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qµ?? = gµ? b? + g?? bµ ? 2gµ? b? ,
hµ?? = ? (aµ d? ? a? dµ )c? + (dµ c? ? d? cµ )a? + (cµ a? c? aµ )d? ;
?
?
kµ? = ?gµ? ? bµ b? , lµ? = ? bµ k? , mµ? = ? kµ k? ,
3.
2 4
?
qµ?? = gµ? b? + g?? bµ ? 2gµ? b? , hµ?? = (gµ? k? ? gµ? k? );
4
kµ? = ?gµ? ? bµ b? , lµ? = ?? (gµ? + bµ b? ), mµ? = ?? ?2 cµ c? ,
?1
4.
1
qµ?? = gµ? b? + g?? bµ ? 2gµ? b? , hµ?? = ? ?1 (gµ? b? ? gµ? b? ).
2
On non-Lie ansatzes and new exact solutions of the classical YME 465

3 Exact solutions of the nonlinear
Yang–Mills equations
Systems (5), (11) are systems of twelve nonlinear second-order ODE with variable
coefficients. That is why, there is a little hope to construct their general solutions.
But it is possible to obtain particular solutions of system (5), which coefficients are
given by the formulae 2–4 from (11).
Consider, as an example, the system of ODE (5) with coefficients given by the
formulae 2 from (11). We look for its solutions in the form
(12)
Bµ = kµ e1 f (?) + bµ e2 g(?), f g = 0,
where e1 = (1, 0, 0), e2 = (0, 1, 0).
Substituting the expression (12) into the above mentioned system, we get
? f g + 2f?g = 0.
f + (?2 ? e2 g 2 )f = 0, (13)
?
The second ODE from (13) is easily integrated
g = ? f ?2 , ? ? R1 , (14)
? = 0.
Substitution of the result obtained into the first ODE from (13) yields the Ermakov-
type equation for f (?)
f + ?2 f ? e2 ?2 f ?3 = 0,
?

which is integrated in elementary functions [11]
1/2 1/2
f = ??2 C 2 + ??2 C 4 ? ?2 e2 ?2 (15)
sin 2|?|? .
Here C = 0 is an arbitrary constant.
Substituting (12), (14), (15) into the corresponding ansatz for Aµ (x), we get the
following class of exact solutions of YME (1):
1/2
Aµ = e1 kµ exp ??cx ? ?w2 ??2 C 2 + ??2 C 4 ? ?2 e2 ?2 ?
1/2
+ e2 ? ??2 C 2 + ??2 C 4 ? ?2 e2 ?2 ?
? sin 2|?|(bx + w1 )
1
?1
? sin 2|?|(bx + w1 ) bµ + kµ w1 .
?
2
In a similar way, we have obtained five other classes of exact solutions of the
Yang–Mills equations
ie?
1/2
Aµ = e1 kµ e?T bx cos w1 + cx sin w1 + w2 Z1/4bx cos w1 +
2
2
+ e2 ?(bx cos w1 + cx sin w1 + w2 ) ?
+ cx sin w1 + w2
1 ?
? cµ cos w1 ? bµ sin w1 + 2kµ ?eT + T bx sin w1 ? cx cos w1 + w3 ;
4
Aµ = e1 kµ e?T C1 ch e? bx cos w1 + cx sin w1 + w2 + C2 sh e? bx cos w1 +
+ cx sin w1 + w2 + e2 ? Cµ cos w1 ? bµ sin w1 +
1 ?
?eT + T (bx sin w1 ? cx cos w1 ) + w3 ;
+ 2kµ
4
466 R.Z. Zhdanov, W.I. Fushchych

2 1/2
Aµ = e1 kµ e?T C 2 bx cos w1 + cx sin w1 + w2 + ?2 e2 C ?2 +
?1
+ e2 ? C 2 (bx cos w1 + cx sin w1 + w2 )2 + ?2 e2 C ?2 ?
1
? bµ cos w1 + Cµ sin w1 ? kµ w1 (bx sin w1 ? cx cos w1 ) ? w2 ;
? ?
2
ie?
(bx + w1 )2 + (cx + w2 )2 + e2 ? ?
Aµ = e1 kµ Z0
2
1
? cµ (bx + w1 ) ? bµ (cx + w2 ) ? kµ w1 (cx + w2 ) ? w2 (bx + w1 ) ;
? ?
2
e?/2
Aµ = e1 kµ C1 (bx + w1 )2 + (cx + w2 )2 +
?e?/2 ?1
?
+ C2 (bx + w1 )2 + (cx + w2 )2 + e2 ? (bx + w1 )2 + (cx + w2 )2
1
? cµ (bx + w1 ) ? bµ (cx + w2 ) ? kµ w1 (cx + w2 ) ? w2 (bx + w1 )
? ? .
2
Here C1 , C2 , C = 0, ?, ? are arbitrary parameters; w1 , w2 , w3 are arbitrary smooth
functions of ? = 1 kx, T = T (?) is a solution of ODE (10).
2
Besides that, we use the following notations:

kx = kµ xµ , bx = bµ xµ , cx = cµ xµ ,
Zs (?) = C1 Js (?) + C2 Ys (?), e1 = (1, 0, 0), e2 = (0, 1, 0),

where Js , Ys are Bessel functions. Thus, we have obtained the wide families of exact
non-Abelian solutions of YME (1).
In conclusion we say a few words about a symmetry interpretation of the ansatzes
(2), (7), (10). Let us consider, as an example, the ansatz determined by the formulae 1
from (9). As a direct computation shows, generators of the three-parameter Lie group
leaving it invariant are of the form

Q1 = k? ?? ,
3
Q2 = b? ?? ? w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ ) Aa? ?Aaµ ,
(16)
a=1
3
Q3 = c? ?? ? 2 w1 (kµ b? ? k? bµ ) + w3 (kµ c? ? k? cµ ) Aa? ?Aaµ .
a=1

Evidently, the system of PDE (1) is invariant under the one-parameter group having
the generator Q1 . But it is not invariant under the groups having the generators Q2 ,
Q3 . At the same time, the system of PDE

?? ? ? Aµ ? ? µ ?? A? + e (?? A? ) ? Aµ ? 2(?? Aµ ) ? A? + (? µ A? ) ? A? +
+ e2 A? ? (A? ? Aµ ) = 0,
Q0 Aµ ? k? ?? Aµ = 0,
Q1 Aµ ? b? ?? Aµ + 2 w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ ) A? = 0,
Q2 Aµ ? c? ?? Aµ + 2 w1 (kµ b? ? k? bµ ) + w3 (kµ c? ? k? cµ ) A? = 0

is invariant under the said group. Consequently, YME (1) are conditionally-invariant
under the Lie algebra (16). It means that the solutions of YME obtained with the help
On non-Lie ansatzes and new exact solutions of the classical YME 467

of the ansatz invariant under the group with the generators (16) can not be found by
the classical symmetry reduction procedure.
As rather tedious computations show, the ansatzes determined by the formulae
2–4 from (9) also correspond to conditional symmetry of YME. Hence it follows,
in particular, that YME should be included into the long list of mathematical and
theoretical physics equations possessing a nontrivial conditional symmetry [12].
Acknowledgments. The authors are indebted for financial support to Commit-
tee for Science and Technologies of Ukraine, Soros and Alexander von Humboldt
Foundations.

1. Actor A., Classical solutions of SU (2) Yang–Mills theories, Rev. Mod. Phys., 1979, 51, ¹ 3,
461–525.
2. Zhdanov R.Z., Lahno V.I., On the new exact solutions of the Yang–Mills equations, Dopovidi Akad.
Nauk Ukrainy, 1994 (to appear).
3. Fushchych W.I., Shtelen W.M., Conformal symmetry and new exact solutions of SU (2) Yang–Mills
equations, Lett. Nuovo Cim., 1983, 38, ¹ 2, 37–40.
4. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear
equations of mathematical physics, Kiev, Naukova Dumka, 1989.
5. Ovsiannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.
6. Ibragimov N.Kh., Transformation groups in mathematical physics, Moscow, Nauka, 1983.
7. Fushchych W.I., Serov N.I., Chopyk V.I., Conditional invariance and nonlinear heat equations, Dopovi-
di Akad. Nauk Ukrainy, Ser. A, 1988, ¹ 9, 17–21.
8. Fushchych W.I., Conditional symmetry of equations of nonlinear mathematical physics, Ukr. Math. J.,
1991, 43, ¹ 11, 1456–1470.
9. Fushchych W.I., Zhdanov R.Z., Nonlinear spinor equations: symmetry and exact solutions, Kiev,
Naukova dumka, 1992.
10. Fushchych W.I., Zhdanov R.Z., Revenko I.V., General solutions of the nonlinear wave equation and
eikonal equation, Ukr. Math. J., 1991, 43, ¹ 11, 1471–1476.
11. Kamke E., Handbook on ordinary differential equations, Moscow, Nauka, 1976.
12. Fushchych W.I., Conditional symmetry of equations of mathematical physics, in Proceedings of the
International Workshop “Modern Group Analysis”, Editors N. Ibragimov, M. Torrisi and A. Valenti,
Kluwer Academic Publishers, 1993, 231–240.
13. Lahno V., Zhdanov R., Fushchych W., Symmetry reduction and exact solution of the Yang–Mills
equations, J. Nonlinear Math. Phys., 1995, 2, ¹ 1, 51–72.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 468–472.

Symmetry and reduction
of nonlinear Dirac equations
R.Z. ZHDANOV, W.I. FUSHCHYCH
We present results of symmetry classification of the nonlinear Dirac equations with
respect to the conformal group C(1, 3) and its principal subgroups. Next we briefly
consider the problem of classical and non-classical symmetry reduction and construction
of exact solutions for the nonlinear Poincar?-invariant Dirac equations. In particular,
e
a class of exact solutions is constructed which can not be in principle obtained within
the framework of the classical Lie approach.

The Dirac equation is a system of four complex partial differential equations of the
form
(i?µ ?xµ ? m)?(x) = 0, (1)
where ? = ?(x0 , x) is a four-component function-column, ?µ are 4 ? 4 Dirac matrices

I 0 0 ?a
?0 = , ?a =
?I ??a
0 0
and ?a are usual 2 ? 2 Pauli matrices.
In fact in the following we do not use an explicit representation of the Dirac
matrices, we use the commutational relations
?
? 1, µ = ? = 0,
?µ ?? + ?? ?µ = 2gµ? = 2 ?1, µ = ? = 1, 2, 3,
?
0, µ = ?
only.
Nonlinear generalizations of the Dirac equation were suggested by Ivanenko [1]
?
[i?µ ?xµ ? m + ?(??)]? = 0 (2)
and by Heisenberg [2]
?
[i?µ ?xµ + ?(??µ ?4 ?)? µ ?4 ]? = 0. (3)
? ? ? ?
?
Here ? = (?0 , ?1 , ??2 , ??3 ) is a four-component function-row, ?4 = ?0 ?1 ?2 ?3 ,
? = const.
The above equations can be obtained in a unified way within the framework of
symmetry approach. For the equation of the form
? (4)
[i?µ ?xµ + F (?, ?)]? = 0
In Proceedings of the International Symposium on Mathematical Physics “Nonlinear, Deformed
and Irreversible Quantum Systems” (15–19 August, 1994, Clausthal, Germany), Editors H.-D. Doebner,
V.K. Dobrev and P. Nattermann, Singapore – New Jersey – London – Hong Kong, World Scientific, 1995,
P. 223–229.
Symmetry and reduction of nonlinear Dirac equations 469

to be physically acceptable generalization of the linear Dirac equation (1) it must obey
the Einstein relativity principle. From mathematical point of view, it means that on
the set of solutions of Eq. (4) some representation of the Poincar? group P (1, 3) is
e
?
to be realized. Consequently, one has to describe all the matrices F (?, ?) such that
Eq. (4) is invariant under the Poincar? group. Furthermore, it is known that the
e
massless Dirac equation is invariant under the 15-parameter conformal group C(1, 3).
Therefore it is of interest to describe nonlinear equations (4) admitting conformal
group. Such procedure is usually called symmetry or group-theoretical classification
of nonlinear equations (4).
First, we give the results of symmetry classification and then turn to the problem
of constructing exact solutions of the nonlinear Dirac equations (4).
Theorem 1. System of partial differential equations (4) is Poincar? invariant iff
e
? ?? ?? (5)
F (?, ?) = F1 (??, ??4 ?) + F2 (??, ??4 ?)?4 ,

where F1 , F2 are arbitrary complex functions.
Theorem 2. System of PDE (4) is invariant under the extended Poincar? group e
? (1, 3) = P (1, 3) ? D(1), where D(1) is a one-parameter group of scale transforma-
?
P
tions generated by the following infinitesimal operator:

k ? R1 , (6)
D = xµ ?µ + k,
?
iff the matrix-function F (??) is of the form (5), Fi being determined by the formulae
? ?? ?
Fi = (??)1/2k Fi (??/??4 ?), (7)
i = 1, 2,
?
with arbitrary complex functions Fi .
Theorem 3. System of PDE (4) is invariant under the 15-parameter conformal group
?
C(1, 3) = P (1, 3) ? K(4), where K(4) is a 4-parameter group of special conformal
?
transformations which is generated by the following infinitesimal operators:
1
Kµ = 2xµ D ? x? x? ? µ + (?µ ?? ? ?? ?µ )x? , (8)
µ = 0, . . . , 3,
2
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