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solutions of the Yang–Mills equations
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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