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?
2
i
? ?1 ?4 ? + i?1 ? = ?1 ,
(6) ?
2?
i
? ?1 ?4 ? + i[?(?0 + ?3 )e??/? ? ?2 ]? = ?1 ,
(7) ?
2?
1 ?1/2
?2 ? + 2i? 1/2 ?2 ? = ?1 ,
(8) i? ?
2
i
? ?3 ?4 ? + i?3 ? = ?1 ,
(9) ?
2?
i
(10) ?0 ?4 ? + i?0 ? = ?1 ,
?
2?
1
i(?0 ? ?3 )?4 ? + i(?0 + ?3 )? = ?1 ,
(11) ?
4
1 ?1
(12) i? (?0 + ?3 )? + i(?0 + ?3 )? = ?1 ,
?
2
i ?1
(13) ? (? + ?4 )(?0 + ?3 )? + i(?0 + ?3 )? = ?1 ,
?
2?
(3.24)
1
(14) i(?0 + ?3 )?4 ? + i(?0 + ?3 )? = ?1 ,
?
2
(15) 2i?1 ? = ?1 ,
?
2i(?2 ? ??1 )? = ?1 ,
(16) ?
i ?1/2
?2 (? ? ?4 )? + 2i? 1/2 ?2 ? = ?1 ,
(17) ? ?
2?
1
i(?0 + ?3 )(1 + ??4 )? + i[?(?0 + ?3 ) + ?0 ? ?3 ]? = ?1 ,
(18) ?
2
i? ?1 (?0 + ?3 )? + i(?0 + ?3 )? = ?1 ,
(19) ?
1
i[?(? + ?) ? ?]?1 (?0 + ?3 ){[2?(? + ?) ? ? ? 1]?4 ? 2? ? ?}? +
(20)
2
+ i(?0 + ?3 )? = ?1 ,
?
1
i[?(? + ?)]?1 (?0 + ?3 )(2? + ? ? ?4 )? + i(?0 + ?3 )? = ?1 ,
(21) ?
2
1
i[?(? + 1)]?1 (2? + 1)(?0 + ?3 )? + i(?0 + ?3 )? = ?1 ,
(22) ?
2
i(?0 + ?3 )? + i[?(?0 + ?3 ) + ?0 ? ?3 ]? = ?1 ,
(23) ?
i(?0 + ?3 )? + i[?2 ? ?(?0 + ?3 )]? = ?1 ,
(24) ?
1
i(?0 + ?3 )? + i (?0 + ?3 )? ?1 + (?0 ? ?3 )?4 ? = ?1 ,
(25) ?
4
1
i(?0 + ?3 )(3 + ??4 )? + i[(?0 + ?3 )? + ?0 ? ?3 ]? = ?1 ,
(26) ?
2
592 W.I. Fushchych, R.Z. Zhdanov

where
?1 ? (F1 + F2 ?4 )?, ? ? d?/d?.
Fi = Fi (??, ??4 ?),
?? ?
To integrate eqs. (3.24) one can again apply group-theoretical methods. In ref.
[29] it was pointed out how to obtain some information about the symmetry of the
reduced PDE by purely algebraic methods (without application of the infinitesimal Lie
method). It is based on the following statement:
Let G be a Lie group of transformations, H be a normal divisor in G. And let
there be a PDE invariant under the group G.
Theorem 5. The equation obtained via reduction with the help of H-invariant solu-
tions admits the factor group G/H.
A proof can be found in ref. [33].
We use the equivalent formulation of this theorem in terms of Lie algebras: If
there is a PDE with the symmetry algebra AG and subalgebra Q which is an ideal in
AG, then the equation obtained via reduction with the help of Q-invariant solutions
admits the Lie algebra AG/Q.
Straightforward application of the above theorem to the three-dimensional algebras
listed in the table 1 is impossible because these algebras are not, in general, ideals in
AP (1, 3). Therefore there arises the intermediate problem of constructing the maximal
subalgebras A1 , . . . , A26 of the algebra AP (1, 3) having the algebras of the table 1 as
ideals.
It is known from the theory of Lie algebras [33] that the algebra Q1 , Q2 , Q3 is
the ideal in the Lie algebra ?1 , ?2 , . . . , ?s iff
[Qi , ?j ] = ?k Qk , ?k = const,
ij ij

where [Qi , ?j ] is the commutator, and summation over repeated indices is understood.
µ? µ
Consequently, the operator ?k Jµ? + ?k Pµ belongs to the algebra Ak iff
[Qi , ?k Jµ? + ?k Pµ ] = ?j Qj ,
µ? µ
(3.25)
i = 1, 2, 3, k = 1, . . . , 26.
ik

Here ?k , ?k and ?j are constants; Q1 , Q2 , Q3 is the triplet of operators in table 1
µ? µ
ik
under number k.
When one calculates the commutators on the left-hand sides of equalities (3.25)
and equates the coefficients to zero at linearly independent operators Jµ? and Pµ , one
µ? µ
obtains a system of algebraic equations for ?i and ?i . The solution of these equations
gives the explicit expression for the basis operators of the algebras A1 to A26 .
The next step is the calculation of the factor algebras {Ai /Qi , i = 1, . . . , 26},
which generate the invariance groups of the reduced equations (3.24). We shall realize
the above scheme for the algebra P0 , P1 , P2 , the remaining algebras being treated in
the same way. To do this one needs the commutation relations of the algebra AP (1, 3)
[31],
[Jµ? , J?? ] = i(gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? ),
(3.26)
[Pµ , J?? ] = i(gµ? P? ? gµ? P? ), [Pµ , P? ] = 0.
Relations (3.25) are rewritten for Q1 = P0 , Q2 = P1 , Q3 = P2 in the following way:
µ? µ
[P0 , ?1 Jµ? + ?1 Pµ ] = ?1 P0 + ?2 P1 + ?3 P2 ,
11 11 11
µ? µ
[P1 , ?1 Jµ? + ?1 Pµ ] = ?21 P0 + ?21 P1 + ?3 P2 ,
1 2
(3.27)
21
µ? µ
[P2 , ?1 Jµ? + ?1 Pµ ] = ?31 P0 + ?31 P1 + ?3 P2 .
1 2
31
Symmetry and exact solutions of nonlinear spinor equations 593

Taking into account relations (3.26) one obtains the following equalities:


2i?1 Pµ = ?1 P0 + ?2 P1 + ?3 P2 ,
11 11 11

?2i?1 Pµ = ?1 P0 + ?2 P1 + ?3 P2 , (3.28)
21 21 21

?2i?1 Pµ = ?1 P0 + ?2 P1 + ?3 P2 ,
31 31 31

µ
03 13 23 01 02 12
whence it follows that ?1 = ?1 = ?1 = 0, ?1 , ?1 , ?1 , ?1 are arbitrary real
parameters. Consequently, the set of linearly independent solutions of the system
(3.27) is exhausted by the operators

J01 , J02 , J12 , P0 , P1 , P2 , P3 = A1 ,

whence one obtains

(3.29)
A1 / P0 , P1 , P2 = J01 , J02 , J12 , P3 .

To construct the invariance algebra of eq. (1) of (3.24) it is necessary to rewrite
the operators (3.29) in the new variables ?, ?. As a result one has

(1) ?0 ?1 , ?0 ?2 , ?1 ?2 , ?? .

The invariance algebras of the other equations of (3.24) are as follows:

(2) ?1 ?2 , ?2 ?3 , ?1 ?3 , ?? ;
1
?1 (?0 + ?3 ), ?2 (?0 + ?3 ), ?1 ?2 , ??? ? ?0 ?3 , ?? ;
(3)
2
(4) ?1 ?2 ;
(5) ?? ;
(6) ?0 ?3 , ?? ;
2??? ? ?0 ?3 ;
(7)
(8) ?0 ?3 ;
(9) ?? , ?1 ?2 ;
(10) ?? , ?1 ?2 ;
(11) ?? , ?1 ?2 ;
1
?2 (?0 + ?3 ), ? ?1 ?1 (?0 + ?3 ), ??? ? ?0 ?3 ;
(12)
2
1
(?1 + ??2 )(?0 + ?3 ), ? ?1 ?1 (?0 + ?3 ), ??? ? ?0 ?3 ;
(13)
2
(14) ?1 (?0 + ?3 ), ?2 (?0 + ?3 ), ?? ;
(15) ?? , ?2 (?0 + ?3 ) ;
(16) ?? ;
(17) ?0 ?3 ;
594 W.I. Fushchych, R.Z. Zhdanov

(18) ?1 ?2 ;
1
??? ? ?0 ?3 , ? ?1 ?1 (?0 + ?3 ), ? ?2 ?2 (?0 + ?3 ), ?1 ?2 ;
(19)
2
[?(? + ?) ? ?]?1 (?0 + ?3 )[(? + ?)?1 ? ?2 ],
(20)
[?(? + ?) ? ?]?1 (?0 + ?3 )(??2 ? ??1 ), ?1 ?2 ;
? ?1 ?1 (?0 + ?3 ), [?(? + ?)](?0 + ?3 )(??2 ? ?1 ) ;
(21) (3.30)
? ?1 ?1 (?0 + ?3 ), (? + 1)?1 (?0 + ?3 )?2 ;
(22)
(23) ?1 ?2 ;
(24) ?? , ?1 (?0 + ?3 ) ;
(25) ?1 ?2 ;
(26) ?1 ?2 .

Here Q1 , . . ., Qs denotes the set of all linear combinations of the operators Q1 , . . ., Qs .
Let us note that the Lie algebras (3.30) are not, in general, the maximal invariance
algebras of the equations of (3.24). As an example we shall consider eq. (3). By direct
verification one can check that this equation is invariant under the infinite-parameter
group of the form
(3.31)
? = ?, ? = exp{[f1 (?)?1 + f2 (?)?2 ](?0 + ?3 )}?,
where fi (?) are arbitrary smooth functions; the Lie group generated by the operators
?1 (?0 + ?3 ), ?2 (?0 + ?3 ) in line (3) of (3.30) is a two-parameter subgroup of the group
(3.31).
Nevertheless, the information obtained about the symmetry of the ODE (3.24)
proves to be very useful while constructing their particular solutions. Besides if an
ODE has a lagrangian then one can construct its first integrals using Noether’s
theorem.
Let us also stress that an arbitrary Poincar?-invariant equation for a spinor field,
e
after being reduced to systems of ODE with the help of the ansatze of table 1,
possesses the symmetry (3.30).
Let us turn to the system (3.14). Substitution of the ans?tze (3.18)–(3.21) into
a
(3.14) gives rise to the following systems of equations for the spinors ?1 , . . . , ?9 :
k = 1:
(1) i?1 ?1 = ?2 (?1 , ?1 );
? ?
1 ?1/2 1/2
(2) iz2 ?2 ?2 + 2i?2 z2 ?2 = ?2 (?2 , ?2 );
? ?
2
?i?0 ?3 = ?2 (?3 , ?3 );
(3) ? ?
?i?1 ?4 = ?2 (?4 , ?4 );
(4) ? ? (3.32)
1 ?1/2 1/2
?2 ?5 + 2iz5 ?2 ?5 = ??2 (?5 , ?5 );
(5) iz5 ? ?
2
?i?1 ?6 = ?2 (?6 , ?6 );
(6) ? ?
1 ?1/2 1/2
?2 ?7 + 2iz7 ?2 ?7 = ??2 (?7 , ?7 );
(7) iz7 ? ?
2
k ? R1 :
(8) i?1 ?8 = ?2 (?8 , ?8 );

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