ñòð. 103 |

ln(x0 + x1 )?

P0 + + K1 ? P1 ?

K0

6? 1/2

? ?arctg(x1 ? x0 )

(1)

? ?(J01 + D ), ? > 0 ? x0 + x1 ?(?)

? ?3

3? ?(2) = ?;

16

? ?3

4? ?(4) ? ?(2) = ?;

16

? ?3

5? ?(4) + ?(2) = ?;

16

6? 4? 2 ?(4) + (4 ? ? 2 )?(2) ? ? = ? ??3 .

4

The general solution of equation 3? is of the form

(c1 ? + c2 )2 ?

?=± + , c1 = 0;

c1 16c1

1v

?=± ??? + c,

2

where c, c1 , c2 are arbitrary constants.

Hence we obtain the following exact solutions of equation (36):

1/4

1 ? 1/2 1/2

u = ±v (x0 + x1 + a2 )2 ? a1 x0 ? x1 + a3

1. ,

2 a1

1/4

1 ? 1/2 1/2

u = ±v (x0 ? x1 + b2 )2 ? b1

2. x0 + x1 + b3 ,

2 b1

1/4

1 ? 1/2 1/2

u=± (x0 ? x1 + c3 )2 + c1 (x0 + x1 + c4 )2 + c2

3. ,

2 c1 c2

where ai , bi , cj , i = 1, 3, j = 1, 4 are arbitrary constants.

430 W.I. Fushchych, R.Z. Zhdanov, O.V. Roman

Besides, the expression

1/2

u = ±?1/4 (x0 ? x1 + c1 )(x0 + x1 + c2 )

(c1 , c2 are arbitrary constants) was proved in Section 4 to be the exact solution of

equation (36).

In conclusion let us note that we can obtain the same solutions using the following

ansatz

?2 = x0 ? x1 ,

u = ?1 (?1 )?2 (?2 ), ?1 = x0 + x1 ,

which reduces equation (36) to the system of ordinary differential equations for the

unknown functions ?1 (?1 ) and ?2 (?2 ), namely

c ?3

?1 =

? ?,

41

(38)

?

?2 = ??3 ,

?

4c 2

where c is an arbitrary constant.

Acknowledgement. The main part of this work for the authors was made by the

financial support by Soros Grant, Grant of the Ukrainian Foundation for Fundamental

Research and the Swedish Institute.

1. Fushchych W.I., Symmetry in problems of mathematical physics, in Algebraic-Theoretical Studies in

Mathematical Physics, Kiev, Inst. of Math., 1981, 6.

2. Bollini C.G., Giambia J.J., Arbitrary powers of d’Alembertians and the Huygens’ principle, J. Math.

Phys., 1993, 34, ¹ 2, 610–621.

3. Fushchych W.I., Selehman M.A., Integro-differential equations invariant under group of Galilei, Poi-

ncare, Schr?dinger and conformal group, Proceedings of the Academy of Sciences of Ukrainy, 1983,

o

¹ 5, 21–24.

4. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of nonlinear

mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993, 436 p.

5. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of the nonlinear many-

dimensional Liouville, d’Alembert and eikonal equations, J. Phys. A: Math. Gen., 1983, ¹ 16, 3645–

3658.

6. Ibragimov N.H., Lie groups in some questions of mathematical physics, Novosibirsk, Novosibirsk

University, 1972, 200 p.

7. Winternitz P., Grundland A.M., Tuszynski J.A., Exact solutions of the multidimensional classical ?6

field equations obtained by symmetry reduction, J. Math. Phys., 1987, 28, ¹ 9, 2194–2212.

8. Fedorchuk V.M., On reduction and some exact solutions of nonlinear wave equation, in Symmetry and

Solutions of Nonlinear Equations of Mathematical Physics, Kiev, Inst. of Math., 1987, 73–76.

9. Fushchych W.I., Barannik L.F., Barannik A.F., Subgroup analysis of Galilei and Poincar? groups and

e

reduction of nonlinear equations, Kiev, Naukova Dumka, 1991, 304 p.

10. Ovsiannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978, 400p.

11. Serov N.I., Conformal symmetry of nonlinear wave equations, in Algebraic-Theoretical Studies in

Mathematical Physics, Kiev, Inst. of Math., 1981, 59.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 431–450.

Symmetry reduction and exact solutions

of the Yang–Mills equations

V.I. LAHNO, R.Z. ZHDANOV, W.I. FUSHCHYCH

We present a detailed account of symmetry properties of SU (2) Yang–Mills equations.

Using a subgroup structure of the Poincar? group P (1, 3) we have constructed all

e

P (1, 3)-inequivalent ansatzes for the Yang-Mills field which are invariant under the

three-dimensional subgroups of the Poincar? group. With the aid of these ansatzes

e

reduction of Yang-Mills equations to systems of ordinary differential equations is carried

out and wide families of their exact solutions are constructed.

1 Introduction

Since Newton’s and Euler’s works, exact solutions of differential equations describing

physical processes were highly estimated. Green, Lame, Liouville, Cayley, Donkin,

Stokes, Kirchhoff, Poincar?, Stieltjes, Forsyth, Volterra, Appel, Macdonald, Weber,

e

Bateman, Whittaker, Sommerfeld and many other famous researchers constructed

different classes of exact solutions of linear Laplace, d’Alembert, heat, and Maxwell

equations.

Nowadays, this constructive branch of mathematical physics is not so popular as

earlier. But if one wants to have some nontrivial information on solutions of basic

motion equations in quantum mechanics, field theory, gravitation theory, acoustics,

and hydrodynamics, then the more intensive research work should be carried out in

order to develop analytical methods of solution of partial differential equations (PDE).

And what is more, unlike the mathematical physics of the 19th century, modern

mathematical physics is essentially nonlinear. It means that all principal equations

of modern physics, biology and chemistry are nonlinear. This fact complicates very

much the problem of constructing their exact solutions (see, e.g. [1] and references

therein).

Up to now, we have comparatively few papers devoted to construction of exact so-

lutions of nonlinear multi-dimensional d’Alembert, Maxwell, Schr?dinger, Dirac, Max-

o

well–Dirac, Yang–Mills equations. Whereas, a huge amount of papers and monographs

are devoted to construction of exact solutions of equations for gravitational field. It

is difficult even to estimate the number of papers and monographs, where the soliton

solutions of the one-dimensional nonlinear KdV, Schr?dinger and Sine-Gordon equa-

o

tions are studied. We are sure that the above mentioned equations should deserve

much more attention of researchers in mathematical physics.

With the present paper we start a series of papers devoted to construction of new

classes of exact solutions of the classical Yang–Mills equations (YME) with the use of

their Lie and non-Lie symmetry. Here we study in detail symmetry reduction of YME

by Poincar?-invariant ansatzes and obtain wide families of its exact Poincar?-invariant

e e

solutions.

J. Nonlinear Math. Phys., 1995, 2, ¹ 1, P. 51–72.

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

By the classical YME, we mean the following nonlinear system of twelve second-

order PDE:

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

(1.1)

+ e2 A? ? (A? ? Aµ ) = 0.

?

Here ?? = ?x? , µ, ? = 0, 3, e = const, Aµ = Aµ (x0 , x1 , x2 , x3 ) is the three-component

vector-potential of the Yang–Mills field (called, for bravity, the Yang–Mills field).

Hereafter, the summation over the repeated indices µ, ? from 0 to 3 is understood.

Raising and lowering the vector indices is performed with the aid of the metric tensor

?

? 1, µ = ? = 0,

?1, µ = ? = 1, 2, 3,

gµ? =

?

0, µ = ?

(i.e. ? µ = gµ? ?? ).

It should be said that there were several reviews devoted to classical solutions

of YME (see [2] and the literature cited there). But, in fact, symmetry properties of

YME were not used. The solutions were obtained with the help of ad hoc substitutions

suggested by Wu and Yang, Rosen, ’t Hooft, Corrigan and Fairlie, Wilczek, Witten

(for more detail, see [2]).

The structure of our paper is as follows. In the second Section we give all

necessary information about symmetry properties of YME and about a solution

generation procedure by virtue of the finite transformations of the symmetry group

admitted by YME. In Section 3 we construct P (1, 3)-inequivalent ansatzes for the

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

group. Section 4 is devoted to reduction of YME to systems of ordinary differential

equations (ODE). Integrating these in Section 5 we construct multi-parameter fami-

lies of exact solutions of YME. In Section 6 we consider some generalizations of

the solutions obtained and, in particular, construct the generalization of Coleman’s

solution.

2 Symmetry and solution generation

for the Yang–Mills equations

It was known long ago that YME are invariant with respect to the group C(1, 3) ?

SU (2), where C(1, 3) is the 15-parameter conformal group having the following

generators:

P µ = ?µ ,

J?? = x? ?? ? x? ?? + Aa? ?Aa ? Aa? ?Aa ,

?

?

(2.1)

D = xµ ?µ ? Aa ?Aa ,

µ µ

Kµ = 2xµ D ? (x? x? )?µ + 2Aaµ x? ?Aa ? 2Aa x? ?Aa ,

?

? µ

and SU (2) is the infinite-parameter special unitary group with the following basis

generator:

Q = (?abc Ab wc (x) + e?1 ?µ wa (x))?Aa . (2.2)

µ µ

Symmetry reduction and exact solutions of the Yang–Mills equations 433

?

In (2.1), (2.2) ?Aa = ?Aa , wc (x) are arbitrary smooth functions, ?abc is the third-

µ µ

order anti-symmetrical tensor with ?123 = 1. Hereafter, summation over the repeated

indices a, b, c from 1 to 3 is understood.

But the fact that the group with generators (2.1), (2.2) is a maximal (in Lie’s

sense) invariance group admitted by YME was established only recently [3] with the

use of a symbolic computation technique. The only explanation for this situation is a

very cumbersome structure of the system of PDE (1.1). As a consequence, realization

of the Lie algorithm of finding the maximal invariance group admitted by YME

demands a huge amount of computations. This difficulty had been overcome with the

aid of computer facilities.

One of the remarkable possibilities provided by the fact that the considered equati-

on admits a nontrivial symmetry group gives the possibility of getting new solutions

from the known ones by the solution generation technique [1, 4]. This technique is

based on the following assertion.

Lemma. Let

µ = 0, n ? 1,

xµ = fµ (x, u, ? ),

ua = ga (x, u, ? ), a = 1, N ,

where ? = (?1 , ?2 , . . . , ?r ) be the r-parameter invariance group of some system of

PDE and Ua (x), a = 1, N be its particular solution. Then the N -component function

ua (x) determined by implicit formulae

(2.3)

Ua (f (x, u, ? )) = ga (x, u, ? ), a = 1, N

is also a solution of the same system of PDE.

To make use of the above assertion we need formulae for finite transformations

generated by infinitesimal operators (2.1), (2.2). We adduce these formulae following

[1, 2].

1. The group of translations (generator X = ?µ Pµ )

xµ = xµ + ?µ , Ad = Ad .

µ µ

2. The Lorentz group O(1, 3)

a) the group of rotations (generator X = ? Jab )

x0 = 0, xc = xc , c = a, c = b,

xa = xa cos ? + xb sin ?,

xb = xb cos ? ? xa sin ?,

Ad = Ad , Ad = Ad , c = a, c = b,

c c

0 0

d d d

Aa = Aa cos ? + Ab sin ?,

Ad = Ad cos ? ? Ad sin ? ;

a

b b

b) the group of Lorentz transformations (generator X = ? J0a )

x0 = x0 cosh ? + xa sinh ?,

xa = xa cosh ? + x0 sinh ?, xb = xb , b = a,

Ad = Ad cosh ? + Ad sinh ?,

a

0 0

Aa = Aa cosh ? + Ad sinh ?, Ad = Ad , b = a.

d d

ñòð. 103 |