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D = xµ ?xµ + u?u .

1?k

Case 2. k = (n + 1 + 2l)/(n + 1 ? 2l), n + 1 = 2l. The maximal invariance group

of (4) is the conformal group C(1, n) generated by the operators (3) and operators

(2l ? n ? 1)

D(1) = xµ ?xµ + u?u ,

(8)

2

= 2xµ D(1) ? (x? x? )?xµ , µ, ? = 0, n.

(1)

Kµ

Theorem 3. Equation (5) has the following symmetry:

Case 1. n = 2l ? 1. The maximal invariance group of (5) is the extended Poincar?

e

group P (1, n) generated by the operators (3) and

D(2) = xµ ?xµ ? 2l?u . (9)

Case 2. n = 2l ? 1. The maximal invariance group of (5) is the conformal group

C(1, n) generated by the operators (3) and operators

Kµ = 2xµ D(2) ? (x? x? )?xµ ,

(2)

(10)

µ, ? = 0, n.

Theorem 4. The maximal invariance group of the equation (6) is generated by the

operators (3) and

Q? = f (x)?u , I = u?u ,

where f (x) is an arbitrary solution of PDE (6).

Theorem 5. The maximal invariance group of the equation (7) is generated by the

operators (3), (8) and

Q? = q(x)?u , I = u?u ,

where q(x) is an arbitrary solution of PDE (7).

Symmetry and some exact solutions of non-linear polywave equations 417

The proof of the Theorems 1–5 carried out by means of the infinitesimal algorithm

of S. Lie [5] requires very cumbersome computations. That is why, we omit it.

An important consequence of the Theorems 1–5 is the following statement.

Theorem 6. The non-linear PDE (1) is invariant under the conformal group C(1, n)

iff it is equivalent to the following

n+1+2l

1.2l u = ?1 u n+1?2l , (11)

n + 1 = 2l;

2. 2l u = ?2 eu , (12)

n + 1 = 2l.

Remark 1. Conformal invariance of the equation (11) was first ascertained in [8]

and that of equation (12) was done in [3] by means of Baker–Campbell–Hausdorff

formulae.

Assuming l = 1 in (11) we obtain the well-known result [3]; that non-linear wave

equation (2) admits the conformal group if it is equivalent to the PDE

n+3

2u = ?u n?1 when n = 1.

Remark 2. When l = 2 it follows from the Theorem 6 that in the four-dimensional

space R(1, 3) there is only one C(1, 3)-invariant equation

22 u = ?eu .

One of the important applications of the Lie groups in mathematical physics is

the finding exact solutions of non-linear PDE. To this end one has to construct so

called invariant solutions [2, 3, 5] which reduce PDE under study to equations with

less number of independent variables (in particular, to ordinary differential equations).

Integrating these one gets exact solutions of the initial PDE. A procedure described

is called symmetry (or group-theoretical) reduction of differential equations. Here

we perform symmetry reduction of the conformally-invariant biwave equation in the

two-dimensional space R(1, 1):

22 u = ?u?3 . (13)

Making use of inequivalent one-dimensional subalgebras of the conformal algebra

AC(1, 1) [9] one can obtain the following C(1, 1)-inequivalent Ans?tze which reduce

a

the equation (13) to ordinary differential equations. For each case the reduced equati-

ons are given:

? = x0 or ? = x1 ,

1. u = ?(?),

?(4) = ???3 ;

? = x2 ? x2 ,

2. u = ?(?), 0 1

? ?3

?(4) ? 2 + 4?(3) ? + 2?(2) = ?;

16

? = x0 ? x1 ,

u = (x0 + x1 )1/2 ?(?),

3.

?

?(2) = ? ??3 ;

4

u = (x0 + x1 )1/(?+1) ?(?), ? = (x0 ? x1 )(x0 + x1 )(??1)/(?+1) ;

4.

(? ? 2)(2? ? 1) (2) ? (? + 1)2 ?3

?(4) ? 2 + 4?(3) ? + ?= ? , ? > 1;

(? ? 1)2 16 (? ? 1)2

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

1

u = exp(x0 ? x1 )?(?), exp ?2(x0 ? x1 ) ,

5. ?= x0 + x1 +

2

9 ? ?3

?(4) ? 2 + 4?(3) ? + ?(2) = ?;

4 64

1/2

u = (x0 ? x1 )2 + 1

6. ?(?), ? = x0 + x1 ,

? ?3

?(2) = ?;

16

1/2

u = (x0 ? x1 )2 + 1 ?(?), ? = x0 + x1 + arctan(x0 ? x1 ),

7.

? ?3

?(4) + ?(2) = ?;

16

1 1 + x0 ? x1

1/2

u = (x0 ? x1 )2 + 1

8. ?(?), ? = x0 + x1 + ln ,

2 1 ? x0 + x1

? ?3

?(4) ? ?(2) = ?;

16

1/2 1/2

u = (x0 ? x1 )2 + 1

9. x0 + x1 ?(?),

? = ln(x0 + x1 ) ? ? arctan(x1 ? x0 ),

?

4? 2 ?(4) + (4 ? ? 2 )?(2) ? ? = ??3 , ? > 0;

4

1/2 1/2

u = (x0 ? x1 )2 + 1 (x0 + x1 )2 + 1

10. ?(?),

? = (? ? 1) arctan(x0 ? x1 ) + (? + 1) arctan(x0 + x1 ),

? ?3

(? 2 ? 1)2 ?(4) + 2(? 2 + 1)?(2) + ? = ? , 0 ? ? < 1.

16

Integration of the reduced equations gives rise to exact solutions of the non-linear

biwave equation (13). Here we present some exact solutions of this equation obtained

with the use of Ans?tze 3 and 6:

a

1/2

u = ±?1/4 x2 ? x2 ,

0 1

1/4

1 ? 1/2

u = ±v (x0 ? x1 )2 ? c1 (x0 + x1 )1/2 , (14)

2 c1

1/4

1? 1/2 1/2

u=± (x0 ? x1 )2 + 1 (x0 + x1 )2 + c2 ,

2 c2

where c1 , c2 are arbitrary constants.

Since the conformal group C(1, 1) is a maximal symmetry group of equation (13),

formulae 1–10 give “maximal” information about its solutions which can be obtained

within the framework of the Lie approach. It means that any solution invariant under

a subgroup of the symmetry group of PDE (13) can be reduced by a transformation

from the group C(1, 1) to one of the Ans?tze 1–10.

a

Acknowledgments. One of the authors (RZZ) is supported by the Alexander von

Humboldt Foundation.

Symmetry and some exact solutions of non-linear polywave equations 419

1. Bollini C.G., Giambia J.J., Arbitrary powers of d’Alembertians and the Huygens’ principle, J. Math.

Phys., 1993, 34, 610.

2. Fushchych W.I., Symmetry in problems of mathematical physics, in Algebraic-Theoretical Studies in

Mathematical Physics, Kiev, Institute of Mathematics, 1981, 6.

3. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

4. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of the nonlinear many-

dimensional d’Alembert, Liouville and eikonal equations, J. Phys. A, 1983, 16, 3645.

5. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.

6. Hereman W., Review of symbolic software for the computation of Lie symmetries of differential

equations, Euromath. Bull., 1994, 1, 45.

7. Mansfield E.L., Clarkson P.A., Applications of the differential algebra package diffgrob2 to reducti-

ons of PDE, in Proceedings of the Fourteenth IMACS World Congress on Computation and Applied

Mathematics, Editor W.F. Ames (Georgia Inst. Tech.), 1994, 1, 336.

8. Serov N.I., Conformal symmetry of nonlinear wave equations, in Algebraic-Theoretical Studies in

Mathematical Physics, Kiev, Institute of Mathematics, 1981, 59.

9. Fushchych W.I., Barannik L.F., Barannik A.F., Subgroup analysis of the groups of Galilei and Poincar?

e

and reduction of nonlinear equations, Kiev, Naukova Dumka, 1991.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 420–430.

Symmetry properties, reduction and exact

solutions of biwave equations

W.I. FUSHCHYCH, R.Z. ZHDANOV, O.V. ROMAN

We have studied symmetry properties of the biwave equations 22 u = F (u) and the

systems of wave equations which are equivalent to them. Reduction of the nonlinear

biwave equations with the use of subalgebras of the extended Poincar? algebra AP (1, 1)

e

and the conformal algebra C(1, 1) was carried out. Some exact solutions of these equati-

ons were obtained.

It was suggested in [1] to describe different physical processes with the help of

nonlinear partial equations of high order, namely

?u ?u

2l u = F u, (1)

.

?xµ ?xµ

Here and further 2 = ? 2 /?x0 ? ? 2 /?x1 ? · · · ? ? 2 /?xn is d’Alembertian in (n + 1)-

dimensional pseudo-Euclidean space R(1, n) with metric tensor gµ? = diag(1, ?1, . . .,

?1), µ, ? = 0, n; 2l = 2(2l?1 ), l ? N; xµ = x? gµ? ; F (·, ·) is an arbitrary smooth

function; u = u(x) is a real function; the summation over the repeated indices from 0

to n is understood.

Equations (1) were considered from different points of view in [2, 3, 4], where the

pseudodifferential equations of type (1) were also studied (in this case l is fractional

or negative).

Assuming l = 1 and F = F (u) in (1) we obtain the standard wave equation

2u = F (u) (2)

which describes a scalar spinless uncharged particle in quantum field theory. Sym-

metry properties of equation (2) were studied in [4, 5, 6] and wide classes of its

exact solutions with certain concrete values of the function F (u) were obtained in

[4, 5, 7, 8, 9].

In this paper we restrict ourselves by considering the biwave equation

22 u = F (u) (3)

which is one of the simplest equations of type (1) of high order (l = 2, F = F (u)).

1 Symmetry classification of the biwave equation

In order to carry out a symmetry classification of equation (3) we shall establish

at first the maximal transformation group admitted by equation (3) provided F (u)

is an arbitrary function. After that we shall determine all the functions F (u) when

equation (3) admits more extended symmetry.

Results of symmetry classification of equation (3) are cited in the following

statements.

Preprint LiTH-MAT-R-95-20, Department of Mathematics, Link?ping University, Sweden, 16 p.

o

Symmetry properties, reduction and exact solutions of biwave equations 421

Lemma 1 The maximal invariance group of equation (3) with an arbitrary function

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

e

? ? ?

? x?

Jµ? = xµ (4)

Pµ = , , µ, ? = 0, n.

?xµ ?x? ?xµ

Theorem 1 All the equations of type (3) admitting more extended invariance algeb-

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

e

1. 22 u = ?1 uk , (5)

?1 = 0, k = 0, 1;

2. 22 u = ?2 eu , (6)

?2 = 0;

3. 22 u = ?3 u, (7)

?3 = 0;

4. 22 u = 0. (8)

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

Theorem 2 The symmetry of the equations (5)–(8) is described in the following

way:

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

k = 0, 1 is the extended Poincar? group P (1, n) generated by the operators (4) and

e

? 4 ?

D = xµ + u.

1 ? k ?u

?xµ

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

n = 3 is the conformal group C(1, n) generated by the operators (4) and

3?n ?

?

D(1) = xµ + u,

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