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L15 = G1 + P0 ? P3 , P0 + P3 , P1 + ?P2 ; L16 = J12 , J03 , P0 + P3 ;
L17 = G1 + P2 , G2 ? P1 + ?P2 , P0 + P3 ; L18 = J03 , G1 , P2 ;
L19 = G1 , J03 , P0 + P3 ; L20 = G1 , J03 + P2 , P0 + P3 ;
L21 = G1 , J03 + P1 + ?P2 , P0 + P3 ; L22 = G1 , G2 , J03 + ?J12 ;
L23 = G1 , P0 + P3 , P1 ; L24 = J12 , P1 , P2 ;
L25 = J03 , P0 , P3 ; L26 = J12 , J13 , J23 ;
L27 = J01 , J02 , J12 .
Here Gi = J0i ? Ji3 (i = 1, 2), ? ? R.
Ansatzes for the Yang–Mills field Aa (x) are of the form (3.13), functions ?µ (x),
µ
µ = 0, 3, w(x) being determined by one of the following formulae:
L1 : ?µ = 0, w = dx; L2 : ?µ = 0, w = ax; L3 : ?µ = 0, w = kx;
?0 = ? ln |kx|, ?1 = ?2 = 0, ?3 = ? ln |kx|, w = (ax)2 ? (dx)2 ;
L4 :
?0 = ? ln |kx|, ?1 = ?2 = ?3 = 0, w = cx;
L5 :
?0 = ?bx, ?1 = ?2 = ?3 = 0, w = cx;
L6 :
?0 = ?bx, ?1 = ?2 = ?3 = 0, w = bx ? ln |kx|;
L7 :
?0 = ? arctan(bx(cx)?1 ), ?1 = ?2 = 0,
L8 :
?3 = ? arctan(bx(cx)?1 ), w = (bx)2 + (cx)2 ;
?0 = ?1 = ?2 = 0, ?3 = ?ax, w = dx;
L9 :
L10 : ?0 = ?1 = ?2 = 0, ?3 = dx, w = ax;
1
?0 = ?1 = ?2 = 0, ?3 = ? kx, w = ax ? dx;
L11 : (3.15)
2
1
?0 = 0, ?1 = (bx ? ?cx)(kx)?1 , ?2 = ?3 = 0, w = kx;
L12 :
2
?0 = ?2 = ?3 = 0, ?1 = 1 cx, w = kx;
L13 : 2
1
?0 = ?2 = ?3 = 0, ?1 = ? kx, w = 4bx + (kx)2 ;
L14 :
4
1
?0 = ?2 = ?3 = 0, ?1 = ? kx, w = 4(?bx ? cx) + ?(kx)2 ;
L15 :
4
?0 = ? ln |kx|, ?1 = ?2 = 0, ?3 = ? arctan(bx(cx)?1 ),
L16 :
w = (bx)2 + (cx)2 ;
1
?0 = ?3 = 0, ?1 = (cx + (? + kx)bx)(1 + kx(? + kx))?1 ,
L17 :
2
Symmetry reduction and exact solutions of the Yang–Mills equations 439

1
?2 = ? (bx ? cxkx)(1 + kx(? + kx))?1 , w = kx;
2
1
?0 = ? ln |kx|, ?1 = bx(kx)?1 , ?2 = ?3 = 0,
L18 :
2
w = (ax) ? (bx) ? (dx)2 ;
2 2

1
?0 = ? ln |kx|, ?1 = bx(kx)?1 , ?2 = ?3 = 0, w = cx;
L19 :
2
1
?0 = ? ln |kx|, ?1 = bx(kx)?1 , ?2 = ?3 = 0, w = ln |kx| ? cx;
L20 :
2
1
?0 = ? ln |kx|, ?1 = (bx ? ln |kx|)(kx)?1 , ?2 = ?3 = 0,
L21 :
2
w = ? ln |kx| ? cx;
1 1
?0 = ? ln |kx|, ?1 = bx(kx)?1 , ?2 = cx(kx)?1 ,
L22 :
2 2
?3 = ? ln |kx|, w = (ax) ? (bx) ? (cx) ? (dx)2 .
2 2 2


Here ax = aµ xµ , bx = bµ xµ , cx = cµ xµ , dx = dµ xµ , µ = 0, 3, kx = ax + dx.
Note. Basis elements of subalgebras L23 , L24 , L25 , L26 , L27 do not satisfy (3.3). That
is why, ansatzes invariant under these subalgebras are partially-invariant solutions
and are not considered here.


4 Reduction of the Yang–Mills equations
In order to reduce YME to ODE it is necessary to substitute ansatz (3.13) into (1.1)
and convolute the expression obtained with Qµ (x). As a result, we get a system of
?
a
twelve nonlinear ODE for functions B? (w) of the form
? ? ?
kµ? B ? + lµ? B ? + mµ? B ? + egµ?? B ? ? B ? + ehµ?? B ? ? B ? +
(4.1)
+ e2 B? ? (B ? ? Bµ ) = 0.
Coefficients of the reduced ODE are given by the following formulae:
?
kµ? = gµ? F1 ? Gµ G? , lµ? = gµ? F2 + 2Sµ? ? Gµ H? ? Gµ G? ,
?
mµ? = Rµ? ? Gµ H? , gµ?? = gµ? G? + g?? Gµ ? 2gµ? G? , (4.2)
hµ?? = (1/2)(gµ? H? ? gµ? H? ) ? Tµ?? ,
where gµ? is a metric tensor of the Minkowski space R(1, 3) and F1 , F2 , Gµ , . . . , Tµ??
are functions on w determined by the relations
F1 = wxµ wxµ , F2 = 2w, Gµ = Q?µ wx? , Hµ = Q?µx? ,
Sµ? = Q? Q??x? wx? , Rµ? = Q? 2Q?? , (4.3)
µ µ
Q? Q??x? Q?? Q? Q??x? Q?µ Q? Q?µx? Q?? .
Tµ?? = + +
µ ? ?

Substituting functions Qµ? (x) from (3.13), where ?µ (x), w(x) are determined by
one of the formulae (3.15) into (4.2), (4.3) we obtain coefficients of the corresponding
systems of ODE (4.1)
L1 : kµ? = ?gµ? ? dµ d? , lµ? = mµ? = 0,
gµ?? = gµ? d? + g?? dµ ? 2gµ? d? , hµ?? = 0;
440 V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych

kµ? = gµ? ? aµ a? , lµ? = mµ? = 0,
L2 :
gµ?? = gµ? a? + g?? aµ ? 2gµ? a? , hµ?? = 0;
kµ? = ?kµ k? , lµ? = mµ? = 0, gµ?? = gµ? k? + g?? kµ ? 2gµ? k? ,
L3 :
hµ?? = 0;
kµ? = 4gµ? w ? aµ a? (w + 1)2 ? dµ d? (w ? 1)2 ? (aµ d? + a? dµ )(w2 ? 1),
L4 :
lµ? = 4(gµ? + ?(bµ c? ? cµ b? )) ? 2kµ (a? ? d? + k? w), mµ? = 0,
gµ?? = (gµ? (a? ? d? + k? w) + g?? (aµ ? dµ + kµ w) ?
? 2gµ? (a? ? d? + k? w)),
[gµ? k? ? gµ? k? ] + ? [(bµ c? ? cµ b? )k? + (b? c? ? c? b? )kµ +
hµ?? =
2
+ (b? cµ ? c? bµ )k? ];
= ?gµ? ? cµ c? , lµ? = ? cµ k? , mµ? = 0,
L5 : kµ?
gµ?? = gµ? c? + g?? cµ ? 2gµ? c? , (gµ? k? ? gµ? k? );
hµ?? =
2
kµ? = ?gµ? ? cµ c? , lµ? = 0,
L6 :
mµ? = ?(aµ a? ? dµ d? ), gµ?? = gµ? c? + g?? cµ ? 2gµ? c? ,
hµ?? = ?[(aµ d? ? a? dµ )b? + (a? d? ? a? d? )bµ + (a? dµ ? aµ d? )b? ];
kµ? = ?gµ? ? (bµ ? kµ )(b? ? k? ), lµ? = ?2(aµ d? ? a? dµ ),
L7 :
mµ? = ?(aµ a? ? dµ d? ),
gµ?? = gµ? (b? ? k? ) + g?? (bµ ? kµ ) ? 2gµ? (b? ? k? ),
hµ?? = ?[(aµ d? ? a? dµ )b? + (a? d? ? a? d? )bµ + (a? dµ ? aµ d? )b? ];
(4.4)
kµ? = ?4w(gµ? + cµ c? ), lµ? = ?4(gµ? + cµ c? ),
L8 :
1
mµ? = ? (?2 (aµ a? ? dµ d? ) + bµ b? ),
w
v
gµ?? = 2 w(gµ? c? + g?? cµ ? 2gµ? c? ),
1 ?
hµ?? = v (gµ? c? ? gµ? c? ) + v ((aµ d? ? a? dµ )b? +
2w w
+ (a? d? ? d? a? )bµ + (a? dµ ? aµ d? )b? );
kµ? = ?gµ? ? dµ d? , lµ? = 0,
L9 :
mµ? = bµ b? + cµ c? , gµ?? = gµ? d? + g?? dµ ? 2gµ? d? ,
hµ?? = a? (bµ c? ? cµ b? ) + aµ (b? c? ? c? b? ) + a? (b? cµ ? c? bµ );
kµ? = gµ? ? aµ a? , lµ? = 0,
L10 :
mµ? = ?(bµ b? + cµ c? ), gµ?? = gµ? a? + g?? aµ ? 2gµ? a? ,
hµ?? = ?[d? (bµ c? ? cµ b? ) + dµ (b? c? ? c? b? ) + d? (b? cµ ? c? bµ )];
kµ? = ?(aµ ? dµ )(a? ? d? ), lµ? = ?2(bµ c? ? cµ b? ), mµ? = 0,
L11 :
gµ?? = gµ? (a? ? d? ) + g?? (aµ ? dµ ) ? 2gµ? (a? ? d? ),
hµ?? = 1 [k? (bµ c? ? cµ b? ) + kµ (b? c? ? c? b? ) + k? (b? cµ ? c? bµ )];
2
?2
1
kµ? = ?kµ k? , lµ? = ? kµ k? , mµ? = ? 2 kµ k? ,
L12 :
w w
gµ?? = gµ? k? + g?? kµ ? 2gµ? k? ,
1
(gµ? k? ? gµ? k? ) +
hµ?? =
2w
Symmetry reduction and exact solutions of the Yang–Mills equations 441

?
((kµ b? ? k? bµ )c? + (k? b? ? k? b? )cµ + (k? bµ ? kµ b? )c? );
+
w
kµ? = ?kµ k? , lµ? = 0, mµ? = ?kµ k? ,
L13 :
gµ?? = gµ? k? + g?? kµ ? 2gµ? k? , hµ?? = ?((kµ b? ? k? bµ )c? +
+ (k? b? ? k? b? )cµ + (k? bµ ? kµ b? )c? );
kµ? = ?16(gµ? + bµ b? ), lµ? = mµ? = hµ?? = 0,
L14 :
gµ?? = 4(gµ? b? + g?? bµ ? 2gµ? b? );
kµ? = ?16[(1 + ?2 )gµ? + (cµ ? ?bµ )(c? ? ?b? )],
L15 :
lµ? = mµ? = hµ?? = 0,
gµ?? = ?4[gµ? (c? ? ?b? ) + g?? (cµ ? ?bµ ) ? 2gµ? (c? ? ?b? )];
v
kµ? = ?4w(gµ? + cµ c? ), lµ? = ?4(gµ? + cµ c? ) ? 2 k? cµ w,
L16 :
v
1
mµ? = ? bµ b? , gµ?? = 2 w(gµ? c? + g?? cµ ? 2gµ? c? ),
w
1 1
hµ?? = [ (gµ? k? ? gµ? k? ) + v (gµ? c? ? gµ? c? )];
2 w
2w + ?
kµ? = ?kµ k? , lµ? = ?
L17 : kµ k? ,
w(w + ?) + 1
mµ? = ?4kµ k? (1 + w(? + w))?2 , gµ?? = gµ? k? + g?? kµ ? 2gµ? k? ,
hµ?? = 1 (? + 2w)(gµ? k? ? gµ? k? )(1 + w(? + w))?1 ?
2
2(1 + w(w + ?))?1 ((kµ b? ? k? bµ )c? + (k? b? ? k? b? )cµ +
+ (k? bµ ? kµ b? )c? );
kµ? = 4wgµ? ? (kµ w + aµ ? dµ )(k? w + a? ? d? ),
L18 :
lµ? = 6gµ? + 4(aµ d? ? a? dµ ) ? 3k? (kµ w + aµ ? dµ ), mµ? = ?kµ k? ,
gµ?? = (gµ? (k? w + a? ? d? ) + g?? (kµ w + aµ ? dµ ) ?
? 2gµ? (k? w + a? ? d? )),
hµ?? = (gµ? k? ? gµ? k? );
kµ? = ?gµ? ? cµ c? , lµ? = 2 k? cµ , mµ? = ?kµ k? ,
L19 :
gµ?? = gµ? c? + g?? cµ ? 2gµ? c? , hµ?? = (gµ? k? ? gµ? k? );
kµ? = ?gµ? ? (cµ ? kµ )(c? ? k? ), lµ? = 2 k? cµ ? 2kµ k? ,
L20 :
mµ? = ?kµ k? ,
gµ?? = gµ? ( k? ? c? ) + g?? ( kµ ? cµ ) ? 2gµ? ( k? ? c? ),
hµ?? = (gµ? k? ? gµ? k? );
kµ? = ?gµ? ? (cµ ? ? kµ )(c? ? ? k? ), lµ? = 2( k? cµ ? ?kµ k? ),
L21 :
mµ? = ?kµ k? ,
gµ?? = ?gµ? (c? ? ? k? ) ? g?? (cµ ? ? kµ ) + 2gµ? (c? ? ? k? ),
hµ?? = (gµ? k? ? gµ? k? );
kµ? = 4wgµ? ? (aµ ? dµ + kµ w)(a? ? d? + k? w),
L22 :
lµ? = 4[2gµ? + ?(bµ c? ? cµ b? ) ? aµ a? + dµ d? ? wkµ k? ],
mµ? = ?2kµ k? ,
gµ?? = (gµ? (a? ? d? + k? w) + g?? (aµ ? dµ + kµ w) ?
? 2gµ? (a? ? d? + k? w),
442 V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych

3
(gµ? k? ? gµ? k? ) ? ?[k? (bµ c? ? cµ b? ) +
hµ?? =
2
+ kµ (b? c? ? c? b? ) + k? (b? cµ ? c? bµ )];

= ?1 for ax + dx < 0.
where kµ = aµ + dµ , = 1 for ax + dx > 0 and


5 Exact solutions of the Yang–Mills equations
When applying the symmetry reduction procedure to the nonlinear Dirac equation,
we succeeded in constructing general solutions of a large part of reduced systems
of ODE. In the case involved we are not so lucky. Nevertheless, we obtain some
particular solutions of equations (4.2), (4.4).
The principal idea of our approach to integration of systems of ODE (4.2), (4.4)
is rather simple and quite natural. It is a reduction of these systems by the number
of components with the aid of ad hoc substitutions. Using this trick we construct
particular solutions of equations 1, 2, 5, 8, 14, 15, 16, 18, 19, 20, 21, 22 (? = 0).
Below we adduce substitutions for Bµ (w) and corresponding equations.

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