ñòð. 110 |

hand, some invariant solutions (namely those determined by expressions (8b), (8d),

(8i), (8j), (8k), (8r), (8v)) can not be connected with other solutions by the curve

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

which is a conditionally-invariant solution of the form (10). A possible explanation of

this fact is that there exist more general conditionally-invariant solutions of YMEs.

The above picture admits an analogy with a case when equation under study has

general solution. In that case, each two solutions can be connected by a curve which

is a solution of the equation. The only exceptions are the singular solutions which

are obtained by some asymptotic procedure. So one can guess that there exists such

collection of conditionally-invariant solutions of YMEs that the majority of invariant

solutions are their particular cases and the remaining ones are obtained from these by

an asymptotic procedure. However, this problem so far is completely open and needs

further investigation.

One last remark is that the procedure suggested yields also some well-known exact

solutions of YMEs. For example, the ansatz for the Yang–Mills field determined by

expressions (2), (3) and (8v) gives rise to the meron and instanton solutions of the

system (1), originally obtained with the help of the Ansatz suggested by ’t Hooft [21],

Corrigan and Fairlie [5] and Wilczek [22] (for more details, see [16]).

Acknowledgments. One of the authors (RZ) is supported by the Alexander von

Humboldt Foundation.

1. Actor A., Rev. Mod. Phys., 1979, 51, 461.

2. Atiyah M.F., Hitchin N.J., Drinfeld V.G., Manin Yu.A., Phys. Lett. A, 1978, 65, 185.

3. Bluman G., Cole J., J. Math. Phys., 1969, 18, 1025.

4. Clarkson P.A., Kruskal M.D., J. Math. Phys., 1989, 30, 2201.

5. Corrigan E., Fairlie D.B., Phys. Lett. B, 1977, 67, 69.

6. Fushchych W.I., Ukr. Math. J., 1991, 43, 1456.

7. Fushchych W.I., in Proceedings of the International Workshop “Modern Group Analysis”, Editors

N. Ibragimov, M. Torrisi and A. Valenti, Dordrecht, Kluwer Academic Publishers, 1993, 231.

8. Fushchych W.I., Serov N.I., Chopyk V.I., Dopovidi Akad. Nauk Ukrainy, Ser. A, 1988, ¹ 9, 17.

9. Fushchych W.I., Shtelen W.M., Lett. Nuovo Cim., 1983, 38, ¹ 2, 37.

10. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear

equations of mathematical physics, Kiev, Naukova Dumka, 1989.

11. Fushchych W.I., Zhdanov R.Z., Nonlinear spinor equations: symmetry and exact solutions, Kiev,

Naukova Dumka, 1992.

12. Fushchych W.I., Zhdanov R.Z., Ukr. Math. J., 1992, 44, 970.

13. Fushchych W.I., Zhdanov R.Z., Revenko I.V., Ukr. Math. J., 1991, 43, 1471.

14. Kamke E., Differentialgleichungen. L?sungmethoden und L?sungen, Leipzig: Akademische Verlags-

o o

gesellschaft, 1961.

15. Kovalyov M., L?gar? M., Gagnon L., J. Math. Phys., 1993, 34, 3245.

ee

16. Lahno V.I., Zhdanov R.Z., Fushchych W.I., J. Nonlinear Math. Phys., 1995, 2, 51.

17. Levi D., Winternitz P., J. Phys. A: Math. Gen., 1989, 22, 2922.

18. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1993.

19. Olver P., Rosenau Ph., Phys.Lett. A, 1986, 112, 107.

20. Ovsiannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.

21. ’t Hooft G., Phys. Rev. D, 1976, 14, 3432.

22. Wilczek F., in Quark Confinement and Field Theory, New York, Wiley, 1977, 211.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 461–467.

On non-Lie ansatzes and new exact solutions

of the classical Yang–Mills equations

R.Z. ZHDANOV, W.I. FUSHCHYCH

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

nary differential equations. With the use of this method, we construct wide families

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

shows that they correspond to conditional symmetry of the equations under study.

1 Introduction

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

of the SU (2) Yang–Mills equations (YME)

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

(1)

+ e2 A? ? (A? ? Aµ ) = 0

is based on the ansatzes for the Yang–Mills field Aµ (x) suggested by Wu and Yang,

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

And what is more, the above ansatzes were obtained in a non-algorithmic way, i.e.,

there was no regular and systematic method for constructing such ansatzes.

Since one has only a few distinct exact solutions of YME, it is difficult to give

their reliable and self-consistent physical interpretation. That is why, the problem of

prime importance is the development of an effective regular approach for constructing

new exact solutions of the system of nonlinear partial differential equations (PDE) (1)

(see also [1]).

A natural approach to construction of particular solutions of YME (1) is to uti-

lize their symmetry properties in the way as it is done in [2–4, 13]. Apparatus

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

of PDE (1) to systems of nonlinear ordinary differential equations (ODE) by using

special ansatzes (invariant solutions) [5, 6]. If one succeeds in constructing general

or particular solutions of the said ODE (which is extremely difficult problem), then

substituting results into the corresponding ansatzes, one gets exact solutions of the

initial system of PDE (1).

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

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

of mathematical physics, see [7, 8] and also [9]). But the prospects of a systematic

and exhaustive study of conditional symmetry of the system of twelve second-order

nonlinear PDE (1) seem to be rather obscure. It should be said that so far we have

no complete description of conditional symmetry of a nonlinear wave equation even

in the case of one space variable.

In [9] we suggested an effective approach to study of conditional symmetry of

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

J. Nonlinear Math. Phys., 1995, 2, ¹ 2, P. 172–181.

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

Poincar?-invariant ansatzes can be represented in the unified form by introducing

e

several arbitrary elements (functions) u1 (x), u2 (x), . . . , uN (x). As a result, we get an

ansatz for the Dirac field which reduces the nonlinear Dirac equation to a system

of ODE provided functions ui (x) satisfy some compatible over-determined system of

nonlinear PDE. After integrating it, we have obtained a number of new ansatzes that

cannot in principle be obtained within the framework of the classical Lie approach.

In the present paper we construct a number of new exact solutions of YME (1)

with the aid of the above described approach.

2 Reduction of YME

In the papers [2, 13] we adduce a complete list of P (1, 3)-inequivalent ansatzes for the

Yang–Mills field which are invariant under three-parameter subgroups of the Poincar? e

group P (1, 3). Analyzing these ansatzes, we come to the conclusion that they can be

represented in the following unified form:

Aµ (x) = Rµ? (x)B ? (?), (2)

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

variable, functions Rµ? (x) are given by

Rµ? (x) = (aµ a? ? dµ d? ) ch ?0 + (aµ d? ? dµ a? ) ch ?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. Risi-

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

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

?

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

substitution of the ansatz (2) into YME yields a system of ordinary differential equati-

ons for the vector function Bµ (?).

By a direct check, one can become convinced of that the following assertion holds

true.

Lemma. Ansatz (2), (3) reduces YME (1) to a system of ODE if the functions ?(x),

?µ (x) satisfy the system of PDE

1. ?xµ ?xµ = F1 (?),

2? = F2 (?),

2.

(4)

3. R?µ ?x? = Gµ (?),

4. R?µx? = Hµ (?),

On non-Lie ansatzes and new exact solutions of the classical YME 463

?

5. Rµ R??x? ?x? = Qµ? (?),

Rµ 2R?? = Sµ? (?),

?

6.

? ? ?

7. 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, the reduced equation has the form

? ? ?

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

(5)

+ ehµ?? B ? B + e B? ? (B ? Bµ ) = 0,

? ? 2 ?

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

Thus, to describe all ansatzes of the form (2), (3) reducing YME to a system of

ODE, one has to construct the general solution of the over-determined system of PDE

(3), (4). Let us emphasize that system (3), (4) is compatible, since ansatzes invariant

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

e

functions F1 , F2 , . . . , Tµ?? .

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

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

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

general solution of the system of equations 1, 2 from (4) is known [10]. With already

known ?(x), we proceed to integration of the linear PDE 3, 4 from (4). Next, we

substitute the results obtained into the remaining equations and get the final form of

the functions ?(x), ?µ (x).

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

make a remark. As a direct check shows, the structure of the ansatzes (2), (7) 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.

It occurs that new (non-Lie) ansatzes are obtained, if functions ?(x), ?µ (x) up to

the equivalence relations (7) have the form

?µ = ?µ (?, b? x? , c? x? ), µ = 0, 3,

(8)

? = ?(?, b? x? , c? x? ),

where ? = 1 k? x? , k? = a? + d? .

2

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

The list of inequivalent solutions of the system of PDE (3), (4) satisfying (8) is

exhausted by the following solutions:

1

k? x? ,

1. ?0 = ?3 = 0, ?=

2

?1 = w0 (?)bµ x + w1 (?)cµ xµ , ?2 = w2 (?)bµ xµ + w3 (?)cµ xµ ;

µ

2. ? = bµ xµ + w1 (?), ?0 = ? cµ xµ + w2 (?) ,

1

?a = ? wa (?), a = 1, 2, ?3 = 0;

?

4

3. ?0 = T (?), ?3 = w1 (?),

? = bµ xµ cos w1 + cµ xµ sin w1 + w2 (?),

1T ?

(?e + T )(bµ xµ sin w1 ? cµ xµ cos w1 ) + w3 (?) sin w1 +

?1 = (9)

4

1

+ w1 (bµ xµ sin w1 ? cµ xµ cos w1 ) ? w2 cos w1 ,

? ?

4

1 ?

?2 = ? (?eT + T )(bµ xµ sin w1 ? cµ xµ cos w1 ) + w3 (?) cos w1 +

4

1

+ w1 (bµ xµ sin w1 ? cµ xµ cos w1 ) ? w2 sin w1 ;

? ?

4

?1

4. ?0 = 0, ?3 = arctg cµ xµ + w2 (?) bµ xµ + w1 (?) ,

1 2 2 1/2

?a = ? wa (?), a = 1, 2, ? = bµ xµ + w1 (?) + cµ xµ + w2 (?)

? .

4

Here ? = 0, ? are arbitrary constants, w0 , w1 , w2 , w3 are arbitrary smooth functions

of ? = 1 kµ xµ , T = T (?) is a solution of the nonlinear ODE

2

2

? + w1 = ?e2T , ? ? R1 .

T + ?eT ?2 (10)

Substitution of the ansatz (2), where Rµ? (x) are given by formulae (3), (9), into

YME yields systems of nonlinear ODE of the form (5), where

1

kµ? = ? kµ k? , lµ? = ?(w0 + w3 )kµ k? ,

1.

4

mµ? = ?4(w0 + w1 + w2 + w3 )kµ k? ? (w0 + w3 )kµ k? ,

2 2 2 2

? ?

1

qµ?? = (gµ? k? + g?? kµ ? 2gµ? k? ),

2

hµ?? = (w0 + w3 )(gµ? k? ? gµ? k? ) + 2 (w1 ? w2 ) (kµ b? ? k? bµ )c? +

+ (bµ c? ? b? cµ )k? + (cµ k? ? c? kµ )b? ;

kµ? = ?gµ? ? bµ b? , lµ? = 0, mµ? = ??2 (aµ a? ? dµ d? ),

2.

(11)

ñòð. 110 |