стр. 2 |

? ?

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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