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1. Bµ = aµ e1 f (w) + bµ e2 g(w) + cµ e3 h(w),
? ?
f ? e2 (g 2 + h2 )f = 0, g + e2 (f 2 ? h2 )g = 0, h + e2 (f 2 ? g 2 )h = 0.
?
2. Bµ = bµ e1 f (w) + cµ e2 g(w) + dµ e3 h(w),
? ?
f + e2 (g 2 + h2 )f = 0, g + e2 (f 2 + h2 )g = 0, h + e2 (f 2 + g 2 )h = 0.
?
?
Bµ = kµ e1 f (w) + bµ e2 g(w), f ? e2 g 2 f = 0, g = 0.
5. ?
8.1. (? = 0) Bµ = kµ e1 f (w) + bµ e2 g(w),
4wf + 4f? ? e2 g 2 f = 0, 4w? + 4g ? w?1 g = 0.
? g ?
8.2. Bµ = aµ e1 f (w) + dµ e2 g(w) + bµ e3 h(w),
2
? + 4f? ? ? f ? 2?e gh + e2 (h2 + g 2 )f = 0,
v
4wf
w w
?2 2?e
g + v f h + e2 (f 2 ? h2 )g = 0,
4w? + 4g +
g ?
(5.1)
w w
2?e
4wh + 4h ? w?1 h + v f g + e2 (f 2 ? g 2 )h = 0.
? ?
w
14.1. Bµ = aµ e1 f (w) + dµ e2 g(w) + cµ e3 h(w),
?
16f ? e2 (h2 + g 2 )f = 0, 16? + e2 (f 2 ? h2 )g = 0,
g
?
16h + e2 (f 2 ? g 2 )h = 0.
?
16f ? e2 g 2 f = 0, g = 0.
14.2. Bµ = kµ e1 f (w) + cµ e2 g(w), ?
1
Bµ = aµ e1 f (w) + dµ e2 g(w) + (1 + ?2 )? 2 (?cµ + bµ )e3 h(w),
15.1.
?
16(1 + ?2 )f ? e2 (h2 + g 2 )f = 0, 16(1 + ?2 )? + e2 (f 2 ? h2 )g = 0,
g
?
16(1 + ?2 )h + e2 (f 2 ? g 2 )h = 0.
1
Bµ = kµ e1 f (w) + (1 + ?2 )? 2 (?cµ + bµ )e2 g(w),
15.2.
?
16(1 + ?2 )f ? e2 f g 2 = 0, g = 0.
?
Symmetry reduction and exact solutions of the Yang–Mills equations 443

16. Bµ = kµ e1 f (w) + bµ e2 g(w),
4wf + 4f? ? e2 g 2 f = 0, 4w? + 4g ? w?1 g = 0.
? g ?
18. Bµ = bµ e1 f (w) + cµ e2 g(w),
4wf + 6f? + e2 g 2 f = 0, 4w? + 6g + e2 f 2 g = 0.
? g ?
19. Bµ = kµ e1 f (w) + bµ e2 g(w),
?
f ? e2 g 2 f = 0, g = 0.
?
20. Bµ = kµ e1 f (w) + bµ e2 g(w),
?
f ? e2 g 2 f = 0, g = 0.
?
21. Bµ = kµ e1 f (w) + bµ e2 g(w),
?
f ? e2 g 2 f = 0, g = 0.
?
22. (? = 0) Bµ = bµ e1 f (w) + cµ e2 g(w),
4wf + 8f? + e2 g 2 f = 0, 4w? + 8g + e2 f 2 g = 0.
? g ?

In the above formulae we use designations e1 = (1, 0, 0), e2 = (0, 1, 0), e3 =
(0, 0, 1).
Thus, combining symmetry reduction by the number of independent variables and
reduction by the number of dependent variables we reduce YME to rather simple ODE.
It is worth reminding that effectiveness of the widely used ansatz for the Yang–Mills
field suggested by t’Hooft et al [2] is closely connected with the fact that it reduces
the system of twelve PDE to one nonlinear wave equation.
Next, we will briefly consider a procedure of integration of equations (5.1).
Substitution f = 0, g = h = u(w) reduces the system of ODE 1 from (5.1) to the
equation

u = e2 u3 , (5.2)
?

which is integrated in elliptic functions [8]. Besides that, ODE (5.2) has a solution
v
which is expressed in terms of elementary functions u = 2(ew ? C)?1 , C ? R1 .
ODE 2 with f = g = h = u(w) reduces to the form u + 2e2 u3 = 0.
?
This equation is also integrated in elliptic functions [8].
Integrating the second equation of system of ODE 5 we get g = C1 w + C2 ,
Ci ? R1 . If C1 = 0, then the constant C2 can be neglected, and we may put C2 = 0.
Provided C1 = 0, the first equation from system 5 reads
?
f ? e2 C1 w2 f = 0.
2
(5.3)

A general solution of ODE (5.3) is given by formula f = w1/2 Z 4 ( ie C1 w2 ).
1
2
Hereafter, we use the designation Z? (w) = C3 J? (w) + C4 Y? (w), where J? , Y? are
Bessel functions, C3 , C4 are arbitrary constants.
In the case C1 = 0, C2 = 0 a general solution of the first equation from system 5
reads f = C3 cosh C2 ew + C4 sinh C2 ew, where C3 , C4 are arbitrary constants.
At last, provided C1 = C2 = 0, a general solution of the first equation from
system 5 has the form f = C3 w + C4 , C3 , C4 ? R1 . v
A general solution of the second ODE from system 8.1 is of the form g = C1 w +
v
C2 ( w)?1 , where C1 , C2 are arbitrary constants.
444 V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych

Substituting the expression obtained into the first equation we get

4w2 f + 4wf? ? e2 (C1 w + C2 )2 f = 0.
? (5.4)

Under C1 , C2 = 0 a solution of ODE (5.4) is not known. In the remaining cases

ie
a) C1 = 0, C2 = 0 f = Z0 C1 w ,
2
eC2 eC2
f = C3 w 2 + C4 w?
b) C1 = 0, C2 = 0 ,
2

c) C1 = 0, C2 = 0 f = C3 ln w + C4 .

Here C3 , C4 are arbitrary constants.
We do not succeed in obtaining particular solutions of system 8.2. Equations 14.1
e
coincide with equations 1, if one changes e by 4 . Similarly, equations 14.2 coincide
e
with equations 5, if one changes e by 4 . Next, equations 15.1 coincide with equations 1
1
and equations 15.2 – with equations 5, if one replaces e by 4 (1 + ?2 )? 2 .
e

System of ODE 16 coincides with system 8.1 and systems 19, 20, 21 – with
system 5. We did not succeed in integrating equations 18.
At last, system 22 (? = 0) with the substitution f = g = u(w) reduces to the form

e2 3
(5.5)
w? + 2u + u = 0.
u ?
4
1
ODE (5.5) is Emden–Fowler equation and the function u = e?1 w? 2 , is its parti-
cular solution.
Substituting the results obtained into corresponding formulae from (5.1) and then
into the ansatz (3.13), we get exact solutions of the nonlinear YME (1.1). Let us
note that solutions of systems of ODE 5, 8.1, 14.2, 15.2, 16, 19, 20, 21 satisfying the
condition g = 0 give rise to Abelian solutions of YME. We do not adduce them and
present only non-Abelian solutions of YME.
v
Aµ = (e2 bµ + e3 cµ ) 2(edx ? ?)?1 ;
1.
v v v ?1
2 2 2
2. Aµ = (e2 bµ + e3 cµ ) ? sn e?dx dn e?dx cn e?dx ;
2 2 2
Aµ = (e2 bµ + e3 cµ )?[cn (e?dx)]?1 ;
3.
4. Aµ = (e1 bµ + e2 cµ + e3 dµ )? cn (e?ax);
v i
Aµ = e1 kµ |kx|?1 cxZ 1 e?(cx)2 + e2 bµ ?cx;
5.
2
4

Aµ = e1 kµ |kx|?1 [?1 cosh(e?cx) + ?2 sinh(e?cx)] + e2 bµ ?;
6.
i
Aµ = e1 kµ Z0 e?((bx)2 + (cx)2 ) + e2 (bµ cx ? cµ bx)?;
7.
2
+ ?2 ((bx)2 + (cx)2 )?
e? e?
Aµ = e1 kµ [?1 ((bx)2 + (cx)2 )
8. ]+
2 2

+ e2 (bµ cx ? cµ bx)?((bx)2 + (cx)2 )?1 ;
Symmetry reduction and exact solutions of the Yang–Mills equations 445
v
1 1 e2
(dµ ? kµ (kx)2 ) + bµ kx + e3 cµ ? sn ?(4bx + (kx)2 ) ?
9. Aµ = e2
8 2 8
v v ?1
e2 e2
? dn ?(4bx + (kx)2 ) ?(4bx + (kx)2 )
cn ;
8 8
1 1
(dµ ? kµ (kx)2 ) + bµ kx + e3 cµ ?
10. Aµ = e2
8 2
v ?1
e 2?
? ? cn (4bx + (kx)2 ) ;
8
1 1
(dµ ? kµ (kx)2 ) + bµ kx + e3 cµ ?
11. Aµ = e2
8 2
v
? 4 2(e(4bx + (kx)2 ) ? ?)?1 ;
ie?
(4bx + (kx)2 )2 + e2 cµ ?(4bx + (kx)2 );
Aµ = e1 kµ 4bx + (kx)2 Z 4
12. 1
8
e?
(4bx + (kx)2 ) +
13. Aµ = e1 kµ ?1 cosh
4
e?
(4bx + (kx)2 )
+ ?2 sinh + e2 cµ ?;
4
1 1
Aµ = e2 dµ ? kµ (kx)2 ? bµ kx +
14.
8 2
1 1
+ e3 ?cµ + bµ + kµ kx (1 + ?2 )? 2 ? (5.6)
2
v
e? 2 1
(4(?bx ? cx) + ?(kx)2 )(1 + ?2 )? 2 ?
? ? sn
8
v
e? 2 1
(4(?bx ? cx) + ?(kx)2 )(1 + ?2 )? 2 ?
? dn
8
v ?1
e? 2 2 ?1
? cn (4(?bx ? cx) + ?(kx) )(1 + ? ) 2
2
;
8
1 1
e2 dµ ? kµ (kx)2 ? bµ kx +
15. Aµ =
8 2
1 1
+ e3 ?cµ + bµ + kµ kx (1 + ?2 )? 2 ?
2
?1
e? 1
(4(?bx ? cx) + ?(kx)2 )(1 + ?2 )? 2
? cn ;
4
1 1
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