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We impose the condition
3/2
(33)
u111 x1 =0
1

on (32). The general solution of (33) is
3/2
u = x1 ?1 + x2 ?2 + x1 ?3 + ?4 (34)
1

with ?i = ?i (x0 , x2 ), i = 1, 2, 3, 4 being arbitrary functions. Using (34) as an ansatz,
equation is reduced to a system with two independent variables
9 22
?3 = 0, ?1 = 0, ?2 = 0, ?4 = (? ) ,
22 22 22
4
(35)
1
3? + ? ? , ? ?2 (1 ? ?).
?1 1 2
?2 2?2
=? =
0 0 2
2
Solving the system (35), we found exact solutions of the starting equation (32) [11].
The above results are only a sample of those already obtained. They illustrate the
very fruitful nature of conditional symmetry and conditional invariance, and I hope
that I have been able to demonstrate that there are new aspects to this concept which
are yet to be exploited fully.
Acknowledgements
I would like to express my heartfelt thanks to Professors M. Torrisi and A. Valenti
for such a well-organized conference and to Professor M. Boffi for enabling me to
attend it. I would also like to thank Dr. P. Basarab-Horwath, Mathematics Depart-
ment, Link?ping University, Sweden, for useful and fruitful discussions during the
o
final preparation of my talk.

1. Fushchych W., On symmetry and partial solutions of some multi-dimensional equations of mathe-
matical physics, in Algebraic-theoretical methods in the problems of mathematical physics, Kiev,
Institute of mathematics, Ukrainian Academy of Sciences, 1983, 4–23.
2. Fushchych W., Nikitin A., Symmetries of Maxwell’s equations, Kiev, Naukova Dumka, 1983;
Dordrecht, Kluwer, 1987.
16 W.I. Fushchych

3. Fushchych W., How can one extend the symmetry of differential equations? in Symmetry and
solutions of non-linear equations in mathematical physics, Kiev, Institute of mathematics, Ukrainian
4. Fushchych W., Tsyfra I., On reduction and solutions of the nonlinear wave-equation with broken
symmetry, J. Phys. A, 1987, 20, L45–L48.
5. Shulha M., Symmetry and some exact solutions of the D’Alembert equation with a nonlinear condi-
tion, in Symmetry and solutions of equations of mathematical physics, Kiev, Institute of Mathemati-
cs, Ukrainian Academy of Sciences, 1985, 36–38.
6. Fushchych W., Serov M., Chopyk W., Conditional invariance and non-linear heat equations, Proc.
Ukr. Acad. Sci., Ser. A, 1988, № 9, 17–21.
7. Fushchych W., Serov M., Conditional invariance and exact solutions of a non-linear equation of
acoustics, Proc. Ukr. Acad. Sci., Ser. A, 1988, № 10, 27–31.
8. Fushchych W., Shtelen W., Serov M., Symmetry analysis and exact solutions of nonlinear equations
of mathematical physics, Kiev, Naukova dumka, 1989; Dordrecht, Kluwer, 1993.
9. Fushchych W., Chopyk W., Conditional invariance of a non-linear Schr?dinger equation, Proc. Ukr.
o
Acad. Sci., Ser. A, 1990, № 4, 30–33.
10. Fushchych W., Conditional symmetry of the equations of mathematical physics, Ukr. Math. J., 1991,
43, № 11, 1456–1470.
11. Fushchych W., Serov M., Amerov T., On the conditional symmetry of the generalized Korteweg-de
Vries equation, Proc. Ukr. Acad. Sci., Ser. A, 1991, № 12, 28–30.
12. Fushchych W., Chopyk W., Myronyuk P., Conditional invariance and exact solutions of three-
dimensional non-linear equations in acoustics, Proc. Ukr. Acad. Sci. Ser. A, 1990, № 9, 25–28.
13. Fushchych W., Shtelen W., Serov M., Popowych R., Conditional symmetry of the linear heat
equation, Proc. Ukr. Acad. Sci., Ser. A, 1992, № 12, 20–26.
14. Fushchych W., Repeta W., Exact solutions of the equations of gas dynamics and non-linear acoustics,
Proc. Ukr. Acad. Sci., Ser. A, 1991, № 8, 35–42.
15. Fushchych W., Zhdanov R., Symmetry and exact solutions of non-linear spinor equations, Phys.
Rep., 1989, 46, 325–365.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 17–20.

Invariants of one-parameter subgroups
of the conformal group C(1, n)
W.I. FUSHCHYCH, L.F. BARANNIK, V.I. LAGNO
Дослiжена структура iнварiантiв одної з основних груп симетрiї математичної фiзи-
ки — конформної групи C(1, n) у просторi Мiнковського R(1, n) для n ? 3. Зокрема,
побудованi noвнi системи iнварiантiв однопараметричних пiдгруп групи C(1, n).

The conformal group C(1, n) of transformations in the Minkovsky space R(1, n)
takes the central place among the invariance Lie groups of mathematical physics [1].
Our interest in the functional invariants of one-parameter subgroups has been mo-
tivated by their application in finding solutions of diferential equations. In Ref. [2]
substitution (ansatz)
(1)
u(x) = f (x)?(?) + g(x)
for construction of exact solutions of multi-dimensional equations is proposed. In
ansatz (1) the unknown function ?(?) depends on the complete system of functional
invariants ?1 , ?2 , . . . , ?m of the one-parameter subgroups of the invariance Lie group
of the given equation.
In this paper complete systems of functional invariants of one-parameter subgroups
of the conformal group C(1, n) (n ? 3) are obtained. It should be noted that analogous
problem for Poincar? groups P (1, n) and P (2, n) in Refs. [3] and [4] is determined.
e
It is known that to each local Lie group corresponds its Lie algebra, in particular,
to group C(1, n) corresponds Lie algebra AC(1, n). Generators P? , J0a , Jab , D, K?
(? = 0, n, a, b = 1, n generate the basis of Lie algebra AC(1, n). We shall consider
the Lie algebra AC(1, n) as algebra of differential operators determined in the space
of scalar functions u(x) (x ? R(1, n)):
?
P? = ??? = ? , J?? = x? ?? ? x? ?? , x? = g?? x? ,
?x?
(2)
g?? = (1, ?1, . . . , ?1) ? ??? , D = ?x? ?? , K? = 2x? D + s2 ??
(s2 ? x? x? = x2 ? x2 ? . . . ? x2 ), ?, ? = 0, n.
n
0 1

It should be noted that the Lie algebra AC(1, n) (2) is an invariance algebra of
many differential equations [1].
The function F (x) (x ? R(1, n)) is the invariant of the on-parameter subgroups of
the group C(1, n) if and only if it is the solution of the differential equation
(3)
Lu = 0,
where L is the corresponding one-dimensional subalgebra of the Lie algebra AC(1, n)
(2) (see, for example, [5]).
Consequently, the problem of finding of invariants of one-parameter subgroups,
of the group C(1, n) is reduced to finding the system of functionally-independent
Доповiдi АН України, 1993, № 3, С. 45–48.
18 W.I. Fushchych, L.F. Barannik, V.I. Lagno

solutions of equation (3). Such systems of functionally-independent solutions of equa-
tion (3) will be called complete systems of invariants (CSI) of the corresponding
one-dimensional subalgebras of the algebra AC(1, n).
Lie algebra AP (1, n) = P? , J?? | ? = 0, n, ? = 1, n is subalgebra the algebra
AC(1, n). CSI of one-dimensional subalgebras of the algebra AP (1, n) are constructed
in [3]. Consequently, we shall describe CSI of the one-dimensional subalgebras of the
factor algebra AC(1, n)/AP (1, n).
One-dimensional subalgebras of the algebra AC(1, n)/AP (1, n) are such algebras
[3, 6]:

D + ?J0n (0 < ? ? 1); L2 = D + J0n + M ;
L1 =
Xt + ?D + ?J0n (? ? ? ? 0, ? = 0);
L3 =
L4 = Xt + ?(D + J0n + M ) (? > 0);
J12 + ?1 J34 + · · · + ? n ?1 Jn?1,n + ?D
L5 = 2

(n ? 0 (mod 2), ? > 0, 0 ? ?1 ? · · · ? ? n ?1 ? 1);
2

L6 = S + T ; L7 = S + T ± M ;
L8 = Xt + ?(S + T ) (? > 0); L9 = S + T + ?Z (? > 0);
L10 = Xt + ?(S + T ) ± M (? > 0);
L11 = Xt + S + T + G1 + P2 ; L12 = Xt + ?(S + T ) + ?Z (?, ? > 0); (4)
L13 = P0 + K0 ; L14 = ?(P0 + K0 ) + J12 (? > 0);
L15 = ?(P0 + K0 ) + J12 + ?1 J34 + · · · + ?s J2s+1,2s+2
(? > 0; 0 < ?1 ? . . . ? ?s ? 1; s = 1, 2, . . . , [(n ? 2)/2]);
L16 = ?(P0 + L0 ) + J12 + ?1 J34 + · · · + ? n?3 Jn?2,n?1 + ? n?1 (Kn ? Pn )
2 2

(? > 0, 0 < ?1 ? · · · ? ? n?1 ? 1);
2

L17 = J12 + ?1 J34 + · · · + ? n?1 (Kn ? Pn )
2

(0 < ?1 ? · · · ? +? n?1 ? 1),
2

where

Xt = ?1 J12 + ?2 J34 + · · · + ?t J2t?1,2t
(?1 = 1, 0 ? ?2 ? · · · ? ?t if t = 1; t = 1, 2, . . . , [(n ? 1)/2]),
M = P0 + Pn , T = 1 (P0 ? Pn ), S = 1 (K0 + Kn ), G1 = J01 ? J1n ,
2 2

Z = J0n ? D. In algebras L16 and L17 value n is an odd number.
Let y = y(x) = x0 + xn , z = z(x) = x0 ? xn , ha = ha (x) = x2 2
2a?1 + x2a ,
?a = ?a (x) = arctg xx2a , ? = ?(x) = x2 + x2 + · · · + x2 .
1 2 n?1
2a?1

Record L : f1 (x), f2 (x), . . . , fs (x) designates that functions f1 (x), f2 (x), . . . , fs (x)
form CSI of the algebra L.
Theorem. Following functions are CSI of one-dimensional subalgebras of the algeb-
ra AC(1, n)/AP (1, n):

zx?1?? , yx1 , x2 x?1 , x3 x?1 , . . . , xn?1 x?1 ;
??1
L1 : 1 1 1 1
Invariants of one-parameter subgroups of the conformal group C(1, n) 19

y ? ln |z|, y ? 2 ln |x1 |, x2 x?1 , x3 x?1 , . . . , xn?1 x?1 ;
L2 : 1 1 1
???? ???
z ? x2t+1 , y ? x2t+1 , ??1 ? ?1 ln |x2t+1 |, ??2 ? ?2 ln |x2t+1 |, . . . ,
L3 :
??t ? ?t ln |x2t+1 |, h1 x?2 , h2 x?2 , . . . , ht x?2 , x2t+2 x?1 ,
2t+1 2t+1 2t+1 2t+1
?1 ?1
x2t+3 x2t+1 , . . . , xn?1 x2t+1 ;
zx?2 , y ? 2 ln |x2t+1 |, ??1 ? ?1 ln |x2t+1 |, ??2 ? ?2 ln |x2t+1 |, . . . ,
L4 : 2t+1
??t ln |x2t+1 |, h1 x?2 , h2 x?2 , . . . , ht x?2 , h2 x?2 , . . . , ht x?2 ,
2t+1 2t+1 2t+1 2t+1 2t+1
?1 ?1 ?1
x2t+2 x2t+1 , x2t+3 x2t+1 , . . . , xn?1 x2t+1 ;
ln h1 ? 2??1 , ?1 ln h1 ? 2??2 , . . . , ? n ?1 ln h1 ? 2?? n ln h1 ? 2?? n ,
L5 : 2 2 2
2 ?1 ?1 ?1 ?1
x0 h1 , h2 h1 , . . . , h3 h1 , . . . , h 2 1 ;
nh

(1 + z 2 )x?2 , y ? z?(1 + z 2 )?1 , x2 x?1 , x3 x?1 , . . . , xn?1 x?1 ;
L6 : 1 1 1 1
2 ?2 ?1
2 ?1
y ± 2 arctg z ? z(1 + z ) ?, (1 + z )x1 , x2 x1 ,
L7 :
x3 x?1 , . . . , xn?1 x?1 ;
1 1
y ? z(1 + z ) ?, (1 + z 2 )h?1 , ??1 ? ?1 arctg z, x2 h?1 , h2 h?1 ,
2 ?1
L8 : 2t+1 1
1 1
?1
h3 h1 , . . . , ht h1 , ?2 ?1 ? ?1 ?2 , ?3 ?1 ? ?1 ?3 , . . . ,
1

?t ?1 ? ?1 ?t , x2t+2 x?1 , x2t+3 x?1 , . . . , xn?1 x?1 ;
2t+1 2t+1 2t+1
2? arctg z + ln(x1 (1 + z ) ), 2? arctg z + ln(y + z?(1 + z 2 )?1 ),
2 ?1
2
L9 :
x2 x?1 , x3 x?1 , . . . , xn?1 x?1 ;
1 1 1
?y ± 2 arctg z ? ?z(1 + z ) ?, ??1 ? ?1 arctg z, (1 + z 2 )h?1 ,
2 ?1
L10 : 1
?1 ?1 ?1 ?1 ?1 ?1
2
x2t+1 h1 , h2 h1 , h3 h1 , . . . , ht h1 , x2t+2 x2t+1 , x2t+3 x2t+1 , . . . ,
xn?1 x?1 , ?2 ?1 ? ?1 ?2 , ?3 ?1 ? ?1 ?3 , . . . , ?t ?1 ? ?1 ?t ;
2t+1
2 ?2
(1 + z )x2t+1 , (x1 + zx2 )(1 + z 2 )?1 , ?2 ? ?2 arctg z,
L11 :
y + 2(x1 + zx2 )(1 + z 2 )?1 arctg z ? z?(1 + z 2 )?1 ,
arctg z ? (x2 ? zx1 )(1 + z 2 )?1 , h2 x?2 , h3 x?2 , . . . , ht x?2 ,
2t+1 2t+1 2t+1
?1 ?1 ?1
x2t+2 x2t+1 , x2t+3 x2t+1 , . . . , xn?1 x2t+1 ;
2? arctg z + ? ln |y ? z?(1 + z 2 )?1 |, 2? arctg z + ? ln h1 (1 + z 2 )?1 ,
L12 :
??1 ? ?1 arctg z, ?2 ?1 ? ?1 ?2 , ?3 ?1 ? ?1 ?3 , . . . , ?t ?1 ? ?1 ?t ,
h2 h?1 , h3 h?1 , . . . , ht h?1 , x2t+2 x?1 , x2t+3 x?1 , . . . , xn?1 x?1 ;
1 1 1 2t+1 2t+1 2t+1
?1 ?1 ?1 ?1
(yz ? ? + 1)x1 , x2 x1 , x3 x1 , . . . , xn x1 ;
L13 :
2??1 ? arctg((yz ? ? ? 1)(2x0 )?1 ), (yz ? ? + 1)x?1 ,
L14 : 3
?1 ?1 ?1 ?1
h1 x3 , x4 x3 , x5 x3 , . . . , xn x3 ;
2??1 ? arctg((yz ? ? ? 1)(2x0 )?1 ), (yz ? ? + 1)2 h?1 , ?1 ?1 ? ?2 ,
L15 : 1
?2 ?1 ? ?3 , . . . , ?s ?1 ? ?s+1 , h2 h1 , h3 h1 , . . . , hs+1 h?1 ,
?1 ?1
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