ñòð. 99 |

(36), and what is more, the operator (34) belongs to the invariance algebra of the

equation (38).

To obtain all the C(1, 1) non-conjugated Ans?tze we use the one-dimensional

a

inequivalent subalgebras of the conformal algebra AC(1, 1) adduced in [11].

Solving for each subalgebra equations (36) we arrive at the collection of C(1, 1)-

invariant Ans?tze which are presented in the Table 3.

a

Table 3.

N Algebra Invariant variable ? Ansatz

1/2

u = (x0 ?x1 )2 + 1 ?

arctg(x1 ? x0 )+

(1)

1? P 0 + K0 1/2

+arctg(x1 + x0 )

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

1/2

u = (x0 ?x1 )2 + 1 ?

(1) (1)

P0 + K0 + ?(K1 ? P1 ) (? ? 1)arctg(x0 ? x1 )+

2? 1/2

+(? + 1)arctg(x1 + x0 )

0<?<1 ? (x0 +x1 )2 + 1 ?(?)

1/2

(x0 ? x1 )2 + 1

u= ?

(1) (1)

3? P 0 + K0 + K1 ? P 1 x0 + x1

??(?)

1/2

x0 + x1 +

(x0 ? x1 )2 + 1

u= ?

(1) (1)

?

1 1 + x0 ? x1

4 2P1 + +

K0 K1

+ ln ??(?)

2 1 ? x0 + x1

1/2

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

u= ?

(1) (1)

?

5 2P1 ? ?

K0 K1

+arctg(x0 ? x1 ) ??(?)

1/2

u = (x0 ? x1 )2 + 1 ?

(1) (1)

ln(x0 + x1 )?

P 0 + K0 + K1 ? P 1 ?

6? 1/2

? ?arctg(x1 ? x0 )

? ?(J01 + D(1) ), ? > 0 ? x0 + x1 ?(?)

We omit subalgebras not containing the conformal operator (39) since they were

considered in the preceding Section.

Substituting Ans?tze obtained in the equation (38) we get the following reduced

a

ODE for a function ?(?):

? ?3

1? ?(4) + 2?(2) + ? = ?;

16

? ?3

2? (?2 ? 1)2 ?(4) + 2(?2 + 1)?(2) + ? = ?;

16

? ?3

3? ?(2) = ?;

16

? ?3

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

16

Symmetry reduction and exact solutions of nonlinear biwave equations 413

? ?3

5? ?(4) + ?(2) = ?;

16

? ?3

6? 4? 2 ?(4) + (4 ? ? 2 )?(2) ? ? = ?.

4

The general solution of the equation 3? is of the form

1v

(c1 ? + c2 )2 ?

?=± ?=± ??? + c,

+ ,

c1 16c1 2

where c, c1 , c2 are arbitrary constants, c1 = 0.

Hence we obtain the following exact solutions of the equation (38):

1/4

1 ? 1/2 1/2

u = ±v (x0 + x1 + a2 )2 ? a1 x0 ? x1 + a3

1. ,

2 a1

1/4

1 ? 1/2 1/2

u = ±v (x0 ? x1 + b2 )2 ? b1

2. x0 + x1 + b3 ,

2 b1

1/4

1 ? 1/2 1/2

u=± (x0 ? x1 + c3 )2 + c1 (x0 + x1 + c4 )2 + c2

3. ,

2 c1 c2

where ai , bi , cj , i = 1, 3, j = 1, 4 are arbitrary constants.

Besides, the expression

1/2

u = ±?1/4 (x0 ? x1 + c1 )(x0 + x1 + c2 )

(c1 , c2 are arbitrary constants) was proved in the Section 4 to be the exact solution

of the equation (38).

Conclusion

Thus, we have shown that the symmetry selection principle is a natural way of classi-

fication of physically admissible nonlinear biwave equations. Requiring an invariance

with respect to the extended Poincar? group picks out very specific nonlinearities (3).

e

And the demand of a conformal invariance yields, in fact, a unique nonlinear PDE

(20), (21).

As equations obtained in this way admit broad Lie symmetry, one can apply the

symmetry reduction procedure to find their exact solutions. An important part of the

said procedure is a construction of special substitutions which reduce the equation

under study to PDE with less number of independent variables. Given a subgroup

classification of the equation under study, a procedure of construction of such substi-

tutions is entirely algorithmic. Of course, there is no guarantee that the reduced

equations can be solved explicitly. But our experience as well as a rich experience of

other groups engaged in the field of group-theoretical, symmetry analysis of nonlinear

partial differential equations evidence that it is almost always possible [2, 5, 14, 16].

The reason is that PDE obtained by means of reduction of some initial PDE admitting

broad Lie symmetry also possess a hereditary symmetry. Moreover, in some excepti-

onal cases this symmetry can be much more extensive than the one of the initial

equation. An example is given in [15], where it is established that some equations

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

obtained by means of reduction of the nonlinear Poincar?-invariant Dirac equation

e

admit infinite-parameter symmetry groups. Since a maximal symmetry group of the

initial equation is the ten-parameter Poincar? group, this symmetry is essentially new.

e

The source of it is the conditional symmetry of the nonlinear Dirac equation [2, 15].

In the present paper we have applied the symmetry reduction procedure to reduce

to ODE the fourth-order nonlinear biwave equations of the form (5) having two

independent variables x0 , x1 and to construct its explicit solutions. A problem of

symmetry reduction of these equations has been completely solved in a sense that

any solution of PDE (5) invariant under a subgroup of the conformal group C(1, 1)

(which is a most extensive group that can be admitted by equation of the form (5))

is equivalent to one of the Ans?tze given in the Tables 1–3. And what is more, these

a

Ans?tze can be applied to reduce any two-dimensional PDE, provided it is invariant

a

under the Poincar?, extended Poincar? and conformal groups having the generators

e e

(2), (10)–(13). But it does not mean that all possibilities to reduce PDE (5) to ODE

are exhausted. New reductions can be obtained by utilizing conditional symmetry

of the biwave equation in the way as it has been done for a number of nonlinear

mathematical physics equations in [2]. This problem is under investigation now.

An another interesting problem is to carry out symmetry reduction of the biwave

equation in the four-dimensional Minkowski space-time. This work is now in progress

and will be reported elsewhere.

Acknowledgments. One of the authors (R.Z. Zhdanov) is supported by the

Alexander von Humboldt Foundation.

1. Heisenberg W., Einf?hrung in die Einheitliche Feldtheorie der Elementarteilchen, Stuttgart, S. Hirzel

u

Verlag, 1967.

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

nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993.

3. Fushchych W.I., Serov N.I., J. Phys. A: Math. Gen., 1983, 16, 3645–3658.

4. Ibragimov N.H., Lie groups in some questions of mathematical physics, University Press, Novosibirsk,

1972.

5. Ovsiannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.

6. Fushchych W.I., in Algebraic-Theoretical Studies in Mathematical Physics, Kiev, Institute of

Mathematics, 1981, 6–11.

7. Bollini C.G., Giambia J.J., J. Math. Phys., 1993, 34, 610–621.

8. Fushchych W.I., Selehman M.A., Dopovidi Acad. Sci. Ukrainy, 1983, ¹ 5, 21–24.

9. Winternitz P., Grundland A.M., Tuszy?ski J.A., J. Math. Phys., 1987, 28, 2194–2212.

n

10. Fedorchuk V.M., in Symmetry and Solutions of Nonlinear Equations of Mathematical Physics, Kiev,

Institute of Mathematics, 1987, 73–76.

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

e

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

12. Serov N.I., in Algebraic-Theoretical Studies in Mathematical Physics, Kiev, Institute of Mathematics,

1981.

13. Svirshchevskii S.R., Differential Equations, 1993, 29, 1538–1547.

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

15. Fushchych W.I.,Zhdanov R.Z., Nonlinear spinor equations: symmetry and exact solutions, Kiev,

Naukova Dumka, 1992.

16. Winternitz P., in Partially Integrable Evolution Equations in Physics, Dordrecht, Kluwer, 1990, 515–

567.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 415–419.

Symmetry and some exact solutions

of non-linear polywave equations

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

We have studied the maximal symmetry group admitted by the non-linear polywave

equation 2l u = F (u). In particular, we establish that equation in question ad-

mits the conformal group C(1, n) if and only if F (u) = ?eu , n + 1 = 2l or

F (u) = ?u(n+1+2l)/(n+1?2l) , n + 1 = 2l. Symmetry reduction for the biwave equation

22 u = ?u?3 is carried out and some exact solutions are obtained.

Recently a number of works (see, e.g., [1, 2, 3]) have appeared pointing out the

possibility to choose linear and non-linear polywave equations

2l u = F (u) (1)

as possible mathematical models describing an uncharged scalar particle in quantum

field theory.

Here 2l = 2(2l?1 ), l ? N; 2 = ?x0 ? ?x1 ? · · · ? ?xn is d’Alembertian in (n + 1)-

2 2 2

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

?1), µ, ? = 0, n; F (u) is an arbitrary smooth function and u = u(x) is a real function

(the case l = 1, n = 1 has been studied earlier [4], that is why we put l + n > 2).

In the following, a summation over the repeated indices from 0 to n is understood,

rising and lowering of the vector indices is performed by means of the tensor gµ? , i.e.

xµ = gµ? x? .

But the fact that the non-linear partial differential equation (PDE) in question is

of high order makes the prospects of studying such a model rather obscure. Using

group properties of equation (1) seems to be the only way to get some non-trivial

information about the said equation and its solutions. It occurs that PDE (1) admits

wide symmetry group which, in fact, is the same as the one of the standard wave

equation

2u = F (u). (2)

The main tool used is the infinitesimal Lie method (see, e.g., [5]). But an appli-

cation of it to study of symmetry properties of equation (1) is by itself a non-trivial

problem in the case l > 1. It should be emphasized that because of arbitrariness

of the order (l) and of the number of independent variables (n) one can not apply

symbolic manipulation programs [6, 7]. We have succeeded in constructing the maxi-

mal symmetry group admitted by equation (1) using the remarkable combinatorial

properties of the prolongation formulae.

Theorem 1. The maximal invariance group of PDE (1) with arbitrary smooth func-

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

e

Jµ? = xµ ?x? ? x? ?xµ , (3)

Pµ = ?xµ , µ, ? = 0, n.

Europhys. Lett., 1995, 31, ¹ 2, P. 75–79.

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

It is established below that the equation of the type (1) admitting the group, which

is more extensive than the Poincar? group, is equivalent up to the change of variables

e

to one of the following equations:

1. 2l u = ?1 uk , (4)

?1 = 0, k = 0, 1;

2. 2l u = ?2 eu , (5)

?2 = 0;

3.2l u = ?3 u, (6)

?3 = 0;

4. 2l u = 0. (7)

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

Maximal invariance groups of the equations (4)–(7) are described by the following

statements.

Theorem 2. Equation (4) has the following symmetry:

Case 1. k = (n + 1 + 2l)/(n + 1 ? 2l), k = 0, 1. The maximal invariance group of

(4) is the extended Poincar? group P (1, n) generated by the operators (3) and

e

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