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N1 = 2im?t + ?aa + ?a ?a ,
1 1 1
N2 = ?m2 ?tt + 2im ?a ?at + ?a ?a ?bb + ?2 ,
?t ?aa + ?a ?b ?ab +
n aa
n n
j
k
(?aa )j?r (?? )k?l?j+r + k(?aa )j (?? )k?j?1 ,
? r l+1?r
Srl (?n)l Cj Ck
Sjk = aa aa
l=0 r=0
k j
k?j (?n) k!
?l l
Rk = Rj (?aa ) (l = 1, 2, 3).
j!(k ? j)!
j=0
Second-order differential invariants of the rotation group O(n) 521

The invariants for the algebras AGII (1, n), AGII (1, n) (m = 0) can be construc-
1
ted similarly to the case of real function. Let us adduce a functional basis for the
algebra AGII (1, n).
2
(1) when ? = 0, then there is a basis consisting of the following expressions:
?2
2
(Sjk )2
N1 N1 ?k?1
?
(Rk )2 N1
l
?+? , 2, , , (l = 1, 2, 4);
k
N2 N2 N1
(2) ? = 0:
?
(Rk )2
l
(Sjk )2
N1
? ?
N1 e(4/?)(?+? ) , N3 e(3/?)(?+? ) ,
, (l = 1, 2, 3), ,
k k
N1 N1 N1
where
N1 = (?t ? ?a ?a )2 + (?tt ? ?a ?at )(? + ?a ?ab rab )
(with {rab } = {?ab }?1 and ?a = rab ?bt ),
N2 = (?t ? ?c ?c )?? ?? rab ? (?? ? ?? ?c )?a ?b rab ,
? ?
ab t c
N3 = (?t ? ?? ) ? ?a (?a ? ?? ) (?a (??ab + ?a ?b ) = ?b ?t + ??bt ),
t a
Rk = Rk (?a , ?ab ), Rk = Rk (?? , ?ab ), Rk = Rk (?a ? ?a , ?ab ),
?
1 2 3
a
Rk = Rk (?a , ?ab ) (?a = (?t ? ?b ?b )(?? rac ? ?c rac ) ? ?b ?d rbd (?a ? ?a )).
? ?
4
c

The proof of this theorem will be easier if we notice that by putting µ = im
in (4.4), we obtain operators similar to the operators (4.2).
The change of variables (4.5) in the adduced invariants allows us to obtain bases
for the algebras AGI and AGII in the representations (4.2) and (4.4). These results
2 2
can also be generalized for the case of several scalar functions.
4.3. Let us present some examples of new invariant equations

1 1 1 1
?t ?aa + ?a ?t + ?a ?b ?ab + ?a ?a ?bb + ?2
?tt + 2µ =
n bb
µ2 n n (4.18)
= (2µ?t + ?a ?a + ?aa )2 F,

1 1 1
?m2 ?tt + 2im ?a ?at + ?a ?a ?bb + ?2 =
?t ?aa + ?a ?b ?ab +
n aa
n n (4.19)
= (2im?t + ?a ?a + ?aa )2 F.

Equations (4.18) and (4.19) are invariant, respectively, under the algebras
AGI (1, n), µ = 0 (4.2), and AGII (1, n), m = 0 (4.4). The F ’s are arbitrary functions
2 2
of the invariants for corresponding algebras.
Evidently, wide classes of invariant equations can be constructed with the adduced
invariants.
5. Conclusion
It is well-known that a mathematical model of physical or some other phenomena
must obey one of the relativity principles of Galilei or Poincar?. Speaking the language
e
of mathematics, it means that the equations of the model must be invariant under the
Galilei or the Poincar? groups. Having bases of differential invariants for these groups
e
522 W.I. Fushchych, I.A. Yegorchenko

(or for the corresponding algebras), we can describe all the invariant scalar equations,
or sort the invariant ones out of a set of equations.
The construction of differential invariants for vector and spinor fields presents
more complicated problems. The first-order invariants for a four-dimensional vector
potential had been found in [18]. The cases of spinor and many-dimensional vector
Poincar?-invariant equations and corresponding bases of invariants are still to be
e
investigated.
Note 5. After having prepared the present paper, we became acquainted with the
article [19] where realizations of the Poincar? group P (1, 1) and the corresponding
e
conformal group were investigated, and all second-order scalar differential equations
invariant under these groups were obtained. Reference [19] contains bases of absolute
differential invariants of the order 2 for the Poincar?, the similitude, and the conformal
e
groups in (1 + 1)-dimensional Minkowski space for various realizations of the corres-
ponding Lie algebras.
Note 6. It was noticed by the referee that an essential misunderstanding arose in the
calculation of second prolongations for differential operators, e.g. in formulae (1.5)
and (1.25).
When we calculate such prolongations with the usual Lie technique (see, e.g., [8]),
we imply that action of an operator of the form X ab ?uab , where X ab are some functi-
ons, is as follows
X ab ?uab (ucd ucd ) = 2X ab uab , ?uab ucd = ?ac ?bd .
With this assumption, ?uab uba = 0, a = b.
Otherwise, the second prolongation of the operator Jab (1.1) will be of the form
2
?
Jab = Jab + Jab ,
?
Jab = ua ?ub ? ub ?ua + uac ?ubc ? ubc ?uac + uab (?ubb ? ?uaa ).
Note 7. The equations which are conditionally invariant with respect to the Poincar?
e
and Galilei algebras were investigated in [20, 21].
Acknowledgement. Authors would like to thank the referees for valuable com-
ments.

1. Lie S., Math. Ann., 1884, 24, 52–89.
2. Tresse A., Acta Math., 1894, 18, 1–88.
3. Vessiot E., Acta Math., 1904, 28, 307–349.
4. Michal A.D., Proc. Nat. Acad. Sci., 1951, 37, 623–627.
5. Fushchych W.I., Yegorchenko I.A., Dokl AN Ukr. SSR, Ser. A, 1989, 4, 29–32.
6. Fuschchych W.I., Yegorchenko I.A., Dokl. AN Ukr. SSR, Ser. A, 1989, 5, 21–22.
7. Spencer A.J.M., Theory of invariants, New York, London, Academic Press, 1971.
8. Ovsyannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982.
9. Olver P., Application of Lie groups to differential equations, New York, Springer-Verlag, 1987.
10. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear
equations of mathematical physics, Kiev, Naukova Dumka, 1989 (in Russian); English version to be
published by Kluwer Publishers, 1993.
11. Bluman G.W., Kumei S., Symmetries and differential equations, New York, Springer Verlag, 1989.
Second-order differential invariants of the rotation group O(n) 523

12. Fuschchych W.I., Yegorchenko I.A., Dokl. AN SSSR, 1988, 298, 347–351.
13. Fuschchych W.I., Serov N.I., Dokl. AN SSSR, 1984, 278, 847.
14. Fushchych W.I., Shtelen W.M., Lett. Nuovo Cimento, 1982, 34, 498.
15. Goff J.A., Amer. J. Math., 1927, 49, 117–122.
16. Niederer U., Helv. Phys. Acta, 1972, 45, 802–810.
17. Fushchych W.I., Cherniha R.M., J. Phys. A, 1985, 18, 3491–3503.
18. Yegorchenko I.A., Symmetry properties of nonlinear equations for complex vector fields, Preprint
89.48, Institute of Mathematics of the Ukr. Acad. Sci, 1989.
19. Rideau G., Winternitz P., J. Math. Phys., 1990, 31, 1095–1105.
20. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, D. Reidel, 1987.
21. Fushchych W.I., Ukrain. Mat. Zh., 1991, 43, 1456.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 524–538.

Conditional symmetry and reduction
of partial differential equations
W.I. FUSHCHYCH, R.Z. ZHDANOV
Sufficient reduction conditions for partial differential equations possessing nontrivial
conditional symmetry are established. The results obtained generalize the classical
reduction conditions of differential equations by means of group-invariant solutions.
A number of examples illustrating the reduction in the number of independent and
dependent variables of systems of partial differential equations are considered.

An analysis of well-known methods for the construction of exact solutions of
nonlinear partial differential equations (PDE) (e.g., method of group-theoretic re-
duction [1, 2], method of differential constraints [3], method of ansatz [4–6]) led
us to conclude that most of these methods involve narrowing the set of solutions,
i.e., out of the whole set of solutions of the particular equations specific subsets
are selected that admit analytic description. In order to implement this approach,
certain additional constraints (expressed in the form of equations) that enable us to
distinguish these subsets must be imposed on the solution set. For obvious reasons,
these additional equations are assumed to be simpler than the initial equations. By
complementing the initial equation with additional constraints, we are usually led
to an over-determined system of PDE. Consequently, there arises the problem of
investigating the consistency of a system of PDE. A second restriction on the choice
of these additional constraints is that the resulting system of PDE possesses broader
symmetry than the initial system of PDE (or simply a different type of symmetry).
In the present paper we establish sufficient conditions for the reduction of differen-
tial equations that generalize the classical reduction conditions of PDE possessing
a nontrivial Lie transformation group. Our concern will be with the following:
UA (x, u, u, . . . , u) = 0, (1)
A = 1, M ,
r
1

?aµ (x, u)u?µ ? ?a (x, u) = 0,
?
(2)
a = 1, N ,
x

where x = (x0 , x1 , . . . , xn?1 ), u(x) = (u0 (x), . . . , um?1 (x)), u = {? s u? /?xµ1 . . . ?xµs ,
s
0 ? µi ? n ? 1}, s = 1, r, are sufficiently smooth functions, N ? n ? 1.
?
UA , ?aµ , ?a
Below summation over repeated indices is understood. Let us introduce the nota-
tion
N n?1
R1 = rank ?aµ (x, u) a=1 µ=0 ,

N n?1 m?1
?
R2 = rank ?aµ (x, u), ?a (x, u) a=1 µ=0 ?=0 .

It is self-evident that R1 ? R2 . We shall prove that the case R1 = R2 leads to
a reduction in the number of independent variables of the PDE (1), while the case
Ukr. Math. J., 1992, 44, N 7, P. 875–886.
Conditional symmetry and reduction of partial differential equations 525

R1 < R2 leads to a reduction in the number of independent and the number of
dependent variables of the PDE (1).
1. Reduction of number of independent variables of PDE. In this section we
assume that R1 = R2 .
Definition 1. The set of first-order differential equations
?
(3)
Qa = ?aµ (x, u)?xµ + ?a (x, u)?u? ,

where ?xµ = ?/?xµ , ?u? = ?/?u? ; ?aµ , ?a are smooth functions, is said to be
?
c
involutive if there exist function fab (x, u) such that:
c
(4)
[Qa , Qb ] = fab Qc , a, b = 1, N .

Here [Q1 , Q2 ] = Q1 Q2 ? Q2 Q1 .
The simplest example of an involutive set of operators is a Lie algebra.
It is well-known that conditions (4) ensure that the over-determined system of
PDE (2) is consistent (Frobenius theorem [7]). The general solution of the system (2)
is given by the formulas

? = 0, m ? 1,
F ? (?1 , ?2 , . . . , ?n+m?R1 ) = 0, (5)

where ?j = ?j (x, u) are functionally independent first integrals of the system of
PDE (2) and F? are arbitrary smooth functions.
By virtue of the condition R1 = R2 , first integrals (say, ?1 , . . . , ?m ) may be chosen
that satisfy the condition
m m?1
det ??j /?u? (6)
= 0.
j=1 ?=0

By solving (5) with respect to ?j , j = 1, . . . , m, we have

(7)
?j = ?j (?m+1 , ?m+2 , . . . , ?m+n?R1 ), j = 1, m,

where ?j are arbitrary smooth functions
Definition 2. Formula (7) is called the ansatz of the field u? = u? (x) invariant
with respect to the involutive set of operators (3) provided (6) is satisfied.
Formula (7) become especially simple and self-evident if

??aµ /?u? = 0, ?a = fa (x)u? ,
? ??

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. 123
( 135 .)



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