<< Ïðåäûäóùàÿ ñòð. 109(èç 122 ñòð.)ÎÃËÀÂËÅÍÈÅ Ñëåäóþùàÿ >>
?1 = ?2 = ?3 = 0,
2
1
b · x(k · x)?1 ,
?0 = ? ln |k · x|, ?1 = ?2 = ?3 = 0,
(8t)
2
? = ln |k · x| ? c · x;

1
(b · x ? ln |k · x|)(k · x)?1 ,
?0 = ? ln |k · x|, ?1 =
(8u)
2
? = ? ln |k · x| ? c · x;
?2 = ?3 = 0,

1 1
b · x(k · x)?1 , ?2 = c · x(k · x)?1 ,
?0 = ? ln |k · x|, ?1 =
(8v)
2 2
?3 = ? ln |k · x|, ? = (a · x) ? (b · x) ? (c · x)2 ? (d · x)2 ,
2 2

where a · x stands for aµ xµ and ? is an arbitrary real constant.
We do not consider reduction of YMEs with the help of the above ans?tze, because
a
it is studied in a great detail in [16].
We concentrate on the cases when the new (non-Lie) ans?tze are obtained. It
a
occurs that the procedure described gives rise to non-Lie ans?tze provided the func-
a
tions ?(x), ?µ (x) within the equivalence relations (7) have the form

?µ = ?µ (?, b? x? , c? x? ), ? = ?(?, b? x? , c? x? ). (9)
456 R.Z. Zhdanov, W.I. Fushchych

The list of inequivalent solutions of system of PDEs (3), (4) satisfying (9) is
exhausted by the following solutions:
1
k · x, ?1 = w0 (?)b · x + w1 (?)c · x,
?0 = ?3 = 0, ?=
(10a)
2
?2 = w2 (?)b · x + w3 (?)c · x;

? = b · x + w1 (?), ?0 = ? c · x + w2 (?) ,
(10b)
1
?a = ? wa (?), a = 1, 2, ?3 = 0,
?
4
?0 = T (?), ?3 = w1 (?), ? = b · x cos w1 + c · x sin w1 + w2 (?),
1T ?
(?e + T )(b · x sin w1 ? c · x cos w1 ) + w3 (?) sin w1 +
?1 =
4
1
+ + w1 (b · x sin w1 ? c · x cos w1 ) ? w2 cos w1 ,
? ? (10c)
4
1T ?
?2 = ? (?e + T )(b · x sin w1 ? c · x cos w1 ) + w3 (?) cos w1 +
4
1
+ w1 (b · x sin w1 ? c · x cos w1 ) ? w2 sin w1 ;
? ?
4
?3 = arctan [c · x + w2 (?)][b · x + w1 (?)]?1 ,
?0 = 0,
(10d)
1 1/2
?a = ? wa (?), a = 1, 2, ? = [b · x + w1 (?)]2 + [c · x + w2 (?)]2
? .
4
Here ? = 0 is an arbitrary constant, ? = ±1, w0 , w1 , w2 , w3 are arbitrary smooth
functions on ? = 1 k · x, T = T (?) is a solution of the nonlinear ODE
2

?
(T + ?eT )2 + w1 = ?e2T , ? ? R1 ,
?2 (11)
where a dot over the symbol denotes differentiation with respect to ?.
Substitution of the ansatz (2), where Rµ? (x) are given by formulae (3), (10), in
the YMEs yields systems of nonlinear ODEs of the form (5), where
1
kµ? = ? kµ k? , lµ? = ?(w0 + w3 )kµ k? ,
4
mµ? = ?4 (w0 + w1 + w2 + w3 )kµ k? ? (w0 + w3 )kµ k? ,
2 2 2 2
? ?
1 (12a)
qµ?? = (gµ? k? + g?? kµ ? 2gµ? k? ),
2
hµ?? = (w0 + w3 )(gµ? k? ? gµ? k? ) +
+ 2(w1 ? w2 ) (kµ b? ? k? bµ ) c? + (bµ c? ? b? cµ )k? + (cµ k? ? c? kµ )b? ;

kµ? = ?gµ? ? bµ b? , lµ? = 0, mµ? = ??2 (aµ a? ? dµ d? ),
qµ?? = gµ? b? + g?? bµ ? 2gµ? b? , (12b)
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? ,
2 4 (12c)
?
= gµ? b? + g?? bµ ? 2gµ? b? , hµ?? = (gµ? k? ? gµ? k? );
qµ??
4
Conditional symmetry and new classical solutions of the Yang–Mills equations 457

lµ? = ?? ?1 (gµ? + bµ b? ), mµ? = ?? ?2 cµ c? ,
kµ? = ?gµ? ? bµ b? ,
(12d)
1
= gµ? b? + g?? bµ ? 2gµ? b? , hµ?? = ? ?1 (gµ? b? ? gµ? b? ).
qµ??
2

3 Exact solutions of the nonlinear
Yang–Mills equations
The systems (5), (12) are systems of twelve nonlinear second-order ODEs with vari-
able 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) whose coefficients are
given by expressions (12b)–(12d).
Consider, as an example, system of ODEs (5) with coefficients given by the
expressions (12b). We seek its solutions in the form
B µ = kµ e1 f (?) + bµ e2 g(?), (13)
f g = 0,
where e1 = (1, 0, 0), e2 = (0, 1, 0).
On substituting the expression (13) into the above mentioned system we get
? f g + 2f?g = 0.
f + (?2 ? e2 g 2 )f = 0, (14)
?
The second ODE from (14) is easily integrated to give
g = ?f ?2 , ? ? R1 , (15)
? = 0.
Substitution of the result obtained in the first ODE from (14) yields the Ermakov-
type equation for f (?)
f + ?2 f ? e2 ?2 f ?3 = 0,
?

which is integrated in elementary functions [14]
1/2
f = ??2 C 2 + ??2 (C 4 ? ?2 e2 ?2 )1/2 sin 2|?|? (16)
.
Here C = 0 is an arbitrary constant.
Substituting (13), (15), (16) into the corresponding ansatz for Aµ (x) we get the
following class of exact solutions of YMEs (1):
Aµ = e1 kµ exp (??c · x ? ?w2 ) ??2 C 2 + ??2 (C 4 ? ?2 e2 ?2 )1/2 ?
1/2
+ e2 ? ??2 C 2 + ??2 (C 4 ? ?2 e2 ?2 )1/2 ?
? sin 2|?|(b · x + w1 )
1
?1
? sin 2|?|(b · x + w1 ) bµ + kµ w1 .
?
2
In a similar way we have obtained five other classes of exact solutions of the
Yang–Mills equations
Aµ = e1 kµ e?T (b · x cos w1 + c · x sin w1 + w2 )1/2 Z1/4 (ie?/2)(b · x cos w1 +
+ c · x sin w1 + w2 )2 + e2 ? (b · x cos w1 + c · x sin w1 + w2 ) ?
?
? cµ cos w1 ? bµ sin w1 + 2kµ [(1/4)(?eT + T )(b · x sin w1 ?
? c · x cos w1 ) + w3 ] ;
458 R.Z. Zhdanov, W.I. Fushchych

Aµ = e1 kµ e?T C1 cosh[e?(b · x cos w1 + c · x sin w1 + w2 )] + C2 sinh[e? ?
? (b · x cos w1 + c · x sin w1 + w2 )] + e2 ? cµ cos w1 ? bµ sin w1 +
?
+ 2kµ [(1/4)(?eT + T )(b · x sin w1 ? c · x cos w1 ) + w3 ] ;
1/2
Aµ = e1 kµ e?T C 2 (b · x cos w1 + c · x sin w1 + w2 )2 + ?2 e2 C ?2 +
?1
+ e2 ? C 2 (b · x cos w1 + c · x sin w1 + w2 )2 + ?2 e2 C ?2 ?
? bµ cos w1 + cµ sin w1 ? (1/2)kµ [w1 (b · x sin w1 ? c · x cos w1 ) ? w2 ] ;
? ?
Aµ = e1 kµ Z0 (ie?/2)[(b · x + w1 )2 + (c · x + w2 )2 ] + e2 ? cµ (b · x + w1 ) ?
? bµ (c · x + w2 ) ? (1/2)kµ [w1 (c · x + w2 ) ? w2 (b · x + w1 )] ;
? ?
Aµ = e1 kµ C1 [(b · x + w1 )2 + (c · x + w2 )2 ]e?/2 + C2 [(b · x + w1 )2 +
+ (c · x + w2 )2 ]?e?/2 + e2 ?[(b · x + w1 )2 + (c · x + w2 )2 ]?1 ?
? cµ (b · x + w1 ) ? bµ (c · x + w2 ) ? (1/2)kµ [w1 (c · x + w2 ) ?
?
? w2 (b · x + w1 )] .
?
Here C1 , C2 , C = 0, ? are arbitrary parameters; w1 , w2 , w3 are arbitrary smooth
functions on ? = 1 k · x; T = T (?) is a solution of ODE (11). In addition, we use the
2
following notations:
k · x = kµ xµ , b · x = bµ xµ , c · x = cµ xµ ,
Zs (?) = C1 Js (?) + C2 Ys (?), e1 = (1, 0, 0), e2 = (0, 1, 0),
where Js , Ys are the Bessel functions.
Thus, we have obtained broad families of exact non-Abelian solutions of YMEs (1).
It can be verified by direct and rather involved computation that the solutions obtained
are not self-dual, i.e. that they do not satisfy self-dual YMEs.

4 Conclusion
Let us say a few words about symmetry interpretation of the ans?tze (2), (3), (10).
a
Consider as an example, the ansatz determined by expressions (10a). As a direct
computation shows, generators of a three-parameter Lie group G leaving it invariant
are of the form
3
Q2 = b? ?? ? 2[w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ )] Aa? ?Aaµ ,
Q1 = k? ?? ,
a=1
(17)
3
Q3 = c? ?? ? 2[w1 (kµ b? ? k? bµ ) + w3 (kµ c? ? k? cµ )] Aa? ?Aaµ .
a=1
Evidently, the system of PDEs (1) is invariant under the one-parameter group G1 ha-
ving the generator Q1 . But it is not invariant under the groups having the generators
Q2 , Q3 . Consider, as an example, the generator Q2 . Acting by the second prolongation
of the operator Q2 (which is constructed in a standard way, see e.g. [18, 20]) on the
system of PDEs (1), after some tedious algebra we obtain the following equality:
Q2 Lµ = 2 w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ ) L? +
2
(18)
+ 2 w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ ) Q1 A? ?
? ?
Conditional symmetry and new classical solutions of the Yang–Mills equations 459

? ? µ (w0 b? + w2 c? )Q1 A? ? k? (w0 Q2 A? + w2 Q3 A? ) ?

? (w0 bµ + w2 cµ )?? Q1 A? ? kµ w0 (w0 b? + w2 c? ) +

+ w2 (w1 b? + w3 c? ) Q1 A? + e (w0 b? + w2 c? )Q1 A? ?

? k? (w0 Q2 A? + w2 Q3 A? ) ? Aµ + 2e(w0 b? A? + w2 c? A? )?Q1 Aµ ?

? 2ek? A? ? (w0 Q2 Aµ + w2 Q3 Aµ ) + eA? ? (w0 bµ + w2 cµ )Q1 A? ?

? ekµ A? ? (w0 Q2 A? + w2 Q3 A? ).

In the above expressions we use the designations
Lµ ? ?? ? ? Aµ ? ? µ ?? A? + e (?? A? ) ? Aµ ? 2(?? Aµ ) ? A? +
+ (? µ A? ) ? A? + e2 A? ? (A? ? Aµ ),
Q1 Aµ ? k? ?? Aµ ,
Q2 Aµ ? b? ?? Aµ + 2 w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ ) A? ,
Q3 Aµ ? c? ?? Aµ + 2 w1 (kµ b? ? k? bµ ) + w3 (kµ c? ? k? cµ ) A?
and by the symbol Q2 we denote the second prolongation of the operator Q2 .
2
As underlined terms in (18) do not vanish on the set of solutions of YMEs,
system of PDEs (1) is not invariant under the Lie transformation group G2 having
the generator Q2 . On the other hand, system
Lµ = 0, Qa Aµ = 0, a = 1, 2, 3
is evidently invariant under the group G2 . The same assertion holds for the Lie
transformation group G3 having the generator Q3 . Consequently, the YMEs are con-
ditionally-invariant with respect to the three-parameter Lie transformation group G =
G1 ? G2 ? G3 . This means that solutions of the YMEs obtained with the help of the
ansatz invariant under the group with generators (17) can not be found by means of
the classical symmetry reduction procedure.
As rather tedious computations show, the ans?tze determined by the expressions
a
(10b)–(10d) also correspond to conditional symmetry of YMEs. Hence it follows, in
particular, that the YMEs should be included into the long list of mathematical and
theoretical physics equations possessing non-trivial conditional symmetry [7].
Another interesting observation is that specifying the arbitrary functions contained
in non-Lie ans?tze in an appropriate way, one can obtain some Lie ans?tze. Really,
a a
expressions (8c), (8l), (8m), (8q) are particular cases of expressions (10a), expressions
(8a), (8e), (8f), (8g), (8n), (8o), (8s), (8t), (8u) are particular cases of expressions
(10b), (10c) and expressions (8h), (8p) are particular cases of the expressions (10d).
So if we denote the invariant solutions of the Yang–Mills equations symbolically by
the dots in some space of solutions of system of PDEs (1), then some of them can be
connected by curves which are conditionally-invariant solutions! Thus, at the first the
distinct glance solutions are the particular cases of more general solutions. A similar
 << Ïðåäûäóùàÿ ñòð. 109(èç 122 ñòð.)ÎÃËÀÂËÅÍÈÅ Ñëåäóþùàÿ >>