , Jµ? = Jµ? ,

? ? ?

, Kµ = 2xµ D(2) ? (x? x? )

D(2) (2)

= xµ + ,

?xµ ?u ?xµ

where the operators Pµ , Jµ? are determined in (2).

3 Symmetry classification of system of wave equations

Introducing a new variable v = 2u in (5) we get a system of partial differential

equations

2u = v,

(22)

2v = F (u),

which is equivalent to the biwave equation (5).

Symmetry properties of the system (22) are investigated by analogy with the

previous Section. That is why, we restrict ourselves to formulating the corresponding

assertions omitting their proofs.

Lemma 2 The maximal invariance group of the system (22) with an arbitrary

function F (u) is the Poincar? group P (1, n) generated by the operators (2).

e

Symmetry reduction and exact solutions of nonlinear biwave equations 407

Theorem 4 Any system of type (22) admitting a more extended invariance algebra

than the Poincar? algebra AP (1, n) is equivalent to one of the following:

e

2u = v,

1.

(23)

2v = ?1 uk , ?1 = 0, k = 0, 1;

2u = v,

2.

(24)

2v = ?2 u, ?2 = 0;

2u = v,

3.

(25)

2v = 0.

Theorem 5 The symmetry of the systems (23)–(25) is described in the following

way:

1. The maximal invariance group of the system (23) is the extended Poincar?

e

group P (1, n) generated by the operators (2) and

? 4 ? 2(1 + k) ?

D = xµ + u + v.

1 ? k ?u 1 ? k ?v

?xµ

2. The maximal invariance group of the system (24) is generated by the opera-

tors (2) and

? ? ? ?

Q1 = u +v , Q2 = v + ?2 u ,

?u ?v ?u ?v

? ?

Q3 = h1 (x) + h2 (x) ,

?u ?v

where (h1 (x), h2 (x)) is an arbitrary solution of the system (24).

3. The maximal invariance group of the system (25) is generated by the opera-

tors (2) and

? ? ? ?

D = xµ + 2u , Q1 = u +v ,

?xµ ?u ?u ?v

? ? ?

Q2 = v , Q3 = q1 (x) + q2 (x) ,

?u ?u ?v

where (q1 (x), q2 (x)) is an arbitrary solution of the system (25).

It follows from the statements above that, unlike the biwave equations, the exten-

ded Poincar? group P (1, n) is the invariance group of the system (22) only in two

e

cases, namely, when the system (22) is equivalent to (23) or (25). Moreover, there are

no systems of the form (22) which are invariant under the conformal group. Therefore,

in the class of Lie operators, the invariance algebras of the biwave equations and the

corresponding systems of the wave equations are essentially different.

408 W.I. Fushchych, O.V. Roman, R.Z. Zhdanov

4 Reduction and exact solutions

of the equation 22 u = ?eu

As follows from the Theorem 2 the maximal invariance group of the equation (7) with

n = 1 is the extended Poincar? group P (1, 1) with the generators

e

? ? ? ?

(26)

P0 = , P1 = , J01 = x0 + x1 ,

?x0 ?x1 ?x1 ?x0

? ? ?

?4 .

D(2) = x0 (27)

+ x1

?x0 ?x1 ?u

To construct exact solutions of the above equation we shall make use of the

symmetry reduction procedure. A principal idea of the said procedure is a special

choice of a solution to be found. This choice is motivated by a representation of

symmetry group admitted. It is known that if an equation admits a Lie transformation

group having a symmetry operator

? ?

X = ? µ (x) (28)

+ ?(x) ,

?xµ ?u

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