стр. 112 |

iff F (?, ?) is of the form (5), (7) with k = 3/2. In formula (8) D is the operator (6)

with k = 3/2, xµ = gµ? x? , ? µ = gµ? ?? , µ, ? = 0, . . . , 3.

Proof of the Theorems 1–3 is carried out with the help of the infinitesimal Lie

algorithm [3, 4].

Thus, there exists rather narrow class of Poincar? invariant equations of the

e

form (4)

?? ?? (9)

[i?µ ?xµ + F1 (??, ??4 ?) + F2 (??, ??4 ?)?4 ]? = 0.

To construct exact solutions of the nonlinear Dirac equation (9) we apply the

symmetry reduction technique.

The general idea of symmetry reduction of PDEs can be formulated in a very

simple and natural way. Since coefficients of Eq. (9) do not depend explicitly on the

variable x0 , we can look for a particular solution which is also independent of x0

(10)

? = ?(x1 , x2 , x3 ).

470 R.Z. Zhdanov and W.I. Fushchych

After substituting (10) into Eq. (9) we get system of PDEs with three independent

variables

(11)

[i?1 ?x1 + i?2 ?x2 + i?3 ?x3 + F1 (??, ??4 ?) + F2 (??, ??4 ?)?4 ]? = 0.

?? ??

But from the group-theoretical point of view independence of Eq. (9) of the va-

riable x0 means that this equation is invariant under the one-parameter group of

displacements with respect to x0

(12)

x0 = x0 + ?, x = x, ? = ?.

Similarly, (10) is a manifold in the space of variables x, ? invariant under the

group of displacements with respect to x0 .

Thus, imposing on the solution to be found requirement of invariance with respect

to the one-parameter group (12) which is a subgroup of the invariance group of Eq. (9)

we reduce it by one independent variable.

Now we turn to the general case. Let Eq. (9) be invariant under the one-parameter

transformation group

(13)

xµ = fµ (x, ?), ? = F (x, ?)?,

where fµ are some real functions and F is a variable 4 ? 4 matrix.

It is known, that there exists such change of variables

(14)

?µ = ?µ (x), ? = B(x)?,

where B(x) is some invertible 4 ? 4 matrix, that the group (13) in the space of

variables ?µ , ? takes the form

(15)

?0 = ?0 + ?, ? = ?, ? = ?.

Consequently, if we make in the initial equation (9) the change of variables (14),

then the equation obtained will be invariant under the one-parameter group of dis-

placements (15). Therefore, a substitution ? = ?(?1 , ?2 , ?3 ) reduce it to a system of

PDEs with three independent variables ?1 , ?2 , ?3 .

In the initial variables the above said substitution reads

(16)

?(x) = A(x)?(?1 (x), ?2 (x), ?3 (x)),

where A(x) = B ?1 (x).

And what is more, substitution of the expression (16) into Eq. (9) reduce it to

a system of PDEs with three independent variables ?1 , ?2 , ?3 .

In fact, we gave a sketch of the proof of the reduction theorem, which is of utmost

importance for applications of Lie transformation groups in mathematical physics.

Namely, solution invariant under the one-parameter subgroup of the invariance group

of the nonlinear Dirac equation reduce it to a system of PDEs with three independent

variables. Obviously, a solution invariant under a three-parameter subgroup of inva-

riance group reduce the nonlinear Dirac equation to a system of ordinary differential

equations (ODEs).

So each three-parameter subgroup of the Poincar? group P (1, 3) gives rise to an

e

Ansatz

(17)

?(x) = A(x)?(?(x)),

Symmetry and reduction of nonlinear Dirac equations 471

which reduces the nonlinear Dirac equation (9) to a system of ODEs for a function

?(?).

In practice it is more convenient to work with Lie algebras. Let the operators

(18)

Qa = ?aµ (x)?xµ + ?a (x), a = 1, 2, 3

form a three-dimensional Lie algebra corresponding to a given three-parameter sub-

group G3 of the group P (1, 3). Then a solution invariant with respect to the group

G3 has the form (18), where function ?(x) and matrix function A(x) are determined

by the following equations:

1. ?aµ (x)?xµ ?(x) = 0, a = 1, 2, 3,

(19)

2. (?aµ (x)?xµ + ?a (x))A(x) = 0, a = 1, 2, 3.

Classification of P (1, 3)-inequivalent subalgebras of the Lie algebra of the Poincar?

e

group P (1, 3) has been carried out in [5]. There are, in particular, 27 inequivalent

three-dimensional subalgebras. Solving for each of these system of PDEs (19) we

obtain 27 Ans?tze reducing the nonlinear Dirac equation to systems of ODEs.

a

Consider, as an example, a subgroup which Lie algebra has the following basis

elements:

1

Q1 = ?x0 , Q2 = ?x3 , Q3 = x2 ?x1 ? x1 ?x2 + ?1 ?2 .

2

Ansatz corresponding to the above algebra has the form

1 x1

?(x) = exp ? ?1 ?2 arctan ?(x2 + x2 ). (20)

1 2

2 x2

Substituting the above Ansatz into Eq. (9) after some tedious transformations we

get a system of ODEs

d? i

+ ? ?1/2 ?2 ? + [F1 (??, ??4 ?) + F2 (??, ??4 ?)?4 ]? = 0.

i?2 ?? ??

d? 2

For the nonlinear equation suggested by Ivanenko F1 = m + (??) and F2 = 0. In

?

such a case the above system is integrated. Substituting the result obtained into the

Ansatz (20) we get an exact solution of the nonlinear equation (2)

1 x1

1

?

?(x) = (x2 + x2 ) 2 ??? 4 exp ? ?1 ?2 arctan exp ?m(x2 + x2 )1/2 ?,

?

1 2 1 2

2 x2

where ? an arbitrary constant four-component column.

Symmetry approach to construction of exact solutions of PDEs is so systematic and

algorithmic that one could get an impression that in this way all Ans?tze reducing

a

Eq. (9) to systems of ODEs can be obtained. Luckily, it is not so. The source of

principally new Ans?tze is the conditional symmetry of Eq. (9).

a

To study conditional symmetry of PDEs one can apply Lie algorithm but the

problem is that the determining equations for coefficients of vector field admitted

are essentially nonlinear. This is a reason why more or less systematic results on

conditional symmetry of PDEs are obtained only for two-dimensional equations.

But we suggested a method making it possible to obtain rich information about

conditional symmetry of such a complex nonlinear model as Eq. (9).

472 R.Z. Zhdanov and W.I. Fushchych

The principal idea is based on the following observation: all Ans?tze invariant

a

under three-parameter subgroups of the group P (1, 3) can be represented in the

following unified form [6]:

?(x) = exp ((?1 ?1 (x) + ?2 ?2 (x))(?0 + ?3 )) ?

(21)

? exp (?0 (x)?0 ?3 + ?3 (x)?1 ?2 ) ?(?(x)).

Specifying the functions ?µ (x), ?(x) we get from (21) all Poincar? invariant Ans?t-

e a

ze mentioned above.

The idea is not to impose ad hoc conditions on the functions ?µ , ?. The only

condition is a requirement that substitution of expression (21) into Eq. (9) yields

a system of ODEs for a four-component function ?(?).

As a result, one gets a system of twelve nonlinear PDEs for five functions. From

the first sight it looks even more complicated than the initial Eq. (9). But the fact that

said system is strongly over-determined enabled us to construct its general solution.

In this way we have obtained not only all Poincar? invariant Ans?tze (which is

e a

quite predictable) but also six principally new classes of Ans?tze which correspond to

a

conditional symmetry of the equation under study.

We adduce, as an example, the following Ansatz

1

(w1 ?1 + w2 ?2 )(?0 + ?3 ) + C(y1 + y2 )?1/2 (y2 ?1 ? y1 ?2 ) ?

2 2

?(x) = exp

2 (22)

1 y1

? (?0 + ?3 ) exp ? ?1 ?2 arctan 2 2

?(y1 + y2 ),

2 y2

where ya = xa + wa , a = 1, 2, wa = wa (x0 + x3 ) are arbitrary functions, C is an

arbitrary constant.

It is readily seen that provided C = 0, w1 = w2 = 0 formula (22) gives the Ansatz

(20) which has been obtained with the use of the invariance group of Eq. (9). This

example demonstrates that invariant solutions are very special cases of conditionally

invariant solutions.

It is important to emphasize a principal difference between invariant and condi-

tionally-invariant Ans?tze. Ansatz (20) invariant under the three-parameter subgroup

a

of the Poincar? group can be used to reduce any Poincar? invariant system of PDE.

e e

But conditionally-invariant Ansatz (22) can be used for Eq. (9) only. It means that

the last Ansatz contains more precise information about structure of solutions of the

equation under study.

Acknowledgments. Participation of the authors in the Symposium was supported

by Arnold Sommerfeld Institute for Mathematical Physics and (R. Zhdanov) by the

Alexander von Humboldt Foundation.

1. Ivanenko D.D., Zhurn. Eksperim. Teoret. Fiziki, 1938, 8, № 3, 1938, 260–266.

2. Heisenberg W., Zs. Naturforsch. A, 1954, 9, 1954, 292–303.

3. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.

4. Fushchych W.I., Zhdanov R.Z., Phys. Rep., 1989, 172, № 4, 1989, 123–174.

5. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., J. Math. Phys., 1975, 16, № 8, 1613–1624.

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

Naukova Dumka, 1992.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 473–479.

Reduction of the self-dual Yang–Mills

?

equations. I. The Poincare group

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

We have obtained a complete description of ansatzes for the vector-potential of the

Yang–Mills field invariant under 3-parameter P (1, 3)-inequivalent subgroups of the Poi-

ncar? group. Using these, we carry out a reduction of the self-dual Yang–Mills equations

e

to system of ordinary differential equations.

Для вектор-потенцiалу поля Янга–Мiллса побудовано повний набiр iнварiантних

вiдносно P (1, 3)-нееквiвалентних пiдгруп групи Пуанкаре анзацiв, з використанням

яких проведено редукцiю самодуальних рiвнянь Янга–Мiлса до систем звичайних

диференцiальних рiвнянь.

Classical SU (2) Yang–Mills equations form a system of twelve nonlinear second-

order partial differential equations (PDE) in the Minkowski space R(1, 3). But one

can obtain an important subclass of solutions by considering the following first-order

system of PDE:

i

?µ??? F ?? , (1)

Fµ? =

2

where Fµ? = ? µ A? ? ? ? Aµ + eAµ ? A? is a tensor of the Yang–Mills field; ?µ =

?/?xµ , ???? is the antisymmetric fourth-order tensor; µ, ?, ?, ? = 0, 3. Hereafter,

the summation over the repeated indices from 0 to 3 is understood, rising and

lowering of the tensor indices is carried out with the help of the metric tensor

gµ? = diag (1, ?1, ?1, ?1) of the Minkowski space.

Equations (1) are called self-dual Yang–Mills equations. They are very interesting

because of the fact that any solution of equations (1) automatically satisfies Yang–

Mills equations (see, e.g. [1]). Moreover, symmetry groups of the Yang–Mills and of

the self-dual Yang–Mills equations are the same. Maximal symmetry group admitted

by equations (1) is the conformal group C(1, 3) supplemented by the gauge group

SU (2) [2].

In the present paper, we carry out a symmetry reduction of the self-dual Yang–

Mills equations (1) by using ansatzes for the vector-potential of the Yang–Mills

Aµ (x) invariant under the three-parameter subgroups of the Poincar? group P (1, 3) ?

e

C(1, 3).

It is known that the problem of classification of inequivalent subgroups of a Lie

transformation group is equivalent to the one of classification of inequivalent sub-

algebras of the Lie algebra (see, e.g. [3, 4]). Complete description of P (1, 3)-inequiva-

lent three-dimensional subalgebras of the Poincar? algebra AP (1, 3) had been obtained

e

in [3].

Укр. мат. журн., 1995, 47, № 4, P. 456–462.

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

To establish correspondence between the three-dimensional subalgebra of the

symmetry algebra of equations (1) having the basis elements

3

?

b

(2)

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

?Abµ

b=1

where {Aa , a = 1, 3, µ = 0, 3}, and the ansatz for Aµ (x) reducing equations (1) to

µ

a system of ordinary differential equations, one has:

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

operators (2) ? = {?i (x, A), i = 1, 13};

(2) to resolve the relations

(3)

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

with respect to the functions Aa .

µ

As proved in [5], the above procedure can be significantly simplified if coefficients

of operators (2) have the following structure:

3

b

Raµ? (x)Ac .

bc

стр. 112 |