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2. 22 u = ?2 eu , (7)
?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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