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. 43
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L22 :
2 2
?1
? = ax ? dx + log |kx| ? (bx) (kx) ,
2

1 1
? = |cx|?1 , ?0 = log |cx|, ?1 = 0, ?2 = ? kx,
L23 :
2 4
?1
2
? = [4bx + (kx) ](cx) ,
1 1
? = |4bx + (kx)2 |?1 , ?0 = log |4bx + (kx)2 |, ?1 = 0, ?2 = ? kx,
L24 :
2 4
2
1
? = ax ? dx + bxkx + (kx)3 [4bx + (kx)2 ]?3 .
6

Here, ? = arctan cx , ?1 = (bx)2 + (cx)2 , ?2 = (ax)2 ? (dx)2 , and ?3 = (ax)2 ?
bx
(bx)2 ? (dx)2 .
Proof. All of the cases are analyzed similarly, so we can limit ourselves to the
subalgebra L2 = J12 + ?D, P0 , P3 .
According to Theorem 1, the entries of column ??1 A are invariants of the subal-
gebra L2 if and only if
?? ?? ?? ?? ??
?x1 ? ?(S12 + ?E) = 0,
+ x2 + ? xµ = 0, = 0.(8)
?x2 ?x1 ?xµ ?x0 ?x3
192 V.I. Lahno, W.I. Fushchych

The last two equations in (8) demonstrate that ? = ?(x1 , x2 ), while the ?rst equation
implies that one can set ?0 = ?2 = 0 in the expression for ?. By the Campbell–
Hausdor? formula, we have, in this case,
?? ?? ??1
= ???µ
?µ + .
?xµ ?xµ ?xµ
Hence, the common factor of ? can be canceled from the left on the left-hand side
of the ?rst equation in (8), which gives an equation whose left-hand side can be
represented as a combination of the matrices E and S12 . Equating the coe?cients in
these combinations to zero, we arrive at the system of equations below:
1 ?? ?? ?? ??
? x2 ? ? x1 ? ? = 0,
x1 + x2
? ?x2 ?x1 ?x1 ?x2
(9)
??1 ??1 ??1 ??1
? x2 ? ? x1 ? 1 = 0,
x1 + x2
?x2 ?x1 ?x1 ?x2
which is equivalent to (8). It is not di?cult to verify that system (9) is satis?ed by
the functions
x2 cx
1 1
? = (x2 + x2 )? 2 = [(bx)2 + (cx)2 ]? 2 , ?1 = arctan = arctan .
1 2
x1 bx
Equations (7) for ?(x) are of the form
?? ?? ?? ?? ??
?x1 + x2 + ? xµ = 0, = 0, = 0.
?x1 ?x2 ?xµ ?x0 ?x1
This implies that
x2 cx
? = log(x2 + x2 ) + 2 arctan = log[(bx)2 + (cx)2 ] + 2 arctan ,
1 2
x1 bx
which proves the theorem.


3 Covariant form of the linear ansatz
and symmetry reduction of SDYME
By Theorem 2, the ansatze that correspond to the subalgebras Lj (j = 1, . . . , 24), are
of the linear form (4), where

?(x) = exp{2?2 (S01 ? S13 )} exp{?1 S12 } exp{??0 S03 } exp{log ?E}.

Thus, it follows that
? ?
[cosh ?0 + 2?2 e??0 ] [sinh ?0 + 2?2 e??0 ]
2 2
2[??2 cos ?1 ] 2[?2 sin ?1 ]
? ?
2[??2 e??0 ] 2[?2 e??0 ]
? ?
[cos ?1 ] [? sin ?1 ]
? = ?? ?,
? ?
[0] [sin ?1 ] [cos ?1 ] [0]
[sinh ?0 + 2?2 e??0 ] [cosh ?0 ? 2?2 e??0 ]
2 2
2[??2 cos ?1 ] 2[?2 sin ?1 ]

where [f ] denotes [f ] = f · I and I is a unit matrix of order 3.
Reduction of self-dual Yang–Mills equations 193

In view of the above, ansatz (4) can be represented in the following form:

A0 = ?[cosh ?0 B0 + sinh ?0 B3 + 2?2 e??0 (B0 ? B3 ) + 2?2 (sin ?1 B2 ? cos ?1 B1 )],
2

A1 = ?[cos ?1 B1 ? sin ?1 B2 ? 2?2 e??0 (B0 ? B3 )],
(10)
A2 = ?[sin ?1 B1 + cos ?1 B2 ],
A3 = ?[sinh ?0 B0 + cosh ?0 B3 + 2?2 e??0 (B0 ? B3 ) + ?2 (sin ?1 B2 ? cos ?1 B1 )],

and, as is not di?cult to verify,

Aµ = aµ A0 + bµ A1 + cµ A2 + dµ A3 ,
B1 = ?b? B ? , B2 = ?c? B ? , B3 = ?d? B ? ,
B0 = a? B ? ,

where aµ , bµ , cµ , and dµ are the µth components of the vectors a, b, c, and d, respec-
tively, given in Section 2.
In these notations, the linear ansatz (10), as well as the linear ansatz (4) can be
represented as

Aµ (x) = ?aµ? (x)B ? (?) = ?{(aµ a? ? dµ d? ) cosh ?0 + (dµ a? ? d? aµ ) sinh ?0 +
+ 2(aµ + dµ )[?2 cos ?1 b? ? ?2 sin ?1 c? + ?2 e??0 (a? + d? )] +
2
(11)
+ (bµ c? ? b? cµ ) sin ?1 ? (cµ c? + bµ b? ) cos ?1 ?
? 2e??0 ?2 bµ (a? + d? )}B ? )}B ? (?).

The values taken by the functions ?, ?0 , ?1 , ?2 , and ? in (11) are given in Theorem 2
for each of the subalgebras Lj (j = 1, . . . , 24).
?
Thus, we have written the P (1, 3)-invariant ansatz for the Aµ (x) ?elds in a mani-
festly covariant form.
Let us note that ansatz (11) can be obtained from (10) by applying the proliferation
formulas that correspond to the Lorentz group AO(1, 3) to the functions Aµ from (10)
with the generators (2) (see, for instance, [14, 15]). Therefore, the vectors a, b, c, and
d can be viewed as a general system of orthonormalized vectors in the Minkowski
space R(1, 3), which can be expressed as

aµ aµ = ?bµ bµ = ?cµ cµ = ?dµ dµ = 1,
aµ bµ = aµ cµ = aµ dµ = bµ cµ = bµ dµ = cµ dµ = 0.

?
The uni?ed form of the P (1, 3)-invariant ansatze derived in (11) allows us to perform
the reduction of SDYME (1) in the general form.
Lemma. The ansatz (11) allows one to reduce SDYME (1) to the system
i
?µ??? T ?? , (12)
Tµ? =
2
where

dB? (?) dBµ (?)
? G? (?) + Hµ (?)B? (?) ?
Tµ? = Gµ (?)
(13)
d? d?
? H? (?)Bµ (?) + Sµ?? (?)B ? (?) + eBµ (?) ? B? (?).
194 V.I. Lahno, W.I. Fushchych

In (13), the functions Gµ (?), Hµ (?), and Sµ?? (?) are determined from

?? ?? ?aµ? ?aµ?
?S??? = aµ a?? ? aµ
H? ?2 = aµ?
?G? = aµ? , , a?? .
?
?
?xµ ?xµ ?x? ?x?

To prove the lemma, it su?ces to substitute ansatz (11) into SDYME (1) and to
contract the resulting expression with the tensor aµ a? , using the fact that aµ? satis?es
??
aµ aµ? = g?? .
?
According to the lemma, the construction of the reduced systems associated with
subalgebras Lj is tantamount to ?nding the functions G? (?), H? (?), and S??? (?)
for every such subalgebra. We skip the cumbersome calculations and give only the
explicit form of these functions for each of the subalgebras Lj in the following list:

G? = 1 (c? ? b? ?), H? = ? 1 b? , S??? = 0,
L1 :
G? = 2(b? + c? ), H? = ?b? , S??? = (b? c? ? b? c? )c? ,
L2 :
v v 1
G? = 2 ?(b? ? 2 ?d? ), H? = ? 2 d? , S??? = v (c? b? ? b? c? )c? ,
L3 :
?
v v 1
G? = 2 ?(b? ? 3 ?a? ), H? = ? 3 a? , S??? = v (c? b? ? b? c? )c? ,
L4 :
?
G? = 1 (c? ? b? ?), H? = ? 1 b? ,
L5 :
S??? = 1 ??1 [b? (d? a? ? d? a? ) ? b? (d? a? ? d? a? )],
G? = 4 (1 ? ?)(a? ? d? ) + 5 (1 + ?)k? ,
L6 :
1
H? = ? 6 [ 5 (a? ? d? ) + 4 k? ],
2
1
S??? = [ 4 (a? ? d? ) ? 5 k? ](a? d? ? a? d? ),
2
1
G? = ?[ 5 ?k? ? ? ? + 7 (1 ? ?)c? ], H? = ? 7 c? ,
L7 :
?1
[c? (a? d? ? d? a? ) ? c? (a? d? ? d? a? )],
S??? = 7?
1
G? = k? ? 4 (a? ? d? ), H? = ? 4 (a? ? d? ),
L8 :
2
1
S??? = 4 [(a? ? d? )(a? d? ? a? d? )],
2
G? = k? ? 2 7 c? , H? = ? 7 c? ,
L9 :
S??? = 7 [c? (a? d? ? d? a? ) ? c? (a? d? ? d? a? )],
G? = 4 [(a? ? d? )? + k? ] ? 2 7 c? ?, H? = ? 7 c? ,
L10 :
S??? = 4 (a? ? d? )(a? d? ) ? a? d? ) ? 7 c? (a? d? ? d? a? ) +
+ 7 c? (a? d? ? d? a? ),
G? = 7 ?(c? ? b? ?), H? = ? 7 c? ,
L11 :
S??? = 7 [c? (a? d? ? d? a? ) ? c? (a? d? ? d? a? )] ? 5 k? (a? d? ? d? a? ),
G? = 2(b? + ?c? ), H? = ?b? ,
L12 :
S??? = c? (c? b? ? c? b? ) ? ?[c? (d? a? ? a? d? ) ? c? (d? a? ? a? d? )],
G? = 4 (? ? ?)(a? ? d? ) + 5 (? + ?)k? ,
L13 :
1
H? = ? 6 [ 4 k? + 5 (a? ? d? )],
2
Reduction of self-dual Yang–Mills equations 195

1 1
[ 4 (a? ? d? ) ? 5 k? ](a? d? ? a? d? ) ? [( 4 (a? ? d? ) ?
S??? =
2 2?
? 5 k? )(b? c? ? c? b? ) ? ( 4 (a? ? d? ) ? 5 k? )(b? c? ? c? b? )],
1
G? = k? ? 4 (a? ? d? ), H? = ? 4 (a? ? d? ),
L14 :
2
1
S??? = 4 [(a? ? d? )(a? d? ? a? d? ) ? (a? ? d? )(b? c? ? c? b? ) +
2
+ (a? ? d? )(b? c? ? c? b? )],
1
G? = 2(b? + ?c? ? k? e 2 ? ), H? = ?b? ,
L15 :
S??? = c? (c? b? ? c? b? ) ? ?[c? (d? a? ? a? d? ) ? c? (d? a? ? a? d? )],
G? = 2[(1 ? ?)b? + ?k? + ?c? ], H? = ?b? ,
L16 :
S??? = c? (c? b? ? c? b? ) ? k? (a? d? ? a? d? ) + b? (d? a? ? a? d? ) ?
? b? (d? a? ? a? d? ),
G? = k? ? 2b? + 2?c? , H? = ?b? ,
L17 :
S??? = b? (d? a? ? a? d? ) ? b? (d? a? ? a? d? ) + c? (c? b? ? c? b? ),
G? = k? + 2c? , H? = ?b? ,
L18 :
S??? = b? (d? a? ? a? d? ) ? b? (d? a? ? a? d? ) + c? (c? b? ? c? b? ),
v
G? = 2b? ? ? 6 ? ?( 4 k? + 5 (a? ? d? )), H? = ?b? ,
L19 :
1v
?[ 4 (a? ? d? ) ? 5 k? ](d? a? ? a? d? ) + c? (b? c? ? c? b? )
S??? =
2

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. 43
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