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+2x1 + ?) ? 1 ?2 ?3 arctg ? exp ? x0 ) ? (x2 + x2 )?1
2 ?
x3
1
??
10 exp ln(x0 + x1 ) x3
0 1
x2 ? x2 ln(x0 + x1 ) ? x2
Symmetry and exact solutions of nonlinear spinor equations




201
x2 x2
x2 + x2 arctg
11 + ?1 x0 + ?2 x1 ?2 x0 + ?1 x1
2 3
exp ? 1 ?2 ?3 arctg
2 x3 x3
?1
12 x?k I x1 x?1 x3 x?1
0 0 x2 x0 0
13 I x0 + x3 x1 x2
14 I x1 x2 x3
15 I x0 x1 x2
Notation: I is the unit 4 ? 4 matrix; a, b, ?, ?1 , ?2 , ?, ?1 , ?2 are arbitrary real numbers.
587
588 W.I. Fushchych, R.Z. Zhdanov

1 ?1
? (?0 ? ?2 )? + (?0 ? ?2 )??1 + ?3 ??2 +
(2)
21
?1
+ [(?0 + ?2 )?1 + (?0 ? ?2 )?3 ?1 ]??3 = ?i?2 (?, ?);
?
3
[?3 + ?(?0 ? ?2 )]??1 + 2?1 ??2 + (2?2 + (?0 ? ?2 )?2 )??3 = ?i?2 (?, ?);
(3) ?
2
1 ?1
? ?4 (?0 ? ?2 )? + (?0 ? ?2 )??1 + [(?0 + ?2 )?1 ? 2? ?1 ?3 ?1 +
(4)
2
?1
+ (?0 ? ?2 )(? ?2 ?3 + ?2 )?1 ]??2 + (??1 ? ?3 ?1 )??3 = ?i?2 (?, ?);
2
?
1
(1 ? 2k)?3 ? + (?1 ?3 )1/2 (?0 ? ?1 ?1 )??1 + 2(?3 + a?2 )??2 +
(5)
2
1/2 ?1/2
+ [2?3 ? (?0 + ?1 ?1 )?3 ?1 ]?3 ??3 = ?i?2 (?, ?);
?
1/2(a+1) 1/2(a+1)
? k(?0 cosh ln ?1 ? ?1 sinh ln ?1
(6) )+
1 1
?1/2(a+1) 1/2
+ ?3 ?2 ? ?
+ (?0 + ?1 )?1
2(a + 1) 2
1/2(a+1) 1/2(a+1)
? 2(a + 1)(?0 cosh ln ?1 ? ?1 sinh ln ?1 )?1 ??1 +
1/2(a+1) 1/2(a+1) 3/2
? ?1 sinh ln ?1 ) ? ?2 ?3 ]??2 +
+ 2[?2 (?0 cosh ln ?1
1/2
+ 2(a?2 + b?3 )?2 ??3 = ?i?2 (?, ?);
?
1 1
1/2 1/2 1/2 1/2
?k(?0 cosh ln ?1 ? ?1 sinh ln ?1 ) + (?0 ? ?1 )?1 + ?3 ?2
(7) ?+
4 2 (3.16)
1/2 1/2 1/2 1/2
+ (?0 + ?1 )?1 ??1 + 2?2 (?0 cosh ln ?1 ? ?1 sinh ln ?1 ? ?3 ?2 )??2 +
1/2
+ 2?2 (b?3 ? ?2 )??3 = ?i?2 (?, ?);?
1 ?1/2 ?1/2
)? + [?1 (?0 + ?1 ) + ?0 ? ?1 )??1 + 2?3 ?2
(8) (?0 + ?1 + ?3 ?3 ?? 2 +
2
?1/2
]??3 = ?i?2 (?, ?);
+ [b(?0 + ?1 ) + ?2 ?2 ?
1
[(1 ? 2k)(?0 + ?1 ) + ?3 ?2 ]? + 2? ?1 ?1 [?(?0 + ?1 ) ? ?0 + ?1 ]??1 +
1/2
(9)
2
1/2 1/2
+ 2?2 (?0 + ?1 ? ?3 ?2 )??2 + 2?2 (?2 + b?3 )??3 = ?i?2 (?, ?); ?
1
(?0 + ?1 )? + [?1 (?0 + ?1 ) + ?0 ? ?1 ]??1 +
(10)
2
+ (?0 + ?1 ? ?2 )??2 + ?3 ??3 = ?i?2 (?, ?);
?
1 ?1/2 ?1/2
1/2
(11) ?3 ?1 ? + 2?3 ?1 ??1 + (?1 ?0 + ?2 ?1 + ?2 ?1 )??2 +
2
+ (?2 ?0 + ?1 ?1 )??3 = ?i?2 (?, ?);
?
?k?0 ? + (?a ? ?0 ?a )??a = ?i?2 (?, ?);
(12) ?
(?0 + ?3 )??1 + ?1 ??2 + ?2 ??3 = ?i?2 (?, ?);
(13) ?
?1 ??1 + ?2 ??2 + ?3 ??3 = ?i?2 (?, ?);
(14) ?
?0 ??1 + ?1 ??2 + ?2 ??3 = ?i?2 (?, ?),
(15) ?
where ??a ? ??/??a , a = 1, 2, 3,
? ? ? ?? ?
?2 ? [(F1 + F2 ?4 )(??)1/2k ]?,
? Fi = Fi (??/??4 ?).
The group-theoretical properties of eqs. (3.16) were investigated in ref. [29]. We
Symmetry and exact solutions of nonlinear spinor equations 589

consider in more detail the PDE (3) and (13)–(15) of (3.16). Using the Lie method
[32–34] one can prove the following statements.
Proposition 1. PDE (13) of (3.16) is invariant under the infinite-parameter Lie
group, its generators being of the form
k = 1:
1? ?
Q1 = ?1 (?1 )??2 + ?2 (?1 )??3 + [?1 (?1 )?1 + ?2 (?1 )?2 ](?0 + ?3 ),
2
1
Q2 = ??2 ??3 + ?3 ??2 + ?1 ?2 ,
2
(3.17)
? 0 (?1 )(?2 ?? + ?3 ?? ) + ?0 (?1 ) +
?
Q3 = ?0 (?1 )??1 + ? 2 3

1?
+ ?0 (?1 )(?1 ?2 + ?2 ?3 )(?0 + ?3 ),
2
Q4 = ?3 (?1 )?4 (?0 + ?3 );
k = 1:
1
Q2 = ??2 ??3 + ?3 ??2 + ?1 ?2 ,
Q1 = ??1 ,
2
1? ?
Q3 = ?1 (?1 )??2 + ?2 (?1 )??3 + [?1 (?1 )?1 + ?2 (?1 )?2 ](?0 + ?3 ),
2
Q4 = ?1 ??1 + ?2 ??2 + ?3 ??3 + k, Q5 = ?3 (?1 )?4 (?0 + ?3 ),
where ?0 , . . . , ?3 are arbitrary smooth functions, a dot means differentiation with
respect to ?1 .
Proposition 2. For k = 1 PDE (14) and (15) of (3.16) are invariant under the
conformal groups C(3) and C(1, 2), respectively.
It is important to note that for k = 3/2 the initial equation (3.14) is not con-
formally invariant. The same statement holds for the infinite-parameter group with
generators (3.17). Consequently for k = 1 the PDE (3.14) is conditionally invariant
under the algebras (3.17), AC(3) and AC(1, 2). Using this fact we have constructed
ansatze which are principally different from ones listed in table 1:
k = 1:
1?
?(x) = ??1 exp ?3 ?4 (?0 + ?3 ) ? (?1 ?1 + ?2 ?2 )(?0 + ?3 ) ?
?
0
2
1?
? ?0 ??1 [?1 (x1 + ?1 ) + ?2 (x2 + ?2 )](?0 + ?3 ) ?
0
2 (3.18)
?
??1 ((x1 + ?1 )/?0 ),
?
?
?exp ? 1 ?1 ?2 arctg x1 + ?1 ?2 ([(x1 + ?1 )2 + (x2 + ?2 )2 ]/?2 ),
? 0
2 x2 + ?2
?0 x0 ? ?1 x1 ? ?2 x2
?
?(x) =
(x2 ? x2 ? x2 )3/2
? 0 1 2
??3 (x0 /(x0 ? x1 ? x2 )),
2 2 2
?
?
?? (x /(x2 ? x2 ? x2 )), (3.19)
41 0 1 2
?
?
?
?exp ? 1 ? ? arctg x1 ? ((x2 + x2 )/(x2 ? x2 ? x2 )2 ),
? 12 5 1 2 0 1 2
2 x2
590 W.I. Fushchych, R.Z. Zhdanov
?
??6 (x1 (x2 )?1 ),
?
?·x
?(x) = 2 3/2 ? (3.20)
?exp ? 1 ?1 ?2 arctg x1 ?7 ((x2 + x2 )/(x2 )2 ),
(x ) ? 1 2
2 x2

k = 1:
1? ?
?(x) = exp ?3 ?4 (?0 + ?3 ) ? (?1 ?1 + ?2 ?2 )(?0 + ?3 ) ?
2
?
??8 (x1 + ?1 ),
? (3.21)
?
?exp ? 1 ?1 ?2 arctg x1 + ?1 ?9 ((x1 + ?1 )2 + (x2 + ?2 )2 ),
?
2 x2 + ?2
In eqs. (3.18)–(3.21) ?0 , . . . , ?3 are arbitrary smooth functions of x0 + x3 , ?1 , . . . , ?9
are new unknown spinors. While obtaining formulae (3.19)–(3.21) we essentially used
the conformally invariant ansatz suggested in refs. [20, 21] and the results of refs.
[24, 29].
Let us turn now to eq. (3) of (3.16). If one chooses ? = ?(?1 , ?2 ) and introduces
the notations
1
?1 = ?3 + ?(?0 ? ?2 ), ?2 = ?1 , z1 = ? 1 , z2 = ?2 ,
2
then one obtains the following PDE:

?1 ?z1 + ?2 ?z2 = ??2 (?, ?), (3.22)
?

where ?2 = ?2 = ?1, ?1 ?2 + ?2 ?1 = 0.
1 2
With the aid of the Lie method [32–34] it is possible to prove that eq. (3.22) is
invariant under the conformal group C(2) if k = 1/2 [consequently, for k = 1/2 the
initial PDE (3.14) is conditionally invariant under the conformal group C(2)]. Using
this fact we have constructed the ansatz that reduces (3.14) to a system of ODE [24]:
1
?(x) = ??1 exp ?1 (?0 ? ?2 )(x0 ? x2 ) ?
2
(3.23)
1
? [?3 + ?(?0 ? ?2 )][x3 + ?(x0 ? x2 )] + ?1 (2x1 + (x0 ? x2 ) ) ?(?),
2
2
where
1
?1 [x3 + ?(x0 ? x2 )] + ?2 [2x1 + (x0 ? x2 )2 ] ??1 ,
?=
2
1
? = [x3 + ?(x0 ? x2 )]2 + [2x1 + (x0 ? x2 )2 ]2 ,
4
?, ?1 , ?2 are constants.
3.2. Reduction of the nonlinear Dirac equation to systems of ODE. To reduce
the nonlinear Dirac equation (3.10) via ansatze from table 1 one has to make rather
cumbersome calculations. Therefore we give the final result, systems of ODE for
?(?), omitting intermediate calculations.
Symmetry and exact solutions of nonlinear spinor equations 591

(1) i?2 ? = ?1 ,
?
(2) i?0 ? = ?1 ,
?
(3) i(?0 + ?3 )? = ?1 ,
?
1
i(?0 + ?3 )? + i[?(?0 + ?3 ) + ?0 ? ?3 ]? = ?1 ,
(4) ?
2
1
(5) i(?0 + ?3 )? + i?2 ? = ?1 ,

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. 136
( 145 .)



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