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e
? ?
vector fields P? , J?? and the extended Poincar? algebra AP (1, n) = AP (1, n) + D .
e
In order to reduce the equation (1) by subalgebras of the algebra AC(1, n), it
is necessary to describe all C(1, n)-nonequivalent subalgebras of this algebra. The
subalgebras K1 and K2 of the algebra AC(1, n) are called as C(1, n)-equivalent ones
if they have the same invariants with respect to C(1, n)-conjugation. Among C(1, n)-
equivalent algebras there exists one (maximal) subalgebra containing all the other
subalgebras. The maximal subalgebras K1 and K2 of the algebra AC(1, n) are equi-
valent if and only if K1 and K2 are C(1, n)-conjugated.
The maximal subalgebras of the rank n of the algebra AP (1, n) with respect to
P (1, n)-conjugation are described in [2]. The maximal subalgebras of the rank n of
? ?
the algebra AP (1, n) with respect to P (1, n)-conjugation are described in [3, 4].
The present article is a continuation of researches which were realized in [3, 4].
The full classification of the maximal subalgebras of the rank n ? 1 of the algebra
?
AC(1, n) which are contained in the algebra AP (1, n) has been carried out in the
present article. Ansatzes corresponding to these subalgebras reduce the equation (1)
to ordinary differential equations.
We will use the notations:
1
(P0 ? Pn ), Ga = J0n ? Jan , a = 1, . . . , n ? 1,
M = P0 + Pn , T=
2
AO[r, s] = Jab | a, b = r, . . . , s , r ? s,
Доповiдi АН України, 1993, № 6, С. 38–41.
6 A.F. Barannyk, W.I. Fushchych

AE[r, s] = Pr , . . . , Ps + AO[r, s], r ? s,
?
AE1 [r, s] = Gr , . . . , Gs + AO[r, s], r ? s.
?

If s > r then AO[r, s] = 0, AE[r, s] = 0 by definition.
Let
? r, s ? N, r ? s, ? ? R.
?(r, s, ?) = Gr + ?Pr , . . . , Gs + ?Ps + AO[r, s],
Let
?
?d,q = U + F,
where F is the diagonal of AO[1, d] ? AO[d + 1, 2d] ? · · · ? AO[(q ? 1)d + 1, qd], and
U is the Abelian algebra which has the basis
G1 + ?1 P1 + ?1 P(q?1)d+1 , . . . , Gd + ?1 Pd + ?1 Pqd ,
Gd+1 + ?2 Pd+1 + ?2 P(q?1)d+1 , . . . , G2d + ?2 P2d + ?2 Pqd ,
·························································
G(q?2)d+1 + ?q?1 P(q?2)d+1 + ?q?1 P(q?1)d+1 , . . . , G(q?1)d +
+ ?q?1 P(q?1)d + ?q?1 Pqd ,
where 0 ? ?1 < ?2 < · · · < ?q?1 , ?1 > 0, ?2 > 0, . . . , ?q?1 > 0.
Results of the work [5] reduce the problem constructing invariants of any subalgeb-
?
ra of the algebra AP (1, n) to the problem of constructing invariants of the irreducible
subalgebras of the orthogonal algebra AO(k) for all k ? n. The latter problem has
no solution in quadratures. Therefore, we shall restrict ourself considering of such
?
subalgebras of the algebra AP (1, n) which projections onto AO[1, n] are subdirect
sums on the algebras AO[r, s]. Moreover, to find real solutions of the equation (1) it
is necessary to exclude from consideration such subalgebras of the algebra AP (1, n)
which with respect to equivalence contain P0 + Pn or P0 . Therefore we prove the
following theorems.
Theorem 1. Let L be the maximal subalgebra of the rank n ? 1 of the algebra
AP (1, n). Then L is C(1, n)-conjugated with one of the following algebras:
1) L1 = AE[1, n ? 1];
2) L2 = AO[1, m] ? AE[m + 1, n], m = 1, . . . , n, n ? 2;
3) L3 = AE1 [1, m] ? AE[m + 1, n ? 1], m = 1, . . . , n ? 1, n ? 2;
4) L4 = AO[1, m] ? AE[m + 1, n ? 1] ? J0n , m = 1, . . . , n ? 1, n ? 3;
5) L5 = AO[0, m] ? AE[m + 1, n ? 1], m = 2, . . . , n ? 1, n ? 3;
6) L6 = AO[0, m]?AO[m+1, q]?AE[q+1, n?1], m = 2, . . . , n?1, q = m+1, . . . , n,
n ? 3;
7) L7 = G1 + P0 ? Pn ? AE[2, n ? 1], n ? 2;
8) L8 = ?(d0 +1, d1 , ?1 )?· · ·??(dt?1 +1, m, ?t )?AE[m+1, n?1], m = 1, . . . , n?1,
n ? 3;
9) L9 = J0n + P1 ? AE[2, n ? 1], n ? 2;
10) L10 = (AE1 [1, m] ? J0n + Pm+1 ) ? AE[m + 2, n ? 1], m = 1, . . . , n ? 2, n ? 3;
11) L11 = J12 + P0 ? AE[3, n], n ? 2.
Theorem 2. Let L be the maximal subalgebra of the rank n ? 1 of the algebra
?
AP (1, n) which has a nonzero projection onto D . Then L is C(1, n)-conjugated
with one of the following algebras:
On maximal subalgebras of the rank n ? 1 of the conformal algebra AC(1, n) 7

1) L1 = (AO[0, d] ? AO[d + 1, m] ? AO[m + 1, q] ? AE[q + 1, n]) + D , d = ?
2, . . . , n ? 2, m = d + 1, . . . , n ? 2, q = m + 1, . . . , n ? 1, 2n ? d + q, n ? 4;
2) L2 = (AO[0, m] ? AE[m + 1, n ? 2]) + D + ?Jn?1,n , m = 2, . . . , n ? 2, n ? 4,
?
? > 0;
3) L3 = (AO[1, m] ? AO[m + 1, q] ? AE[q + 1, n]) + D , m = 2, . . . , n ? 2,
?
q = m + 2, . . . , n, 2m ? q, n ? 2;
4) L4 = (AE1 [1, m] ? AE[m + 1, n ? 3]) + Jn?2,n?1 + cJ0n , D + ?J0n , m =
?
1, . . . , n ? 3, n ? 4, c > 0, ? ? 0;
5) L5 = (AO[1, m] ? AO[m + 1, q] ? AE[q + 1, n ? 1]) + D, J0n , m = 1, . . . , n ? 2,
?
q = m + 1, . . . , n ? 1, 2m ? q, n ? 3;
6) L6 = AE[3, n ? 1] + J12 + cJ0n , D + ?J0n , c > 0, ? ? 0, n ? 3;
?
?
7) L7 = (AE1 [1, d]?AO[d+1, m]?AE[m+1, n?1]) + D+?J0n , d = 1, . . . , n?2,
m = d + 1, . . . , n ? 1, n ? 3, ? ? 0;
8) L8 = (AO[1, m] ? AE[m + 1, n ? 1]) + D + ?J0n , m = 1, . . . , n ? 1, n ? 2,
?
? ? 0;
9) L9 = ( G1 +2T ?AO[2, m]?AE[m+1, n?1]) + 2D ?J0n , m = 2, . . . , n?1,
?
n ? 3;
10) L10 = (AE1 [1, d] ? AO[d + 1, m] ? AE[m + 1, n ? 1]) + D + J0n + M ,?
d = 1, . . . , n ? 2, m = d + 1, . . . , n ? 1, n ? 3;
11) L11 = (AO[1, m] ? AE[m + 1, n ? 1]) + D + J0n + M , m = 1, . . . , n ? 1,
?
n ? 2;
12) L12 = (AE1 [1, m] ? AE[m + 1, n ? 3]) + Jn?2,n?1 + ?M, D + J0n + M ,
?
m = 1, . . . , n ? 3, n ? 4, ? ? 0;
13) L13 = (AE1 [1, m] ? AE[m + 1, n ? 3]) + Jn?2,n?1 + M, D + J0n , m =
?
1, . . . , n ? 3, n > 4;
14) L14 = AE[3, n ? 1] + J12 + ?M, D + J0n + M , n ? 3, ? ? 0;
?
15) L15 = AE[3, n ? 1] + J12 + M, D + J0n , n ? 3;
?
16) L16 = (?d,q ? AE[dq + 1, n ? 1] + D ? J0n , d ? 2, n ? 5;
?
17) L17 = (?(d0 + 1, d1 , ?1 ) ? ?(d1 + 1, d2 , ?2 ) ? · · · ? ?(dt?1 + 1, dt , ?t ) ? AO[dt +
1, m] ? AE[m + 1, n ? 1]) + D ? J0n , where d0 = 0, ?1 < ?2 < · · · < ?t , t > 1,
?
m = 1, . . . , n ? 2, n ? 3;
18) L18 = (?d,q ? ?(l0 + 1, l1 , µ1 ) ? ?(l1 + 1, l2 , µ2 ) ? · · · ? ?(lt?1 + 1, lt , µt ) ?
AE[lt + 1, n ? 1]) + D ? J0n , where µ1 < µ2 < · · · < µt , t ? 1, l0 = dq.
?
L1 –L11 and L1 –L18 of the theorems 1 and 2 respectively and to carry out a
reduction of the equation (1). Consider, for example, the subalgebra L17 . The ansatz
t
1
u = ?(x0 + xm ] + x2i?1 +1 + · · · x2i ?(?) ?
2
x0 ? xm + ?i d d
i=1
? ? ··· ? x2 , ? = x0 ? xm ,
x2t +1 m?1
d

corresponds to this subalgebra. This ansatz reduces the equation (1) to equation
?? ? ? = 0. Using the solution of this equation we find the following solution of the
?
equation (1):
t
1
u = ?(x0 + xm ] + x2i?1 +1 + · · · + x2i ?
2
x0 ? xm + ?i d d
i=1
? (x0 ? xm + C) ? x2t +1 ? · · · ? x2 .
m?1
d
8 A.F. Barannyk, W.I. Fushchych

1. Fushchych W.I, Shtelen W.M., The symmetry and some exact solutions of the relativistic eikonal
equation, Lett. Nuovo Cimento, 1982, 34, 498–502.
2. Grundland A.M., Harnad L., Winternitz P., Symmetry reduction for nonlinear relativistically invari-
ant equations, J. Math. Phys., 1984, 25, № 4, 791–806.
3. Фущич В.И., Баранник Л.Ф., Максимальные подалгебры ранга n ? 1 алгебры AP (1, n) и реду-
кция нелинейных волновых уравнений. I, Укр. мат. журн., 1990, 42, № 11, 1260–1256.
4. Фущич В.И., Баранник Л.Ф., Максимальные подалгебры ранга n ? 1 алгебры AP (1, n) и реду-
кция нелинейных волновых уравнений. II, Укр. мат. журн., 1990, 42, № 12, 1693–1700.
5. Баранник А.Ф., Баранник Л.Ф., Фущич В.И., Редукция многомерного пуанкаре-инвариантного
нелинейного уравнения к двумерным уравнениям, Укр. мат. журн., 1991, 43, № 10, 1314–1323.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 9–16.

Conditional symmetries of the equations
of mathematical physics
W.I. FUSHCHYCH
We briefly present the results of research in conditional symmetries of equations of
mathematical and theoretical physics: the Maxwell, D’Alembert, Schr?dinger and KdV
o
equations, as well as the equations of heat conduction and acoustics. Exploiting condi-
tional symmetry, we construct a wide class of exact solutions of these equations, which
cannot be obtained by the classical method of Sophus Lie.

1. Introduction
The concept and terminology of conditional symmetry and conditional invariance
were introduced and developed in the series of articles [1–11] (see also Mathematical
Reviews for the years 1983–1993). Later, this concept was exploited by other authors
for the construction of solutions of various non-linear equations of mathematical phy-
sics. It turned out that nearly all the basic non-linear equations of mathematical
physics have non-trivial conditional symmetry [2, 9, 10].
We understand the conditional symmetry of an equation as being a symmetry (local
or non-local) of some non-trivial subset of its solution set (the formal definition of the
idea of conditional symmetry can be found in Appendix 4 of [2] and in the article [3]).
The general definition of conditional symmetry as the symmetry of a subset of the
set of solutions is non-constructive and requires further specification: the analytical
description of a condition (as an equation) on the solutions of the given equation,
which extend or alter the symmetry of the starting equation. Therefore, the basic
problem in the investigation of conditional symmetries is that of describing those
supplementary equations which increase or change the symmetry of the beginning
equation. This is very complex, non-linear problem in general (even in the case of
quite simple non-linear equations), which can often be significantly more complicated
than constructing solutions of the equation at hand. It is thus meaningful to talk of
the conditional symmetry of some class of equations.
Non-trivial conditional symmetries of a PDE (partial differential equation) allows
us to obtain in explicit form such solutions which can not be found by using the
symmetries of the whole set of solutions of the given PDE. Moreover, conditional
symmetries increase significantly the class of PDEs for which we can construct
ansatzes which reduce these equations to (systems of) ODEs (ordinary differential
equations). As a rule, the reduced equations one obtains from conditional symmetries
are significantly simpler than those found by reduction using symmetries of the full
set of solutions. This allows us to construct exact solutions of the reduced equations.
Looking back, we can say today, that many mathematicians, mechanicians and
physicists, such as Euler, D’Alembert, Poincar?, Volterra, Whittaker, Bateman, impli-
e
citly used conditional symmetries for the construction of exact solutions of the linear
in Proceedings of the International Workshop “Modern Group Analysis: Advanced Analytical and
Computational Methods in Mathematical Physics”, Editors N.H. Ibragimov, M. Torrisi, A. Valenti,
Dordrecht, Kluwer Academic Publishers, 1993, P. 231–239.
10 W.I. Fushchych

wave equation. Some well-known solutions of this equation can not be obtained by
using only Lie symmetries of the full solution set.
2. Conditional symmetry of Maxwell’s equation
We shall first consider the first pair of Maxwell’s equations
?E ?H
= rot H, = ?rot E. (1)
?t ?t
The maximal invariance algebra (in the sense of Lie) of these equations is studied
in [2]. The basis elements of this algebra ?0 , ?a , Jab , D are
? ?
Jab = xa ?b ? xb ?a + sab ,
?0 = , ?a = , a, b = 1, 2, 3,
0 ?xa (2)
?x
D = xµ ?µ + const,
sab are 6 ? 6 matrices realizing a representation of the group O(3). Thus the sys-
tem (1) is invariant under the four-dimensional translations ?µ , the rotations Jab and
scale transformations D, but it is not invariant under the Lorentz boosts
J0a = x0 ?a ? xa ?0 + s0a , (3)
x0 = t,
the matrices s0a , sab realizing a representation of the Lorentz group O(1, 3).
Theorem 1 ([2] 1983, [15] 1987). The system (1) is conditionally invariant under
the Lorentz boosts (3) if and only if the solutions of (1) satisfy the conditions
div E = 0, div H = 0. (4)
It is evident from this theorem, that the concept of conditional invariance of
a PDE is natural, and leads us, by purely group-theoretic means, to the fundamental,
overdetermined system of Maxwell’s equations.
3. Conditional symmetry of the wave equation
We now examine the non-linear D’Alembert equation
2u = F (u), (5)
u = u(x0 , x1 , x2 , x3 ),
F (u) being an arbitrary, smooth function. Equation (5) has conformal symmetry
C(1, 3) if and only if F = ?u3 or F = 0 (see for instance [8, 10]). This is the
maximal symmetry of all of the solution set of equation (5). For an arbitrary function,
(5) admits only the symmetry groups P (1, 3).
Theorem 2 ([5], 1985). Equation (5), with F = 0 is conditionally invariant under
the infinite-dimensional algebra with basis elements
? ?
X = ? µ (x, u) (6)
+ ?(x, u) ,
µ
?x ?u
? µ (x, u) = c00 (u)xµ + cµ? (u)x? + dµ (u), (7)
?(x, u) = ?(u),

where c00 (u), cµ? (u), dµ (u), ?(u) are arbitrary functions of u, if one imposes the
condition
?u ?u
(8)
= 0.
?xµ ?xµ
Conditional symmetries of the equations of mathematical physics 11

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