ñòð. 44 |

G? = 5 ?[(1 + ?)k? ? ? 2? + 8 (1 ? ?)(a? ? d? )? 2? ],

L20 :

1 1 1

H? = ? 5 [k? ? ? 2? + 8 (a? ? d? )? 2? ],

2

1 1 1

S??? = 5 [ (k? ? ? 2? + 8 (a? ? d? )? 2? )(a? d? ? d? a? ) +

2?

1

+ b? (k? b? ? k? b? )? ? 2? ],

G? = k? , H? = ? 9 [c? ? ? b? ],

L21 :

S??? = 9 [(c? ? ? b? )(a? d? ? d? a? ) ? (c? ? ? b? )(a? d? ? d? a? )] +

+ c? (k? b? ? k? b? ) ? c? (k? b? ? k? b? ),

1

G? = a? ? d? + 5 k? , H? = ? 5 k? ,

L22 :

2

1

S??? = 5 [b? (k? b? ? k? b? ) ? k? (a? d? ? d? a? )],

2

G? = 7 (4b? ? ?c? ), H? = ? 7 c? ,

L23 :

1 1

S??? = 7 [c? (a? d? ? d? a? ) ? c? (a? d? ? d? a? )] ? k? (k? b? ? k? b? ),

2 2

1

G? = |?| k? + 2 10 (a? ? d? ) ? 12 10 ?b? , H? = ?4 10 b? ,

L24 :

2

1

S??? = 2 10 [b? (a? d? ? d? a? ) ? b? (a? d? ? d? a? )] ? k? (k? b? ? k? b? ).

2

= ?1 for ? < 0. The values of the functions ? for every

Here, k = 1 for ? > 0 and k

k are given in Table 1.

196 V.I. Lahno, W.I. Fushchych

Table 1

k ? k ?

(ax) ? (dx)2

2

1 bx 6

2 dx 7 cx

8 (ax) ? (bx)2 ? (dx)2

2

3 ax

ax ? dx cxkx ? bx

4 9

4bx + (kx)2

5 kx 10

4 On the exact real solutions of SDYME

Before we proceed to analyzing the reduced systems and constructing their exact

solutions, let us make the following remark. Whereas the YME and SDYME are real

in four-dimensional Euclidean space, in Minkowski space, the YME are a system of

real second-order PDE, while SDYME (1) are a system of complex ?rst-order PDE.

Therefore, self-dual solutions to YME in Minkowski space are, in general, complex,

which is an undesirable property.

On the other hand, the systems of PDE that represent SDYME (1) (and, hence,

the reduced systems (12) and (13), as well) are not completely de?ned. Moreover, the

symmetry reduction of SDYME preserves their symmetric form, which allows one to

address the problem of ?nding real solutions of these equations. Clearly, the necessary

condition for building real solutions of the systems of equations (12) and (13) is given

by the equations

(14)

Tµ? = 0,

which lead us to another system of ?rst-order ODE, this time an overdetermined one.

By imposing additional conditions on the functions Bµ , we have succeeded, in some

cases, in reducing system (14) to an integrable form and in obtaining nontrivial real

non-Abelian solutions of SDYME (1). In what follows, we describe these cases in some

detail.

We use the notation e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). In order to

restore the explicit form of systems (13) and (14), we choose a = (1, 0, 0, 0), b =

(0, 1, 0, 0), c = (0, 0, 1, 0), and d = (0, 0, 0, 1).

The case of the L1 algebra. Let us set B0 = ?0 B and B3 = ?3 B, where ?0 and

?3 are arbitrary real constants such that ?2 + ?2 = 0. Equations (13) and (14) take

0 3

the following form:

dB

+ eB2 ? B = 0,

1

d?

dB2 dB1

+ 1 B2 ? eB1 ? B2 = 0, (15)

1? +1

d? d?

dB

+ 1 B + eB ? B1 = 0.

1?

d?

Further, let us assume that, in (15), B = gm (?)em , B1 = hm (?)em , and B 2 = f (?)e2 ,

m = 1, 2, 3. Then the ?rst two equations of (15) yield the following system for the

Reduction of self-dual Yang–Mills equations 197

functions gm , hm , and f :

dg1 dh1 dg2

+ ef g3 = 0, + ef h3 = 0, = 0,

1 1 1

d? d? d? (16)

df dh2 dg3 dh3

? eg1 f = 0, ? eh1 f = 0.

1? +1 + 1 f = 0, 1 1

d? d? d? d?

We set f = C? ?1 in (16), with C being an arbitrary constant. Then, g2 = C1 and

h2 = C2 , where C1 and C2 are arbitrary constants, while the functions g1 , g3 , h1 ,

and h3 are to be determined from two similar systems of equations, which amounts

to solving the Euler equations. In particular, the system of equations for g1 , g3 reads

dg1 dg3

+ eC? ?1 g3 = 0, ? eC? ?1 g1 = 0,

1 1

d? d?

from which we have the equation

d2 g3 dg3

?2 + e2 C 2 g3 = 0,

+?

2

d? d?

whose general solution is given by

g3 = C3 sin(eC log |?| + C4 ),

and, thus,

cos(eC log |?| + C4 ).

g1 = 1 C3

Similarly, we obtain

cos(eC log |?| + C6 ). h3 = C5 sin(eC log |?| + C6 ).

h1 = 1 C5

where C3 , C4 , C5 , and C6 are arbitrary integration constants.

Finally, having checked the last of the equations in (15), we obtain the following

solution:

B0 = ?0 B, B3 = ?3 B, B = gm (?)em , B1 = hm (?)em , B2 = f (?)e2 ,

where

g1 = ? 1 C3 cos(eC1 log |?| + C2 ), g2 = C3 ,

g3 = ?C3 sin(eC1 log |?| + C2 ), h1 = ± 1 e?1 sin(eC1 log |?| + C2 ), (17)

h2 = ?C1 , h3 = ?e?1 cos(eC1 log |?| + C2 ), f = C1 ? ?1 ,

and C1 , C2 , and C3 are arbitrary constants.

The case of the L9 algebra. Let B0 = B3 = B and B1 = 0. Then the systems

of equations (13) and (14) reduce to the equation

dB dB2

+ 2 7 B + eB ? B2 = 0. (18)

2 +

7

d? d?

Let us set B2 = f (?)e2 and B = g(?)e1 + h(?)e3 . Then it follows from (18) that

dg df dh

+ 2 7 g ? ef h = 0,

2 = 0, 2 + 2 7 h + ef g = 0,

7 7

d? d? d?

198 V.I. Lahno, W.I. Fushchych

which is solved by the functions

eC1 eC1

g = e?? C2 sin ??

? + C3 , (19)

f = C1 , ? + C3 , h= 7e C2 cos

2 2

where C1 , C2 , and C3 are arbitrary integration constants.

The case of the L17 algebra. Setting B0 = B3 = B, we obtain the following

reduction of the system of equations (14):

dB1 dB

+ 2B + eB ? B1 = 0,

+2

d? d?

dB dB2

? + eB2 ? B = 0, (20)

2?

d? d?

dB2 dB1

+ 2B2 ? eB1 ? B2 = 0.

2 + 2?

d? d?

In (20), we set B1 = ?1 e1 , B = f (?)e2 + g(?)e3 , and B2 = h(?)e2 + u(?)e3 , where

?1 = 0 is an arbitrary constant. Then the functions f , g, h, and u can be determined

from the system of equations

df dh dg

+ 2g ? e?1 f = 0,

2 + 2f + e?1 g = 0, 2 + 2h + e?1 u = 0, 2

d? d? d?

du df dh dg du

+ 2u ? e?1 h = 0, hg ? uf = 0, 2? ? ?

2 = 0, 2? = 0.

d? d? d? d? d?

The general solution of the ?rst four equations is given by the functions

?1 e ?1 e

f = C1 e?? cos g = C1 e?? sin

? + C2 , ? + C2 ,

2 2

?1 e ?1 e

h = C3 e?? cos u = C3 e?? sin

? + C4 , ? + C4 ,

2 2

where C1 , C2 , C3 , and C4 are arbitrary constants. Having checked the last three

equations of the system, we arrive at the following solution of (20):

B0 = B3 = B = f e2 + ge3 , B2 = he2 + ue3 , B1 = C3 e1 ,

where

eC3 eC3

f = C1 e?? cos ? + C2 , g = C1 e?? sin ? + C2 ,

2 2

(21)

eC3 eC3

h = 2?C1 e?? cos ? + C2 , u = 2?C1 e?? sin ? + C2 ,

2 2

and C1 , C2 , and C3 are arbitrary constants, with C3 = 0.

The case of the L18 algebra. In this case, we set B0 = 1 B2 = B3 = B. Then

2

Eqs. (14) reduce to the equation

dB

+ 2B + eB ? B1 = 0. (22)

d?

Reduction of self-dual Yang–Mills equations 199

In (22), let B = ?e3 , B1 = gm (?)em , m = 1, 2, 3, and ? = 0 be an arbitrary constant.

Then we have the equations

dg1 dg2 dg3

? e?g2 = 0, + e?g1 = 0, + 2? = 0,

d? d? d?

whose general solution is given by the functions

g3 = ?2?? + C3 ,

g1 = C1 sin(e?? + C2 ), g2 = C1 cos(e?? + C2 ),

with C1 , C2 , and C3 being arbitrary integration constants. Thus, we have constructed

the following solution to (22):

1

B0 = B2 = B3 = C4 e3 ,

2 (23)

B1 = C1 sin(eC4 ? + C2 )e1 + C1 cos(eC4 ? + C2 )e2 + (C3 ? 2C4 ?)e3 ,

where C1 , C2 , C3 , and C4 are arbitrary integration constants, with C4 = 0.

Inserting the solutions of the reduced equations found in (17), (19), (21), and

(23) into ansatz (10), we obtain, respectively, the following exact real solutions of

SDYME (1):

A0 = ?0 |bx|?1 [? 1 C3 cos(eC1 log |cx(bx)?1 | + C2 )e1 + C3 e2 ?

(1)

? C3 sin(eC1 log |cx(bx)?1 | + C2 )e3 ],

A1 = |bx|?1 [± 1 e?1 sin(eC1 log |cx(bx)?1 | + C2 )e1 ? C1 e2 ?

? e?1 cos(eC1 log |cx(bx)?1 | + C2 )e3 ],

?1

A2 = 1 C1 (cx) e2 ,

?3 |bx|?1 [? 1 C3 cos(eC1 log |cx(bx)?1 | + C2 )e1 + C3 e2 ?

A3 =

? C3 sin(eC1 log |cx(bx)?1 | + C2 )e3 ],

1

A0 = A3 = (cx)2 e?kx C2 sin eC1 (kx ? 2 log |cx|) + C3 e1 +

(2)

2

1

eC1 (kx ? 2 log |cx|)|C3 e3 ,

ñòð. 44 |