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(9)

?

= 2xµ D(1) ? (x? x? )

(1)

Kµ .

?xµ

2. (a) The maximal invariance group of equation (6) when n = 3 is the extended

Poincar? group P (1, n) generated by the operators (4) and

e

? ?

?4 .

D(2) = xµ (10)

?xµ ?u

(b) The maximal invariance group of equation (6) when n = 3 is the conformal

group C(1, n) generated by the operators (4) and

?

Kµ = 2xµ D(2) ? (x? x? )

(2)

(11)

.

?xµ

3. The maximal invariance group of equation (7) is generated by the operators (4)

and

? ?

Q = h(x) , I=u ,

?u ?u

where h(x) is an arbitrary solution of equation (7).

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

4. The maximal invariance group of equation (8) is generated by the operators

(4), (9) and

? ?

Q = q(x) , I=u ,

?u ?u

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

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

mal algorithm of S. Lie [4, 10]. Since it requires very cumbersome computations we

only give a general scheme of the proof.

In the Lie approach the infinitesimal operator of equation (3) invariance group is

of the form

? ?

X = ? µ (x, u) (12)

+ ?(x, u) .

?xµ ?u

The invariance criterion of equation (3) under group generated by the opera-

tors (12) is

X (2 u ? F (u))

2

(13)

= 0,

22 u=F (u)

4

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

4

Splitting equation (13) 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,

(14)

= · · · = ?n ,

0 1 n

?0 = ?1

(3 ? n)?00 ,

?

2??u = ? = 0, n,

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

0

(15)

Besides, when n = 1, there are additional equations:

(16)

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

that do not follow from equations (14) and (15).

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

µ

on.

System (14) is a system of Killing equations. The general solution of equations

(14), (16) is of the form:

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

(17)

? = ((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 (17) into the classifying equation (15) and splitting it with respect

to u we obtain statements of Lemma 1 and Theorems 1, 2 according to the form

of F (u).

Symmetry properties, reduction and exact solutions of biwave equations 423

It follows from the statements proved that the equation of type (1) is invariant

under the extended Poincar? group P (1, n) iff it is equivalent one of equations (5), (6)

e

or (8). Let us note that the analogous result was obtained for the wave equations (2)

in [5].

The following statement also is the consequence of the Theorems but since it is

important we adduce it as a Theorem.

Theorem 3 Equation (3) is invariant under the conformal group C(1, n) iff it is

equivalent to the following:

1. 22 u = ?1 u(n+5)/(n?3) , (18)

n = 3;

2. 22 u = ?2 eu , (19)

n = 3.

Let us note that conformal invariance of equation (18) was first ascertained in [11]

and that of equation (19) was done in [4] by means of Baker–Campbell–Hausdorff

formulae.

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

biwave equation (8), when n = 3, which is the consequence of Theorems 2 and 3.

Corollary There exist two nonequivalent representations of the Lie algebra of the

conformal group C(1, n) on the set of solutions of equation (8) [1, 3, 4]:

(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 (4).

2 Symmetry classification of system

of wave equations

Introducing a new variable v = 2u in (3) we get the system of partial differential

equations

2u = v,

(20)

2v = F (u),

which is equivalent to the biwave equation (3).

Symmetry properties of the system (20) are investigated by analogy with the

previous Section. So we only formulate statements analogous to the preceding ones

without proving them.

Lemma 2 The maximal invariance group of the system (20) with an arbitrary

function F (u) is the Poincar? group P (1, n) generated by the operators (4).

e

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

Theorem 4 All the systems of type (20) admitting more extended invariance algebra

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

e

2u = v,

1.

(21)

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

2u = v,

2.

(22)

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

2u = v,

3.

(23)

2v = 0.

Theorem 5 The symmetries of the systems (21)–(23) is described in the following

way:

1. The maximal invariance group of the system (21) is the extended Poincar?e

group P (1, n) generated by the operators (4) and

? 4 ? 2(1 + k) ?

D = xµ + u + v.

1 ? k ?u 1 ? k ?v

?xµ

2. The maximal invariance group of the system (22) is generated by the opera-

tors (4) and

? ? ? ? ? ?

Q1 = u + v , Q2 = v + ?2 u , Q3 = h1 (x) + h2 (x) ,

?u ?v ?u ?v ?u ?v

where (h1 (x), h2 (x)) is an arbitrary solution of the system (22).

3. The maximal invariance group of the system (23) is generated by the opera-

tors (4) 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 (23).

It follows from the foregoing statements that unlike the biwave equations, the

extended Poincar? group P (1, n) is the invariance group of the system (20) only in

e

two cases, namely, when (20) is equivalent to (21) or (23). Moreover, the system

(20) is not invariant under the conformal group for any functions F (u). Therefore,

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

corresponding systems of wave equations are essentially different.

3 Reduction and exact solutions

of the equation 22 u = ?eu

As follows from Theorem 2 the maximal invariance group of the equation (6), when

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

e

? ? ? ?

? x1

J01 = x0 (24)

P0 = , P1 = , ,

?x0 ?x1 ?x1 ?x0

Symmetry properties, reduction and exact solutions of biwave equations 425

? ? ?

?4 .

D(2) = x0 (25)

+ x1

?x0 ?x1 ?u

It is known that if an equation admits the symmetry operator

? ?

X = ? µ (x) (26)

+ ?(x)

?xµ ?u

then its solutions can be found in the form [4]:

(27)

u(x) = ?(?) + g(x).

For the substitution (27) to be an ansatz for the equation with the symmetry

operator (26), the functions ?(x) and g(x) are to satisfy the following conditions:

?? ?g(x)

? µ (x) ? µ (x)

= 0, = ?(x).

?xµ ?xµ

To obtain all the P (1, 1)-nonequivalent ansatzes (27) we have to describe all the

nonequivalent one-dimensional subalgebras of the Lie algebra AP (1, 1) spanned by the

operators (24) and (25) (see [4, 9]). In the paper we make use of classification given

in [9] and omitting rather cumbersome computations we write P (1, 1)-nonequivalent

ansatzes in Table 1.

Table 1.

N Algebra Invariant variables ? Ansatz

1? D ? J01 x0 + x1 u = ?(?) ? 2 ln(x0 ? x1 )

4

(1 + ?) ln(x1 ? x0 )?

2? D + ?J01 , ? = ?1 u = ?(?) ? ln(x0 + x1 )

? (1 ? ?) ln(x0 + x1 ) ?+1

ln(x0 ? x1 + 1/2)?

3? D ? J01 + P0 u = ?(?) ? 2 ln(x0 ? x1 + 1/2)

? 2(x0 + x1 )

4? x2 ? x2 u = ?(?)

J01 0 1

5? u = ?(?)

P0 x1

6? P0 + P1 x0 ? x1 u = ?(?)

Remark. Inequivalent subalgebras listed in Table 1 are built by taking account of the

obvious fact that equation (6) is invariant under the transformations of the form:

x0 > x0 , x0 > x1 ,

and (28)

x1 > ?x1 ; x1 > x0 .

Substituting ansatzes obtained in (6) we get the following equations for the functi-

on ?(?):

1? 0 = ?e? ,

? 2?

2? ?(4) (?2 ? 1)2 + 2?(3) ?(1 ? ?2 ) ? ?(2) (1 ? ?2 ) = exp ? + ,

16 ?+1

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