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Lemma 6.1. Let ?, ?, ? ? R, ? > 0, ? ? 0, ? = 0, and F run through the full
system of representatives of the classes of O(1, 4)-conjugated subalgebras of the
algebra AO(1, 4) [4]. The subalgebras of the algebra AO(1, 4) ? D are exhausted
?
with respect to O(1, 4) conjugation by the algebras F , F ? D and the following
algebras:
J12 + ?D ; J12 + cJ34 + ?D (0 < c ? 1); J04 + ?D ; J12 + cJ04 + ?D (c > 0);
G3 +D ; G3 ?J12 +?D ; J12 +?D, J34 +?D ; J04 +?D, J12 +?D ; J04 , J12 +?D ;
G3 + D, J12 + ?D ; G3 , J12 + ?D ; G1 + D, G2 ; G3 , J04 + ?D ;
G3 , J12 + cJ04 + ?D (c > 0); G3 , J04 + ?D, J12 + ?D ; G3 , J04 , J12 + ?D ;
G1 , G2 , J12 + ?D ; G1 , G2 , J04 + ?D ; G1 , G2 , J12 + cJ04 + ?D (c > 0);
?
On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)
e 171

G1 + D, G2 , G3 ; G1 , G2 , G3 ? J12 + ?D ; J03 , J04 , J34 , J12 + ?D ;
J12 + J34 , J13 ? J24 , J23 + J14 , J34 + ?D ; G1 , G2 , J12 + ?D, J04 + ?D ;
G1 , G2 , J12 , J04 + ?D ; G1 , G2 , G3 + D, J12 + ?D ; G1 , G2 , G3 , J12 + ?D ;
G1 , G2 , G3 , J04 + ?D ; G1 , G2 , G3 , J12 + cJ04 + ?D (c > 0);
J12 , J13 , J23 , J04 + ?D ; G1 , G2 , G3 , J12 + ?D, J04 + ?D ;
G1 , G2 , G3 , J12 , J13 , J23 , J04 + ?D .
Lemma 6.1 is proved with the Goursat method [25] and the result on the classifi-
cation of subalgebras of the algebra AO(1, 4) [4].
Theorem 6.1. Let ?(?) be the system of representatives of the classes of conjugated
?
subalgebras of the algebra AO(1, 4) (respectively, AO(1, 4)) found in Lemma 6.1.
?
The splitting subalgebras of the algebra AP (1, 4) are exhausted with respect to
?
P (1, 4) conjugation by the following algebras:
(1) W ? , where F ? ?, W ? U , and [F, W ] ? W ;
+F
+? ? ?
(2) W ? F , where F ? ? and the projection of F onto AO(1, 4) coincides with
F , F ? ?;
(3) J12 , J34 + ?D : P1 , P2 , P0 , P1 , P2 (? > 0);
(4) G1 + ?D, G2 + ?D : M, P1 , M, P1 + ?P3 , M, P1 , P3 , M, P1 + ?P3 , P2
(? > 0, ? ? 0, ? ? 0, ?2 + ? 2 = 0);
(5) G1 + ?D, G2 + ?D, G3 , M, P1 (? ? 0, ? ? 0, ?2 + ? 2 = 0);
(6) G1 + ?D, G2 , G3 + ?D, M, P1 , P2 (? ? 0, ? ? 0, ?2 + ? 2 = 0).
?
Proof. Let F be the subdirect sum of F ? ? and D, and W a subspace of U
?
invariant under F . Then [F, W ] ? W and on the contrary, if [F, W ] ? W then
?
[F , W ] ? W . Therefore we can use the results on the classification of the splitting
?
subalgebras of AP (1, 4) [9]. Only the cases of the algebras F ? ? simplified by
O(1, 4) automorphisms demand an additional consideration. Such algebras correspond
to the algebra F coinciding with J12 , J34 , G1 , G2 , or G1 , G2 , G3 . If, for example,
?
F = G1 + ?1 D, G2 + ?2 D, G3 + ?3 D then this algebra must be simplified using
transformations contained in the normalizer of M, P1 , M, P1 , P2 , respectively, in
the group of O(1, 4) automorphisms. The theorem is proved.
?
We conceive the classification of nonsplitting subalgebras of AP (1, 4) with respect
?
to P (1, 4) conjugation by virtue of the known classification of the nonsplitting sub-
algebras of AP (1, 4) with respect to P (1, 4) conjugation [11]. The application of the
automorphism exp(?D) allows us to substitute one of the continuous parameters by
the translation generators onto 1.
Let (i1 , . . . , iq ) = Pi1 , . . . , Piq ; (a?b) = Pa + ?Pb (? > 0); (04) = M .
?
Theorem 6.2. The nonsplitting subalgebras of the algebra AP (1, 4) are exhausted
?
with respect to P (1, 4) conjugation by the nonsplitting subalgebras of the algebra
AP (1, 4) and the following algebras:
J04 ? D + P0 : 0, (1), (04), (1,2), (04,1), (1,2,3), (04,1,2), (04,1,2,3);
J12 + c(J04 ? D + P0 ) : 0, (04), (3), (04,3), (1,2), (1,2,3), (04,1,2), (04,1,2,3)
(c > 0);
J04 + D + M, J12 + ?M : 0, (3), (1,2), (1,2,3) (? > 0);
J04 + D, J12 + M : 0, (3), (1,2), (1,2,3); J04 + D + M, J12 : 0, (3), (1,2), (1,2,3);
J04 ? D + P0 , J12 + ?P0 : (04), (04,3), (04,1,2), (04,1,2,3) (? ? 0);
J04 ? D, J12 + P0 : (04), (04,3), (04,1,2), (04,1,2,3);
172 L.F. Barannik, W.I. Fushchych

J04 ? 2D, G3 + P0 : (04), (04,1), (04,1?3), (04,3), (04,1?3,2), (04,1,2), (04,1,3),
(04,1,2,3);
J04 ? 2D, G3 + P0 ? P4 : 0, (1), (1,2); J04 ? D, G3 + P1 : 0, (04), (04,3), (0,3,4);
J04 ? D, G3 + P2 : (1), (04,1), (04,1?3), (04,1,3), (0,1,3,4);
G3 + ?P1 , J04 ? D + P0 , M, P3 (? > 0); J04 ? D + P0 , G3 + ?P2 , M, P1 , P3
(? > 0);
G3 , J04 ? D + P0 : (04,3), (04,1,3), (04,1,2,3); G3 , J04 + D + M : 0, (1), (1,2);
G3 + P0 , J12 + c(J04 ? 2D) : (04), (04,3), (04,1,2), (04,1,2,3) (c > 0);
G3 + P0 ? P4 , J12 + c(J04 ? 2D) : 0, (1,2) (c > 0);
G3 , J12 + c(J04 ? D + P0 ) : (04,3), (04,1,2,3); G3 , J12 + c(J04 + D + M ) : 0, (1,2);
G3 + P0 , J12 , J04 ? 2D : (04), (04,3), (04,1,2), (04,1,2,3);
G3 + P0 ? P4 , J12 , J04 ? 2D : 0, (1,2);
G3 , J12 + ?P0 , J04 ? D + P0 : (04,3), (04,1,2,3) (? ? 0);
G3 , J12 + P0 , J04 ? D : (04,3), (04,1,2,3);
G3 , J12 + ?M, J04 + D + M : 0, (1,2) (? ? 0); G3 , J12 + M, J04 + D : 0, (1,2);
G1 , G2 + P0 , J04 ? 2D : (04,1), (04,1,2), (04,1,2?3), (04,1,3), (04,1,2,3);
G1 + P3 , G2 + µP2 + ?P3 , J04 ? D (µ > 0, ? ? 0); G1 + P3 , G2 , J04 ? D ;
G1 , G2 + P2 + ?P3 , J04 ? D (? ? 0); G1 , G2 + P2 , J04 ? D, P3 ;
G1 + P2 + ?P3 , G2 ? P1 + µP2 + ?P3 , J04 ? D, M (µ > 0, ? > 0 ? ? = 0, ? ? 0);
G1 + P2 + ?P3 , G2 ? P1 , J04 ? D, M (? ? 0); G1 + P3 , G2 , J04 ? D, M ;
G1 + ?P3 , G2 + P2 + ?P3 , J04 ? D, M (? > 0 ? ? = 0, ? ? 0);
G1 + P2 , G2 ? P1 + µP2 , J04 ? D, M, P3 (µ ? 0); G1 , G2 + P2 , J04 ? D, M, P3 ;
G1 + P2 , G2 ? P1 + µP2 , J04 ? D, M, P3 (µ ? 0);
G1 + ?P2 + ?P3 , G2 + P3 , J04 ? D, M, P1 (? > 0 ? ? = 0, ? ? 0);
G1 + P2 + ?P3 , G2 , J04 ? D, M, P1 (? ? 0); G1 + P3 , G2 , J04 ? D, M, P1 ;
G1 + ?P2 + ?P3 , G2 + P3 , J04 ? D, M, P1 + ?P3 (? > 0);
G1 + P2 + ?P3 , G2 , J04 ? D, M, P1 + ?P3 (? > 0);
G1 + P3 , G2 , J04 ? D, M, P1 + ?P3 (? > 0); G1 + P3 , G2 , J04 ? D, M, P1 , P2 ;
G1 + P2 , G2 , J04 ? D, M, P1 , P3 ; G1 , G2 + P3 , J04 ? D, M, P1 + ?P3 , P2 (? > 0);
G1 +P3 , G2 , J04 ?D, P0 , P1 , P2 , P4 ; G1 +?P3 , G2 , J04 ?D+P0 , M, P1 , P2 (? ? 0);
G1 , G2 , J04 ? D + P0 , M, P1 , P2 , P3 ; G1 , G2 , J04 + D + M ; G1 , G2 , J04 + D +
M, P3 ;
G1 + P2 , G2 ? P1 , J12 + c(J04 ? D) : (04), (04,3) (c > 0);
G1 , G2 , J12 + c(J04 ? D + P0 ), M, P1 , P2 , sP3 (c > 0, s = 0, 1);
G1 , G2 , J12 + c(J04 + D + M ) : 0, (3) (c > 0);
G1 , G2 , J12 + P0 , J04 ? D, M, P1 , P2 , sP3 (s = 0, 1);
G1 , G2 , J12 + M, J04 + D : 0, (3);
G1 , G2 , J12 + ?P0 , J04 ? D + P0 , M, P1 , P2 , sP3 (? ? 0, s = 0, 1);
G1 + P2 , G2 ? P1 , J12 , J04 ? D, M, sP3 (s = 0, 1);
G1 , G2 , J12 + ?M, J04 + D + M : 0, (3) (? ? 0);
G1 , G2 , G3 + P0 , J04 ? 2D, M, P1 , P2 , sP3 (s = 0, 1);
G1 , G2 + P2 , G3 + ?P3 , J04 ? D ; G1 , G2 + P2 , G3 + ?P3 , J04 ? D, M ;
G1 + P2 + ?P3 , G2 ? P1 + µP2 + ?P3 , G3 + ?P1 + ?P2 + ?P3 , J04 ? D, M (µ ? 0,
? > 0 ? ? = 0, ? ? 0);
G1 + P2 + ?P3 , G2 ? P1 , G3 + ?P1 + ?P3 , J04 ? D, M (? ? 0);
G1 + ?P2 , G2 + P3 , G3 ? P2 , J04 ? D, M, P1 (? ? 0);
?
On subalgebras of the Lie algebra of the extended Poincar? group P (1, n)
e 173

G1 + ?P2 + ?P3 , G2 + P3 , G3 ? P2 + µP3 , J04 ? D, M, P1 (µ > 0, ? > 0 ? ? = 0,
? ? 0);
G1 + ?P2 + ?P3 , G2 , G3 + P3 , J04 ? D, M, P1 (? > 0 ? ? = 0, ? ? 0);
G1 + P2 , G2 , G3 , J04 ? D, M, P1 ; G1 + P3 , G2 , G3 , J04 ? D, M, P1 , P2 ;
G1 , G2 , G3 , J04 ? D + P0 , M, P1 , P2 , P3 ; G1 , G2 , G3 , J04 + D + M ;
G1 , G2 , G3 + P0 , J12 + c(J04 ? 2D), M, P1 , P2 , sP3 (c > 0, s = 0, 1);
G1 + P2 , G2 ? P1 , G3 + ?P3 , J12 + c(J04 ? D), M (c > 0);
G1 + P2 , G2 ? P1 , G3 , J12 + c(J04 ? D), M, P3 (c > 0);
G1 , G2 , G3 + P3 , J12 + c(J04 ? D) : 0, (04);
G1 , G2 , G3 , J12 + c(J04 + D + M ) (c > 0);
J12 , J13 , J23 , J04 ? D + P0 : 0, (04), (1,2,3), (04,1,2,3);
G1 , G2 , G3 + P0 , J12 , J04 ? 2D, M, P1 , P2 , sP3 (s = 0, 1);
G1 , G2 , G3 , J12 + P0 , J04 ? D, M, P1 , P2 , P3 ;
G1 , G2 , G3 , J12 + ?P0 , J04 ? D + P0 , M, P1 , P2 , P3 (? ? 0);
G1 + P2 , G2 ? P1 , G3 + ?P3 , J12 , J04 ? D, M ;
G1 + P2 , G2 ? P1 , G3 , J12 , J04 ? D, M, P3 ; G1 , G2 , G3 + P3 , J12 , J04 ? D : 0, (04);
G1 , G2 , G3 , J12 + M, J04 + D ; G1 , G2 , G3 , J12 + ?M, J04 + D + M (? ? 0);
G1 , G2 , G3 , J12 , J13 , J23 , J04 ? D + P0 , M, P1 , P2 , P3 ;
G1 , G2 , G3 , J12 , J13 , J23 , J04 + D + M .

7. Conclusions
The results of the present paper may be summarized in the following way.
?
(1) The maximal Abelian subalgebras of the algebra AP (1, n) have been explicitly
found in Corollary 1 to Theorem 4.2 and Corollary 1 to Theorem 5.1.
?
(2) The full classification of one-dimensional subalgebras of algebra AP (1, n) is
contained in Corollary 2 to Theorem 4.2, Corollary 2 to Theorem 5.1 and Proposition
5.10.
?
(3) The completely reducible subalgebras of AO(1, n) which possess only splitting
?
extensions in the algebra AP (1, n) have been picked out. We have established in
? ?
Theorem 3.1 that the description of the splitting subalgebras F of AP (1, n) whose
?
projection F onto AO(1, n) does not have any invariant isotropic subspaces in the
space of translations or annul such subspaces, could be reduced to the description of
the irreducible parts of the algebra F .
(4) A number of assertions on the subalgebras of the algebra U ? +K has been
? n). These assertions concern
proved where K is the normalizer of P0 + Pn in AO(1,
the following matters: The splittability of all extensions of the subalgebra L ? K in
?
AP (1, n) or in some other algebras (Propositions 4.1, 4.2, 5.1, and 5.2); the decompo-
sition of invariant subspaces into a direct sum of its projections onto certain subspaces
(Propositions 5.3, 5.5, 5.6, 5.7, and 5.8); the explicit description of some classes of
?
the conjugated subalgebras of the algebra AP (1, n) (Theorem 4.1, Propositions 4.4
and 5.4).
?
(5) The full classification with respect to P (1, 4) conjugation of the nonsplitting
?
subalgebras of AP (1, 4) which are nonconjugate to the subalgebras of AP (1, 4) has
been carried out.
Note added in proof: In Refs. [26–28] the subalgebras of the algebra AP (1, n)
were used to construct the exact solutions of many-dimensional nonlinear d’Alembert
and Dirac equations. The invariants of subgroups of the generalized Poincar? groupe
174 L.F. Barannik, W.I. Fushchych

P (1, n) were constructed in Ref. [29]. A number of general results on continuous
subgroups of pseudoorthogonal pseudounitary groups had been obtained [30].
Acknowledgment. We are grateful to the referee for his valuable remarks.

1. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1975, 16, 1597.
2. Patera J., Winternitz P., Sharp R.T., Zassenhaus H., Can. J. Phys., 1976, 54, 950.
3. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1975, 16, 1615.
4. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1976, 17, 717.
5. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., J. Math. Phys., 1977, 18, 2259.
6. Burdet G., Patera P., Perrin M., Winternitz P., J. Math. Phys., 1978, 19, 1758.
7. Beckers J., Patera J., Perroud M., Winternitz P., J. Math. Phys., 1977, 18, 72.
8. Burdet G., Patera J., Perrin M., Winternitz P., Ann. Sci. Math. Quebec, 1978, 2, 81.
9. Fedorchuk V.M., Ukrainian Math. J., 1979, 31, 717.
10. Fedorchuk V.M., Ukrainian Math. J., 1981, 33, 696.
11. Fushchych W.I., Barannik A.F., Barannik L.F., Fedorchuk V.M., J. Phys. A: Math. Gen., 1985, 18,
2893.
12. Fushchych W.I., Barannik A.F., Barannik L.F., The continuous subgroups of the generalized Galilei
group. I, Preprint 85.19, Mathematical Institute, Kiev, 1985.
13. Fushchych W.I., Barannik A.F., Barannik L.F., Ukrainian Math. J., 1986, 38, 67.
14. Lassner W., Acta Phys. Slovaca, 1973, 23, 193.
15. Bacry H., Combe Ph., Sorba P., Rep. Math. Phys., 1974, 5, 145.
16. Bacry H., Combe Ph., Sorba P., Rep. Math. Phys., 1974, 5, 361.
17. Sorba P., J. Math. Phys., 1976, 17, 941.
18. Fushchych W.I., The symmetry of mathematical physics problems, in Algebraic-Theoretical Studies
in Mathematical Physics, Mathematical Institute, Kiev, 1981 (in Russian).
19. Fushchych W.I., On symmetry and particular solutions of some multidimensional equations of
mathematical physics, in Algebraic-Theoretical Methods in Mathematical Physics Problems, Ma-
thematical Institute, Kiev, 1983 (in Russian).
20. Grundland A.M., Harnad J., Wintemitz P., J. Math. Phys., 1984, 25, 791.
21. Nikitin A.G., Fushchych W.I., Jurik I.I., Theor. Math. Phys., 1976, 26, 206.
22. Fushchych W.I., Nikitin A.G., J. Phys. A: Math. Gen., 1980, 13, 2319.
23. Jacobson N., Lie algebras, Dover, New York, 1962.
24. Goto M., Grosshans F.D., Semisimple Lie algebras, Dekker, New York, 1978.
25. Coursat E., Ann. Sci. Ecole Norm. Sup., 1889, 3, 69.
26. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Reidel, Dordrecht, 1987.
27. Fushchych W.I., Serov N.I., J. Phys. A: Math. Gen., 1983, 16, 3645.
28. Fushchych W.I., Shtelen W.M., J. Phys. A: Math. Gen., 1985, 18, 271.
29. Barannik L.F., Fushchych W.I., The invariants of subgroups of the generalized Poincar? group
e
P (1, n), Preprint 86.86, Mathematical Institute, Kiev,1986.
30. Barannik A.F., Fushchych W.I., On continuous subgroups of pseudoorthogonal and pseudounitary
groups, Preprint 86.87, Mathematical Institute, Kiev,1986.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 175–179.

The symmetry and exact solutions of some
multidimensional nonlinear equations
of mathematical physics
W.I. FUSHCHYCH

We discuss exact solutions of some nonlinear equations obtained in collaboration
with Shtelen W.M., Serov M.I., Zhdanov R.Z. (Institute of Mathematics, Kiev).
We consider the following equations:
— The nonlinear wave equation
pµ pµ u + F1 (u) = 0, (0.1)
u = u(x) scalar, x = (x0 , x1 , . . . , xn ) ? R(1, n), F1 (u) twice differentiable, pµ =
i?/?xµ .
— The generalized Monge–Ampere equation
det(uµ? ) = F2 (x, u, u),
1
(0.2)
?2u ?u ?u ?u
u=
uµ? = , , ,..., ,
?xµ ?x? ?x0 ?x1 ?xn
1

F2 smooth. With F2 = 0, we get the Monge–Ampere equation used in differential
geometry and, especially at present, in quantum field theory.
— The multidimensional hyperbolic analog of the Euler–Lagrange equation for
minimal surfaces or the Born–Infeld equation
2u(1 ? u? u? ) + uµ? uµ u? = 0,
(0.3)
2 2 2
?u ?u ?u
? ? ··· ?
?
u? u = .
?x0 ?x1 ?xn
— The nonlinear Schr?dinger equation
o
p2
p0 ? a
(0.4)
u + F3 (u)u = 0,

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