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Aµ = e2 dµ ? kµ (kx)2 ? bµ kx +
16.
8 2
1 1
+ e3 ?cµ + bµ + kµ kx (1 + ?2 )? 2 ?
2
v 1
? 4 2(1 + ?2 ) 2 [e(4(?bx ? cx) + ?(kx)2 )]?1 ;
4(?bx ? cx) + ?(kx)2 ?
17. Aµ = e1 kµ
446 V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych

ie? 1
(4(?bx ? cx) + ?(kx)2 )2 (1 + ?2 )? 2
? Z1 +
8
4

1 1
?cµ + bµ + kµ kx ?(4(?bx ? cx) + ?(kx)2 )(1 + ?2 )? 2 ;
+ e2
2
e? 1
(1 + ?2 )? 2 (4(?bx ? cx) + ?(kx)2 ) +
18. Aµ = e1 kµ ?1 cosh
4
e? 1
(1 + ?2 )? 2 (4(?bx ? cx) + ?(kx)2
+ ?2 sinh +
4
1 1
+ e2 ?cµ + bµ + kµ kx ?(1 + ?2 )? 2 ;
2
ie?
Aµ = e1 kµ |kx|?1 Z0 ((bc)2 + (cx)2 ) + e2 (bµ cx ? cµ bx)?;
19.
2
Aµ = e1 kµ |kx|?1 [?1 ((bx)2 + (cx)2 ) + ?2 ((bx)2 + (cx)2 )?
e? e?
20. ]+
2 2

+ e2 (bµ cx ? cµ bx)?((bx)2 + (cx)2 )?1 ;
v ie?
Aµ = e1 kµ |kx|?1 cxZ 4 (cx)2 + e2 (bµ ? kµ bx(kx)?1 )?cx;
21. 1
2
Aµ = e1 kµ |kx|?1 [?1 cosh(?ecx) + ?2 sinh(?ecx)] + e2 (bµ ? kµ bx(kx)?1 )?;
22.
ie?
Aµ = e1 kµ |kx|?1 ln |kx| ? cxZ 1 (ln |kx| ? cx)2 +
23.
2
4

+ e2 (bµ ? kµ bx(kx)?1 )?(ln |kx| ? cx);
Aµ = e1 kµ |kx|?1 [?1 cosh(?e(ln |kx| ? cx)) + ?2 sinh(?e(ln |kx| ? cx))] +
24.
+ e2 (bµ ? kµ bx(kx)?1 )?;
ie?
Aµ = e1 kµ |kx|?1 ? ln |kx| ? cxZ 4 (? ln |kx| ? cx)2 +
25. 1
2
+ e2 (bµ ? kµ (bx ? ln |kx|)(kx)?1 )?(? ln |kx| ? cx);
Aµ = e1 kµ |kx|?1 [?1 cosh(?e(? ln |kx| ? cx)) +
26.
+ ?2 sinh(?e(? ln |kx| ? cx))] + e2 (bµ ? kµ (bx ? ln |kx|)(kx)?1 )?;
1
Aµ = {e1 (bµ ? kµ bx(kx)?1 ) + e2 (cµ ? kµ cx(kx)?1 )}e?1 (xµ xµ )? 2 ;
27.
Aµ = {e1 (bµ ? kµ bx(kx)?1 ) + e2 (cµ ? kµ cx(kx)?1 )}f (xµ xµ ),
28.
wf + 2f? + (e2 f 3 /4) = 0, w = xµ xµ = (ax)2 ? (bx)2 ? (cx)2 ? (dx)2 .
?

In the above formulae Z? (w) is the Bessel function; sn, dn, cn are Jacobi elliptic
v
functions having the modulus 22 ; ?, ?1 , ?2 = const.
In the present paper we do not analyze in detail the obtained solution. We only note
that the solutions numbered by 27 is nothing more but the meron solution of YME [2].
In the Euclidean space meron and instanton solutions were obtained by Alfaro, Fubini,
Furlan [9] and Belavin, Polyakov, Schwartz, Tyupkin [10] with the use of the ansatz
suggested by ’t Hooft [11], Corrigan and Fairlie [12] and Wilczek [13].
Another important point is that we can obtain new exact solutions of YME by
applying to solutions (5.6) the solution generation technique. We do not adduce
corresponding formulae because of their cumbersomity.
Symmetry reduction and exact solutions of the Yang–Mills equations 447

6 Some generalizations
It was noticed in [14] that group-invariant solutions of nonlinear PDE could provide us
with rather general information about the structure of solutions of the equation under
study. Using this fact, we constructed in [4, 14] a number of new exact solutions
of the nonlinear Dirac equation which could not be obtained by symmetry reduction
procedure. We will demonstrate that the same idea will be effective for constructing
new solutions of YME.
Solutions of YME numbered by 7, 8, 19, 20 can be presented in the following
unified form:
(6.1)
Aµ = kµ B(kx, cx) + bµ C(kx, cx),
where kx = kµ xµ , cx = cµ xµ , kµ = aµ + dµ .
Substituting the ansatz (6.1) into YME and splitting the equality obtained with
respect to linearly-independent four-vectors with components kµ , bµ , cµ , we get
1. Cw1 w1 = 0,
C ? Cw1 = 0, (6.2)
2.
Bw1 w1 + eCw0 ? C + e2 C ? (C ? B) = 0.
3.
Here we use designations w0 = kx, w1 = cx.
A general solution of the first two equations from (6.2) is given by one of the
formulae
I. C = f (w0 ),
II. C = (w1 + v0 (w0 ))f (w0 ),

where v0 , f are arbitrary smooth functions.
Consider the case C = f (w0 ). Substituting this expression into the third equation
from (6.2) we have
Bw1 w1 + efw0 ? f + e2 f (f B) ? e2 f 2 B = 0. (6.3)
Since equations (6.3) do not contain derivatives of B with respect to w0 , they can
be considered as a system of ODE with respect to the variable w1 . Multiplying (6.3)
by f we arrive at the relation (B f )w1 w1 = 0, whence
(6.4)
B f = v1 (w0 )w1 + v2 (w0 ).
In (6.4) v1 , v2 are arbitrary smooth enough functions.
With account of (6.4) system (6.3) reads
Bw1 w1 ? e2 f 2 B = ef ? fw0 ? e2 (v1 w1 + v2 )f .
The above linear system of ODE is easily integrated. Its general solution is given
by the formula
B = g(w0 ) cosh e|f |w1 + h(w0 ) sinh e|f |w1 +
(6.5)
+ e?1 |f |?2 fw0 ? f + |f |?2 (v1 w1 + v2 )f ,

where g, h are arbitrary smooth functions.
448 V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych

Substituting (6.5) into (6.4) we get the following restrictions on the choice of the
functions g, h:
(6.6)
f g = 0, f h = 0.
Thus, provided Cw1 = 0, a general solution of the system of ODE (6.3) is given
by the formulae (6.5), (6.6). Substituting (6.5) into the initial ansatz (6.1) we obtain
the following family of exact solutions of YME:

Aµ = kµ {g(kx) cosh e|f |cx + h(kx) sinh e|f |cx +
?
+ e?1 |f |?2 f ? f + (v1 (kx)cx + v2 (kx))f } + bµ f

where f (kx), g(kx), h(kx), v1 (kx), v2 (kx) are arbitrary smooth functions satisfying
? df
(6.6), f = d?0 .
The case C = (w1 + v0 (w0 ))f (w0 ) is treated in analogous way. As a result, we
obtain the following family of exact solutions of YME:

ie
1
|f |(cx + v0 (kx))2 +
Aµ = kµ (cx + v0 (kx)) 2 g(kx)J 1
2
4

ie
|f |(cx + v0 (kx))2
+ h(kx)Y 1 +
2
4

?
+ (v1 (kx)cx + v2 (kx))f + e?1 |f |?2 f ? f + bµ (cx + v0 (kx))f ,

where f (kx), g(kx), h(kx), v0 (kx), v1 (kx), v2 (kx) are arbitrary smooth functions
satisfying (6.6), J 1 (w), Y 1 (w) are the Bessel functions.
4 4
Another effective ansatz for the Yang–Mills field is obtained if one replaces in (6.1)
cx by bx
(6.7)
Aµ = kµ B(kx, bx) + bµ C(kx, bx).

Substitution of (6.7) into YME yields the following system of PDE for B, C:
Bw1 w1 ? Cw0 w1 ? e(B ? Cw1 + 2Bw1 ? C + C ? Cw0 ) + e2 C ? (C ? B) = 0.(6.8)

We succeeded in integrating system (6.8), provided C = f (w0 ). Substituting the
result obtained into (6.7), we come to the following family of exact solutions of YME:

Aµ = kµ {(g + |f |?1 g ? f bx) cos(e|f |bx) + (h + |f |?1 h ? f bx) sin(e|f |bx) +
?
+ e?1 |f |?2 f ? f + (v1 (kx)bx + v2 (kx))f } + bµ f ,

where f (kx), g(kx), h(kx), v1 (kx), v2 (kx) are arbitrary smooth functions.
Besides that, we obtained the following class of exact solutions of YME:
Aµ = kµ e1 v0 (kx)u2 (bx) + bµ e2 u(bx),
where e1 = (1, 0, 0), e2 = (0, 1, 0); v0 (kx) is an arbitrary smooth function; u(bx) is a
solution of the nonlinear ODE u = e2 u5 , which is integrated in elliptic functions.
?
Symmetry reduction and exact solutions of the Yang–Mills equations 449

In conclusion of this Section we will obtain a generalization of the plane-wave
Coleman solution [15] ]
(6.9)
Aµ = kµ (f (kx)bx + g(kx)cx).

It is not difficult to verify that (6.9) satisfy YME with arbitrary f , g.
Evidently, solution (6.9) is a particular case of the ansatz
(6.10)
Aµ = kµ B(kx, bx, cx).
Substituting (6.10) into YME we get
(6.11)
Bw1 w1 + Bw2 w2 = 0,
where w1 = bx, w2 = cx.
Integrating the Laplace equations (6.11) and substituting the result obtained into
(6.10) we have
Aµ = kµ (U (kx, bx + icx) + U (kx, bx ? icx)).

Here U (kx, z) is an arbitrary analytical with respect to z function. Choosing U =
2 (f (kx) ? ig(kx))z we get Coleman solution (6.9).
1

7 Conclusion
Thus, starting from the invariance of YME under the Poincar? group we have obtained
e
wide families of its exact solutions including arbitrary functions. In our future papers
we intend to describe exact solutions of YME invariant under the extended Poincar? e
group and conformal group.
Besides that, we will study exact solutions which correspond to the conditional
and non-local symmetries of the Yang–Mills equations (1.1)
Acknowledgments. One of the authors (Wilhelm Fushchych) is indebted to DKNT
of Ukraina for financial support (project ¹ 11. 3/42). R.Z. Zhdanov was supported
by the Alexander von Humboldt Foundation.

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450 V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych

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Conditional symmetry and new classical
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