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4? = ?{m(?0 ?3 + 1) + i? ? ?m/4k [?(?0 + ?3 ) + ?0 ? ?3 ]}?,
?

where ? = ??C 1/2k . Writing this equality in components we obtain a system of ODE
of the form
2? ?1 = ?m?1 + i? ? ? ?3 ,
2? ?0 = i? ? ?+1 ?2 ,
? ?
(3.51)
2? ?2 = ?m?2 + i? ? ? ?0 , 2? ?3 = i? ? ?+1 ?1 , ? = ?m/4k.
? ?
Symmetry and exact solutions of nonlinear spinor equations 599

It is not difficult to convince oneself that (3.51) is equivalent to the following system
of ODE:
1 1
? 2 ?0 + (m ? 2?)? ?0 + ? 2 ? 2?+1 ?0 = 0,
? ?
2 4
1 1
? 2 ?3 + (m ? 2?)? ?3 + ? 2 ? 2?+1 ?3 = 0,
? ?
2 4
2i ?? 0 2i
?2 = ? ? ? , ?1 = ? ? ?? ?3 .
? ?
? ?
The first and the second equations of this system are Bessel-type equations.
For ? = ?1/2 their general solutions are determined by the formulae

?0 = ? (2+??m)/4 [?0 J? (z) + ?2 Y? (z)],
(3.52)
?3 = ? (2+??m)/4 [?3 J? (z) + ?1 Y? (z)],

where J? , Y? are Bessel functions, z = ? ? (2?+1)/2 /(?+1), ? = (2??m+2)/2(1+2?).
Consequently
i(m ? 2? ? 2) ???1 0
[? J? (z) + ?2 Y? (z)] ?
?2 = ? (2+2??m)/4 ?
2?
dJ? (z) dY? (z)
? i? ?1/2 ?0 + ?2 ,
dz dz
(3.53)
i(m ? 2? ? 2) ???1 3
[? J? (z) + ?1 Y? (z)] ?
?1 = ? (2+2??m)/4 ?
2?
dJ? (z) dY? (z)
? i? ?1/2 ?3 + ?1 ,
dz dz
where ?µ = const, µ = 0, 1, 2, 3. Formulae (3.52), (3.53) determine the general soluti-
on of the initial nonlinear system (3.50) if the following condition holds:

?? ? ?0? ?2 + ?2? ?0 + ?3? ?1 + ?1? ?3 = C? ?m/2 .
?

Substitution of (3.52), (3.53) into this formula yields the following equality:
2i(2? + 1) 0 2?
(? ? ? ?2 ?0? + ?3 ?1? ? ?1 ?3? )? ?m/2 = C? ?m/2 ,
??
where we used the well-known identity for Bessel functions
dY? dJ?
W [J? , Y? ] ? J? ? Y? = 2/?z.
dz dz
Comparing both sides of the equality one obtains
2i(2? + 1) 0? 2
(? ? ? ?2? ?0 + ?3 ?1? ? ?1 ?3? ),
C=
??
whence it follows that
2k/(2k+1)
i(m ? 2k) 0? 2
? (? ? ? ?0 ?2? + ?3? ?1 ? ?1? ?3 ) (3.54)
C= .
?k?
600 W.I. Fushchych, R.Z. Zhdanov

For ? = ?1/2 (k = m/2) one has to consider three cases,

(m ? 1)2 ? 4? 2 = 0, m = 2, 3;
(1)
(2) ? = 0, m = 1;
? = ?(m ? 1)/2, ? = ±1.
(3)

The general solution of the system (3.50) is given by the following formulae:
2i
(?+ ?3 ? ?+ + ?? ?1 ? ?? )? ?1/2 ,
?1 = ?
?0 = ?0 ? ?+ + ?2 ? ?? ,
(1)
? (3.55)
2i
? = ? (?+ ?0 ? ?+ + ?? ?2 ? ?? )? ?1/2 ,
2 3 3 ?+ 1 ??
? =? ? +? ? ,
?
where
1
1?m± (m ? 1)2 ? 4? 2 ,
?± =
4
?0 , . . . , ?3 are arbitrary complex constants; ? satisfies the equality (?1)m i(?0? ?2 ?
?0 ?2? + ?3? ?1 ? ?3 ?1? )[(m ? 1)2 ? 4? 2 ]1/2 = ? m+1 ??m ;

1 1
?0 = ?0 cos ? ln ? + ?2 sin
(2) ? ln ? ,
2 2
1 1
?1 = ?i? ?1/2 ?1 cos ? ln ? ? ?3 sin ? ln ? ,
2 2
(3.56)
1 1
?2 = ?i? ?1/2 ?2 cos ? ln ? ? ?0 sin ? ln ? ,
2 2
1 1
?3 = ?3 cos ? ln ? + ?1 sin ? ln ? ,
2 2

where ?0 , . . . , ?3 are constants; ? satisfies the equality ? = i?(?0 ?2? ??2 ?0? +?3 ?1? ?
?1 ?3? );

?0 = ? (1?m)/4 (?0 + ?2 ln ?),
(3)
1 4i? ?(m+1)/4 1
i(m ? 1)? ?1/2 ?3 +
?1 = ? ?,
1?m
2?
(3.57)
1 4i? ?(m+1)/4 2
i(m ? 1)? ?1/2 ?0 +
?2 = ? ?,
1?m
2?
? = ±1,
?3 = ? (1?m)/4 (?3 + ?1 ln ?),

while the following equality holds:
m+1
m?1
?? ? +? ? ?? ? )=?
0 2? 0? 2 3 1? 3? 1
(?1)m .
2i(? ?
2??

So the general solution of the system (3.50) [and consequently, of the systems (4),
(23) and (26) of (3.24) (? = 0)] is given by formulae (3.52), (3.53), for k = m/2 and
by formulae (3.55)–(3.57) for k = m/2.
Symmetry and exact solutions of nonlinear spinor equations 601

Let us turn now to eqs. (3.32). The systems of ODE (1), (3), (4), (6) and (8) are
integrated in the same way as eqs. (1) and (2) of (3.24). As a result one has
?1 = exp[i??1 (??)1/2 z1 ]?; ?2 = exp[i?(??)1/2 ?0 z3 ]?;
? ?
(3.58)
?j = exp[?i?(??)1/2 ?1 zj ]?, j = 4, 6; ?8 = exp[i?(??)1/2k ?1 z8 ]?.
? ?
Equations (2), (5), (7) and (9) of (3.32) coincide with ODE (8) of (3.24) up to
the sign of the nonlinear term ?(??)1/2k ?. Using this fact one easily obtains their
?
general solutions,
?1/4 1/4
exp[?2i?(??)1/2 ?2 z2 ]?;
?2 (z2 ) = z2 ?
?1/4 1/4
exp[2i?(??)1/2 ?2 zj ]?, j = 5, 7;
?j (zj ) = zj ? (3.59)
2i?k
?1/4 (2k?1)/2k
(??)1/2k ?2 z9
?9 (z9 ) = z9 exp ? ?.
1 ? 2k
Besides we have succeeded in integrating ODE (1) of (3.33) (for k = 1/2) and (2)
of (3.33). The final result has the form
i?
?(?2 ) = exp (??)(?3 + a?2 )?2 ?,
?
2(1 + a2 )
?1/4
?(?2 ) = ?2 [f1 + ?3 f2 + (?0 + ?1 )f3 + ?3 (?0 + ?1 )f4 ]?,
where the functions fi (?) are determined by the following equalities:
k = 1/2:
? ?
f1 = cosh(? ?2 ), f2 = i sinh(? ?2 ),
?2 ?2
1
sinh(2? z ? )dz ? sinh(? ?2 )
? ?
cosh(2? z ? )dz ,
f3 = i cosh(? ?2 )
4
(3.60)
?2 ?2
1
f4 = i ? sinh(? ?2 )
? ? ?
cosh(2? z ? )dz ,
sinh(2? z )dz + cosh(? ?2 )
4
2k ? 1
2?k(??)1/2k
?
?= , ?= ;
2k ? 1 4k
k = 1/2:
1 ? ? /2 1 ? ? /2
?? /2 ?? /2
(2 ?2 + 2?? ?2 i(2 ?2 ? 2?? ?2
f1 = ), f2 = ),
2 2
?? /2
2?? ?2
? /2
2? ?2
1 1/2
?
f3 = i?2 ,
1 ? 2? (3.61)
4 2? + 1
?? /2
2?? ?2
? /2
2? ?2
1 1/2
f4 = ?2 + , ? = ?(??).
?
1 ? 2?
4 2? + 1

The possibility of integrating the nonlinear systems of ODE (3.24), (3.22) and
(3.33) in quadratures is closely connected with the nontrivial symmetry admitted by
these equations. And this property, in its turn, is connected with the large invariance
?
group admitted by the initial equation [in the present case the group P (1, 3)]. That is
why, when the symmetry properties of the equations are better, the group-theoretical
602 W.I. Fushchych, R.Z. Zhdanov

methods of constructing exact solutions are more effective. It is worth noting that
other classical methods of constructing particular solutions (separation of variables,
d’Alembert method and so on) use explicitly or implicitly the symmetry properties of
PDE [30].
Substitution of the above results into the corresponding ans?tze in tables 1 and 2
a
or into the ans?tze (3.18)–(3.21) yields the exact solutions of the nonlinear Dirac–
a
Heisenberg equation (3.34):
k ? R1 :
?1 (x) = exp[i?(??)1/2k ?3 x3 ]?;
?
?2 (x) = exp[?i?(??)1/2k ?0 x0 ]?;
?
1
?3 (x) = exp ? (?0 + ?3 )?1 (x0 + x3 ) ?
2
1
? exp i?(??)1/2k ?1 [2x1 + (x0 + x3 )2 ] ?;
?
2
1
?4 (x) = exp ? (?0 + ?3 )?1 (x0 + x3 ) ?
2
i?
? exp (??)1/2k (?2 ? ??1 )[2(x2 ? ?x1 ) ? ?(x0 + x3 )2 ] ?;
?
2)
2(1 + ?
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