ñòð. 108 |

R.Z. ZHDANOV, W.I. FUSHCHYCH

We suggest an effective method for reducing the Yang–Mills equations to systems of

ordinary differential equations. With the use of this method we construct the extensive

families of new exact solutions of the Yang–Mills equations. Analysis of the solutions

thus obtained shows that they correspond to the conditional (non-classical) symmetry

of the equations under study.

1 Introduction

A majority of papers devoted to construction of explicit form of the exact solutions of

SU (2) Yang–Mills equations (YMEs)

?? ? ? Aµ ? ? µ ?? A? + e (?? A? ) ? Aµ ? 2(?? Aµ ) ? A? +

(1)

+(? µ A? ) ? A? + e2 A? ? (A? ? Aµ ) = 0

are based on the ans?tze for the Yang–Mills field Aµ (x) suggested by Wu and

a

Yang, Rosen, ’t Hooft, Corrigan and Fairlie, Wilczek, Witten (see [1] and references

therein). There were further developments for the self-dual YMEs (which form the

first-order system of nonlinear partial differential equations such that system (1) is its

differential consequence). Let us mention the Atiyah–Hitchin–Drinfeld–Manin method

for obtaining instanton solutions [2] and its generalization due to Nahm. However,

the solution set of the self-dual YMEs is only a subset of solutions of YMEs (1)

and the problem of construction of new non self-dual solutions of system (1) is, in

fact, completely open (see, also [1]). As the development of new approaches to the

construction of exact solutions of YMEs is a very interesting mathematical problem,

it may also be of importance for physics. The reason is that all famous mathematical

models of elementary particles such as solitons, instantons, merons are quite simply

particular solutions of some nonlinear partial differential equations.

A natural approach to construction of particular solutions of YMEs (1) is to utilize

their symmetry properties in the way as it is done in [9, 10, 16] (see, also [15],

where the reduction of the Euclidean self-dual YMEs is considered). The apparatus

of the theory of Lie transformation groups makes it possible to reduce system of

partial differential equations (PDEs) (1) to systems of nonlinear ordinary differential

equations (ODEs) by using special ans?tze (invariant solutions) [10, 18, 20]. If one

a

succeeds in constructing general or particular solutions of the said ODEs (which is

an extremely difficult problem), then on substituting the results in the corresponding

ans?tze one gets exact solutions of the initial system of PDEs (1).

a

Another possibility of construction of exact solutions of YMEs is to use their condi-

tional (non-Lie) symmetry (for more details about conditional symmetry of equations

of mathematical physics, see [6, 8] and also [10, 12]) which has much in common with

J. Phys. A: Math. Gen., 1995, 28, P. 6253–6263.

452 R.Z. Zhdanov, W.I. Fushchych

a “non-classical symmetry” of PDEs by Bluman and Cole [3] (see also [17, 19]) and

“direct method of reduction of PDEs” by Clarkson and Kruskal [4]. But the prospects

of a systematic and exhaustive study of conditional symmetry of system of twelve

second-order nonlinear PDEs (1) seem to be rather remote. It should be said that so

far there is no complete description of conditional symmetry of the nonlinear wave

equation even in the case of one space variable.

A principal idea of the method of ans?tze, as well as of the direct method of

a

reduction of PDEs, is a special choice of the class of functions to which a possible

solution should belong. Within the framework of the above methods, a solution of

system (1) is sought in the form

Aµ = Hµ x, B ? (?(x)) , µ = 0, 3,

where Hµ are smooth functions chosen in such a way that substitution of the above

expressions into the Yang–Mills equations results in a system of ODEs for “new”

unknown vector-functions B ? of one variable ?. However, the problem of reduction

of YMEs posed in this way seemed to be hopeless. Really, if we restrict ourselves to

the case of a linear dependence of the above ansatz on B ?

Aµ (x) = Rµ? (x)B ? (?), (2)

where B ? (?) are new unknown vector-functions, ? = ?(x) is a new independent

variable, then a requirement of reduction of (1) to a system of ODEs by virtue of (2)

gives rise to a system of nonlinear PDEs for 17 unknown functions Rµ? , ?. What is

more, the system obtained is no way simpler than the initial Yang–Mills equations (1).

It means that some additional information about the structure of the matrix function

Rµ? should be input into the ansatz (2). This can be done in various ways. But the

most natural one is to use the information about the structure of solutions provided

by the Lie symmetry of the equation under study.

In [11] we suggest an effective approach to the study of conditional symmetry of

the nonlinear Dirac equation based on its Lie symmetry. We have observed that all

Poincar?-invariant ans?tze for the Dirac field ?(x) can be represented in the unified

e a

form by introducing several arbitrary elements (functions) u1 (x), u2 (x), . . . , uN (x).

As a result, we get an ansatz for the field ?(x) which reduces the nonlinear Dirac

equation to system of ODEs provided functions ui (x) satisfy some compatible over-

determined system of nonlinear PDEs. After integrating it, we have obtained a number

of new ans?tze that cannot in principle be obtained within the framework of the

a

classical Lie approach.

In the present paper we will demonstrate that the same idea proves to be fruitful

for obtaining new (non-Lie) reductions of YMEs and for constructing new exact

solutions of system (1).

2 Reduction of YMEs

In the paper [16] we have obtained a complete list of P (1,3)-inequivalent ans?tze

a

for the Yang–Mills field which are invariant under the three-parameter subgroups of

the Poincar? group P (1,3). Analyzing these ans?tze we come to conclusion that they

e a

can be represented in the unified form (2), where B ? (?) are new unknown vector

Conditional symmetry and new classical solutions of the Yang–Mills equations 453

functions, ? = ?(x) is a new independent variable and functions Rµ? (x) are given by

the formulae

Rµ? (x) = (aµ a? ? dµ d? ) cosh ?0 + (aµ d? ? dµ a? ) sinh ?0 + 2(aµ + dµ ) ?

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

(3)

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

2 2

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

In (3) ?µ (x) are some smooth functions and what is more ?a = ?a (?, bµ xµ , cµ xµ ),

a = 1, 2; ? = 1 kµ xµ = 1 (aµ xµ + dµ xµ ); aµ , bµ , cµ , dµ are arbitrary constants

2 2

satisfying the following relations:

aµ aµ = ?bµ bµ = ?cµ cµ = ?dµ dµ = 1,

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

Hereafter, summation over the repeated indices from 0 to 3 is understood. Rai-

sing and lowering of the indices is performed with the help of the tensor gµ? =

diag(1, ?1, ?1, ?1), e.g. Rµ = g?? R?µ .

?

A choice of the functions ?(x), ?µ (x) is determined by the requirement that

substitution of the ansatz (2) in the YMEs yields a system of ODEs for the vector

function B µ (?).

By the direct check one can convince one self that the following assertion holds

true.

Lemma. Ansatz (2), (3) reduces YMEs (1) to system of ODEs iff the functions ?(x),

?µ (x) satisfy the following system of PDEs:

(4a)

?xµ ?xµ = F1 (?),

2? = F2 (?), (4b)

(4c)

R?µ ?x? = Gµ (?),

(4d)

R?µx? = Hµ (?),

?

(4e)

Rµ R??x? ?x? = Qµ? (?),

Rµ 2R?? = Sµ? (?),

?

(4f)

? ? ?

(4g)

Rµ R??x? R?? + R? R??x? R?µ + R? R?µx? R?? = Tµ?? (?),

where F1 , F2 , Gµ , . . ., Tµ?? are some smooth functions, µ, ?, ? = 0, 3. And what is

more, a reduced equation has the form

? ? ?

kµ? B ? + lµ? B ? + mµ? B ? + eqµ?? B ? ? B ? + ehµ?? B ? ? B ? +

(5)

+ e2 B ? ? (B ? ? B µ ) = 0,

where

kµ? = gµ? F1 ? Gµ G? ,

?

lµ? = gµ? F2 + 2Qµ? ? Gµ H? ? Gµ G? ,

?

mµ? = Sµ? ? Gµ H? , (6)

qµ?? = gµ? G? + g?? Gµ ? 2gµ? G? ,

1

hµ?? = (gµ? H? ? gµ? H? ) ? Tµ?? .

2

454 R.Z. Zhdanov, W.I. Fushchych

Thus, to describe all ans?tze of the form (2), (3) reducing the YMEs to a system

a

of ODEs one has to construct the general solution of the over-determined system of

PDEs (3), (4). Let us emphasize that system (3), (4) is compatible, since the ans?tze

a

for the Yang–Mills field Aµ (x) invariant under the three-parameter subgroups of the

Poincar? group satisfy equations (3), (4) with some specific choice of the functions

e

F1 , F2 , . . . , Tµ?? [16].

Integration of system of nonlinear PDEs (3), (4) demands a huge amount of

computations. That is why we present here only the principal idea of our approach to

solving the system (3), (4). When integrating it we use essentially the fact that the

general solution of system of equations (4a), (4b) is known [13]. With ?(x) already

known we proceed to integration of linear PDEs (4c), (4d). Next, we substitute the

results obtained in the remaining equations (4) and get the final form of the functions

?(x), ?µ (x).

Before presenting the results of integration of system of PDEs (3), (4) we make

a remark. As the direct check shows, the structure of the ansatz (2), (3) is not altered

by the change of variables

? > ? = T (?), ?0 > ?0 = ?0 + T0 (?),

?1 > ?1 = ?1 + e?0 T1 (?) cos ?3 + T2 (?) sin ?3 ,

(7)

?2 > ?2 = ?2 + e?0 T2 (?) cos ?3 ? T1 (?) sin ?3 ,

?3 > ?3 = ?3 + T3 (?),

where T (?), Tµ (?) are arbitrary smooth functions. That is why, solutions of system

(3), (4) connected by the relations (7) are considered as equivalent.

Integrating the system of PDEs within the above equivalence relations we obtain

the set of ans?tze containing the ones equivalent to the Poincar?-invariant ans?tze.

a e a

We list below the corresponding expressions for the functions ?µ , ?:

? = d · x; (8a)

?µ = 0,

? = a · x; (8b)

?µ = 0,

? = k · x; (8c)

?µ = 0,

?0 = ? ln |k · x|, ?1 = ?2 = 0, ?3 = ? ln |k · x|,

(8d)

? = (a · x)2 ? (d · x)2 ;

?0 = ? ln |k · x|, ? = c · x; (8e)

?1 = ?2 = ?3 = 0,

?0 = ?b · x, ? = c · x; (8f)

?1 = ?2 = ?3 = 0,

?0 = ?b · x, ? = b · x ? ln |k · x|; (8g)

?1 = ?2 = ?3 = 0,

?0 = ? arctan(b · x/c · x), ?1 = ?2 = 0,

(8h)

?3 = ? arctan(b · x/c · x), ? = (b · x)2 + (c · x)2 ;

?3 = ?a · x, ? = d · x; (8i)

?0 = ?1 = ?2 = 0,

?3 = d · x, ? = a · x; (8j)

?0 = ?1 = ?2 = 0,

Conditional symmetry and new classical solutions of the Yang–Mills equations 455

1

?3 = ? k · x, ? = a · x ? d · x; (8k)

?0 = ?1 = ?2 = 0,

2

1

(b · x ? ?c · x)(k · x)?1 , ? = k · x; (8l)

?0 = 0, ?1 = ?2 = ?3 = 0,

2

1

c · x, ? = k · x; (8m)

?0 = ?2 = ?3 = 0, ?1 =

2

1

?1 = ? k · x, ? = 4b · x + (k · x)2 ; (8n)

?0 = ?2 = ?3 = 0,

4

1

?1 = ? k · x, ? = 4(?b · x ? c · x) + ?(k · x)2 ; (8o)

?0 = ?2 = ?3 = 0,

4

?0 = ? ln |k · x|, ?1 = ?2 = 0,

(8p)

?3 = ? arctan(b · x/c · x), ? = (b · x)2 + (c · x)2 ;

1

(c · x + (? + k · x)b · x)(1 + k · x(? + k · x))?1 ,

?0 = ?3 = 0, ?1 =

2 (8q)

1

?2 = ? (b · x ? c · xk · x)(1 + k · x(? + k · x))?1 , ? = k · x;

2

1

b · x(k · x)?1 ,

?0 = ? ln |k · x|, ?1 = ?2 = ?3 = 0,

(8r)

2

? = (a · x) ? (b · x) ? (d · x)2 ;

2 2

1

b · x(k · x)?1 ,

?0 = ? ln |k · x|, ? = c · x; (8s)

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