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

3. The group of scale transformations (generator X = ? D)

Ad = Ad e?? .

xµ = xµ e? , µ µ

4. The group of conformal transformations (generator X = ?µ K µ )

xµ = (xµ ? ?µ x? x? )? ?1 (x),

Ad = [gµ? ?(x) + 2(xµ ?? ? x? ?µ + 2?? x? ?µ x? ? x? x? ?µ ?? ? ?? ? ? xµ x? ]Ad? .

µ

5. The group of gauge transformations (generator X = Q)

xµ = xµ ,

w

Ad = Ad cos w + ?dbc Ab nc sin w + 2nd nb Ab sin2 +

µ µ µ µ

2

1d 1

+ e?1 n ?µ w + (?µ nd ) sin w + ?dbc (?µ nb )nc .

2 2

In the above formulae ?(x) = 1 ? ?? x? + (?? ? ? )(x? x? ), na = na (x) is a unit

vector determined by the equality wa (x) = w(x)na (x), a = 1, 3.

Using the Lemma it is not difficult to obtain formulae for generating solutions of

YME by the above transformation groups. We adduce them omitting derivation (see

also [3]).

1. The group of translations

Aa (x) = ua (x + ? ).

µ µ

2. The Lorentz group

Ad (x) = aµ ud (ax, bx, cx, dx) + bµ ud (ax, bx, cx, dx) +

µ 0 1

+ cµ ud (ax, bx, cx, dx) + dµ ud (ax, bx, cx, dx).

2 3

3. The group of scale transformations

Ad (x) = e? ud (xe? ).

µ µ

4. The group of conformal transformations

Ad (x) = [gµ? ? ?1 (x) + 2? ?2 (x)(xµ ?? ? x? ?µ + 2?? x? ?µ x? ?

µ

? x? x? ?µ ?? ? ?? ? ? xµ x? )]ud? ((x ? ? (x? x? ))? ?1 (x)).

5. The group of gauge transformations

w

Ad (x) = ud cos w + ?dbc ub nc sin w + 2nd nb ub sin2 +

µ µ µ µ

2

1d 1

+ e?1 n ?µ w + (?µ nd ) sin w + ?dbc (?µ nb )nc .

2 2

Here ud (x) is an arbitrary given solution of YME; Ad (x) is a new solution of YME;

µ µ

? , ?µ are arbitrary parameters; aµ , bµ , cµ , dµ are arbitrary parameters satisfying the

equalities

aµ aµ = ?bµ bµ = ?cµ cµ = ?dµ dµ = 1,

aµ bµ = aµ cµ = aµ dµ = bµ cµ = bµ dµ = cµ dµ = 0.

Besides that, we use the following designations: x + ? = {xµ + ?µ , µ = 0, 3},

ax = aµ xµ .

Thus, each particular solution of YME gives rise to a multi-parameter family of

exact solutions by virtue of the above solution generation formulae.

Symmetry reduction and exact solutions of the Yang–Mills equations 435

3 Ansatzes for the Yang–Mills field

A key idea of the symmetry approach to the problem of reduction of PDE is a special

choice of the form of a solution. This choice is dictated by a structure of the symmetry

group admitted by the equation under study.

In the case involved, to reduce YME by N variables one has to construct ansatzes

for the Yang–Mills field Aa (x) invariant under N -dimensional subalgebras of the

µ

algebra with the basis elements (2.1), (2.2) [1, 5]. Since we are looking for Poi-

ncar?-invariant ansatzes reducing YME to systems of ODE, N is equal to 3. Due to

e

invariance of YME under the Poincar? group P (1, 3), it is enough to consider only

e

subalgebras which can not be transformed one into another by group transformati-

on, i.e. P (1, 3)-inequivalent subalgebras. Complete description of P (1, 3)-inequivalent

subalgebras of the Poincar? algebra was obtained in [6] (see also [7]).

e

According to the classical symmetry approach, to construct the ansatz invariant

under the invariance algebra having the basis elements

b

(3.1)

Xa = ?aµ (x, A)?µ + ?aµ (x, A)?Ab , a = 1, 3,

µ

where A = {Aa , a = 1, 3, µ = 0, 3}, one has

µ

1) to construct a complete system of functionally-independent invariants of the

operators (3.1) ? = {wi (x, A), i = 1, 13};

2) to resolve relations

(3.2)

Fj (w1 (x, A), . . . , w13 (x, A)) = 0, j = 1, 13

with respect to the function Aa .

µ

As a result, one gets the ansatz for the field Aa (x) which reduces YME to the

µ

system of twelve nonlinear ODE.

Note. Equalities (3.2) can be resolved with respect to Aa , a = 1, 3, µ = 0, 3 if the

µ

condition

3 3

(3.3)

rank ?aµ (x, A) =3

a=1 µ=0

holds. If (3.3) does not hold, the above procedure leads to partially-invariant solu-

tions [5], which are not considered in the present paper.

In [1, 4] we established that the procedure of construction of invariant ansatzes

could be essentially simplified if coefficients of operators Xa have the following

structure:

?aµ = ?bc (x)Ac

b

(3.4)

?aµ = ?aµ (x), aµ? ?

(i.e. basis elements of the invariance algebra realize the linear representation). In this

case, the invariant ansatz for the field Aa (x) is searched for in the form

µ

Aa (x) = Qab (x)B? (w(x)).

b

(3.5)

µ µ?

Here B? (w) are arbitrary smooth functions and w(x), Qab (x) are particular solutions

b

µ?

of the system of PDE

?aµ wxµ = 0, a = 1, 3,

(3.6)

(?a? ?? ? ?bc )Qcd = 0, µ = 0, 3, a, b, d = 1, 3.

aµ? ??

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

Basis elements of the Poincar? algebra Pµ , J?? from (2.1) evidently satisfy the

e

conditions (3.4) and besides the equalities

?aµ = ?aµ? (x)Ab ,

b

(3.7)

a, b = 1, 3, µ = 0, 3

?

hold.

This fact permits further simplification of formulae (3.5), (3.6). Namely, the ansatz

for the Yang–Mills field invariant under the 3-dimensional subalgebra of the Poincar? e

algebra with basis elements of the form (3.1), (3.7) should be looked for in the form

Aa = Qµ? (x)B? (w(x)),

a

(3.8)

µ

a

where B? (w) are arbitrary smooth functions and w(x), Qµ? (x) are particular solutions

of the following system of PDE:

(3.9)

?aµ wxµ = 0, a = 1, 3,

?a? ?? Qµ? ? ?aµ? Q?? = 0, (3.10)

a = 1, 3, µ, ? = 0, 3.

Thus, to obtain the complete description of P (1, 3)-inequivalent ansatzes for the

field Aa (x) invariant under 3-dimensional subalgebras of the Poincar? algebra, one

e

µ

has to integrate the over-determined system of PDE (3.9), (3.10) for each P (1, 3)-

inequivalent subalgebra. Let us note that compatibility of (3.9), (3.10) is guaranteed

by the fact that operators X1 , X2 , X3 form a Lie algebra.

Consider, as an example, the procedure of constructing ansatz (3.8) invariant under

the subalgebra P1 , P2 , J03 . In this case system (3.9) reads

wx1 = 0, wx2 = 0, x0 wx3 + x3 wx0 = 0,

whence w = x2 ? x2 .

0 3

Next, we note that coefficients ?1µ? , ?2µ? of the operators P1 , P2 are equal to

zero, while coefficients ?3µ? form the following (4 ? 4) matrix

0 0 0 1

0 0 0 0

3

?3µ? =

µ,?=0

0 0 0 0

1 0 0 0

(we designate this constant matrix by the symbol S).

With account of the above fact, equations (3.10) take the form

x0 Qx3 + x3 Qx0 ? SQ = 0, (3.11)

Qx1 = 0, Qx2 = 0,

3

where Q = Qµ? (x) µ,?=0 is a (4 ? 4)-matrix.

From the first two equations of system (3.11) it follows that Q = Q(x0 , x3 ). Since

S is a constant matrix, a solution of the third equation can be looked for in the form

(see, for example, [4])

Q = exp {f (x0 , x3 )S}.

Substituting this expression into (3.11) we get

(x0 fx3 + x3 fx0 ? 1) exp {f S} = 0

Symmetry reduction and exact solutions of the Yang–Mills equations 437

or, equivalently,

x0 fx3 + x3 fx0 = 1,

whence f = ln(x0 + x3 ).

Consequently, a particular solution of equations (3.11) reads

Q = exp {ln(x0 + x3 )S}.

Using an evident identity S = S 3 we get the following equalities:

?

(n!)?1 (ln(x0 + x3 ))n S n =

Q=

n=0

= I + S[ln(x0 + x3 ) + (3!)?1 (ln(x0 + x3 ))3 + · · ·] +

+ S 2 [(2!)?1 (ln(x0 + x3 ))2 + (4!)?1 (ln(x0 + x3 ))4 + · · ·] =

= I + S sinh(ln(x0 + x3 )) + S 2 (cosh(ln(x0 + x3 )) ? 1),

where I is a unit (4 ? 4)-matrix.

Substitution of the obtained expressions for functions w(x), Qµ? (x) into (3.8) yi-

elds the ansatz for the Yang–Mills field Aa (x) invariant under the algebra P1 , P2 , J03

µ

Aa = B0 (x2 ? x2 ) cosh ln(x0 + x3 ) + B3 (x2 ? x2 ) sinh ln(x0 + x3 ),

a a

0 0 3 0 3

A1 = B1 (x0 ? x3 ), A2 = B2 (x0 ? x3 ),

a a2 2 a a2 2

(3.12)

Aa = B3 (x2 ? x2 ) cosh ln(x0 + x3 ) + B0 (x2 ? x2 ) sinh ln(x0 + x3 ).

a a

3 0 3 0 3

a

Substituting (3.12) into YME we get a system of ODE for functions Bµ . If we will

succeed in constructing its general or particular solutions, then substituting it into

formulae (3.12) we get an exact solution of YME. But such a solution will have an

unpleasant feature: independent variables xµ will be included into it in asymmetrical

way. At the same time, in the initial equation (1.1) all independent variables are on

equal rights. To remove this defect one has to apply solution generation procedure

by transformations from the Lorentz group. As a result, we will obtain an ansatz for

the Yang–Mills field in the manifestly-covariant form with symmetrical dependence

on xµ .

In the same way, we construct the rest of ansatzes invariant under three-dimen-

sional subalgebras of the Poincar? algebra. They are represented in the unified form

e

Aa (x) = {(aµ a? ? dµ d? ) cosh ?0 + (dµ a? ? d? aµ ) sinh ?0 +

µ

+ 2(aµ + dµ )[(?1 cos ?3 + ?2 sin ?3 )b? + (?2 cos ?3 ? ?1 sin ?3 )c? +

(3.13)

+ (?1 + ?2 )e??0 (a? + d? )] + (bµ c? ? b? cµ ) sin ?3 ?

2 2

? (cµ c? + bµ b? ) cos ?3 ? 2e??0 (?1 bµ + ?2 cµ )(a? + d? )}B a? (w).

Here ?µ , µ = 0, 3, w are some functions whose explicit form is determined by the

choice of a subalgebra of the Poincar? algebra AP (1, 3).

e

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

Below, we adduce a complete list of 3-dimensional P (1, 3)-inequivalent subalgebras

of the Poincar? algebra following [7]

e

L1 = P0 , P1 , P2 ; L2 = P1 , P2 , P3 ;

L3 = P0 + P3 , P1 , P2 ; L4 = J03 + ?J12 , P1 , P2 ;

L5 = J03 , P0 + P3 , P1 ; L6 = J03 + P1 , P0 , P3 ;

L7 = J03 + P1 , P0 + P3 , P2 ; L8 = J12 + ?J03 , P0 , P3 ;

L9 = J12 + P0 , P1 , P2 ; L10 = J12 + P3 , P1 , P2 ;

L11 = J12 + P0 ? P3 , P1 , P2 ; L12 = G1 , P0 + P3 , P2 + ?P1 ;

L14 = G1 + P0 ? P3 , P0 + P3 , P2 ;

L13 = G1 + P2 , P0 + P3 , P1 ;

(3.14)

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