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g = ?g 3 ,
?µ = const, (14)
bµ = ?µ g(?), ? ?? = 0

(? — постоянная, ? — постоянный спинор).
Таким образом, получаем конформно инвариантные решения систем (1)–(3):
?x ?x c
?(x) = exp{i?(??)?}?, ?= , ?(x) = ,
(x? x? )2 x? x? x? x? (15)
? = (? ? ?? )?1 ?1 (??)1/3 + ?2 c , ?3 c3 = ?2 ??,
? ?
?x
?(x) = exp{i?(??)?}?,
(x? x? )2
(16)
?1
? = (? ?? )
? 1/3 2 ? 2 1/5
?[??)
? + µ(? (? ?? ) ??)
? ,

?µ ?x
? 2xµ ? g = ?g 3 , (17)
Aµ (x) = g(?), ?? = 0, ?
x? x? (x x? )2

g(?) — эллиптические функции.
Аналогичным образом, получаются трансляционно инвариантные решения си-
стем (1)–(3). Они имеют вид соответственно
?(x) = exp{?i?(?k)(kx)}?, ?(x) = c,
(18)
? = (k ? k? )?1 ?1 (??)1/3 + ?2 c , ?3 c3 = ?2 ??,
? ?

?(x) = exp{?i?(?k)(kx)}?,
(19)
?1
? = (k k? ) + µ ? (k k? ) (??)
? 1/3 2 ? 2 1/5
?[??)
? ? ,

g = ?g 3 . (20)
Aµ (x) = ?µ g(kx), ?k = 0, ?

Решения (18)–(20) можно использовать для получения других семейств решений
систем (1)–(3) по формулам [2, 3].
Для конформных преобразований (4) формулы генерирования новых решений
имеют вид
xµ ? cµ x2
?c (x )
?(x) = 1 ? 2cx + c2 x2 ,
?н (x) = , xµ = ,
?(x) ?(x)
1 ? ?x?c (21)
Aн (x) = gµ? /?(x) + 2/? 2 (x)[cµ x? ? c? xµ +
?н (x) = ?c (x ), µ
?(x)
+2cxxµ c? ? c2 xµ x? ? x2 cµ c? ) Aµ (x ).
c

В заключение отметим, что с помощью использованного здесь метода найдены
многопараметрические семейства точных решений нелинейного уравнения Дирака
[2, 3], эйконала [4], Янга–Миллса [5], уравнений квантовой электродинамики с
самодействием электромагнитного поля [6].
520 В.И. Фущич, В.М. Штелень

1. Фущич В.И., Симметрия в задачах математической физики, В кн.: Теоретико-алгебраические
исследования в математической физике, Киев, Ин-т математики АН УССР, 1981, 6–28.
2. Фущич В.И., Штелень В.М., Об инвариантных решениях нелинейного уравнения Дирака, ДАН
СССР, 1983, 269, № 1, 88–92.
3. Fushchych W.I., Shtelen W.М., On some exact solutions of the nonlinear Dirac equation, J. Phys.
A, 1983, 16, № 2, 271–278.
4. Fushchych W.I., Shtelen W.М., The symmetry and some exact solutions of relativistic eikonal
equation, Lett. Nuovo Cim., 1982, 34, № 15, 498–502.
5. Fushchych W.I., Shtelen W.М., Conformal symmetry and new exact solutions of SU (2) Yang–Mills
theory, Lett. Nuovo Cim., 1983, 38, № 2, 37–40.
6. Fushchych W.I., Shtelen W.М., On some exact solutions of the nonlinear equations of quantum
electrodynamics, Phys. Lett. B, 1983, 128, № 3/4, 215–217.
7. Боголюбов Н.Н., Ширков Д.В., Квантовые поля, М., Наука, 1980, 320 с.
8. Barut А.O., Bo-Wei Xu, Derivation and uniqueness of vacuum solutions of conformally invariant
coupled nonlinear field equations, Phys. Lett. B, 1981, 102, № 1, 37–39.
9. Barut A.O., Bo-Wei Xu, New exact solution of coupled nonlinear field equations, Physica D, 1982,
6, № 2, 137–139.
10. Гейзенберг В., Нелинейное спинорное уравнение, В кн.: Нелинейная квантовая теория поля, М.,
Изд-во иностр. лит., 1959, 63–75.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 521–523.

On nonlocal transformations
W.I. FUSHCHYCH, W.M. SHTELEN
A procedure of finding finite transformations generated by a linear arbitrary-order di-
fferential operators is presented. Dirac equation is shown to be Galilei invariant with
the nonlocal law of transformation of the ?-function.

At the present time special interest in the study of the invariance properties
of partial differential equations (PDE) excite nonlocal symmetries such as contact,
Lie–B?cklund [1] non-Lie [2, 3]. Recently it was shown [3] that many fundamental
a
equations of theoretic physics possess an additional (non-Lie) invariance. The basis
elements of such invariance algebras are arbitrary order differential operators even
pseudo-differential, while the Lie symmetry is generated by first-order differential
operators only. It will be noted that for systems of linear PDE non-Lie symmetry
generated by finite-order differential operators can be obtained by the Lie–B?cklund
a
approach [1], but with more formidable calculations. In other words, the non-Lie
method [2, 3] applicable to systems of linear PDE gives the same results as Lie–
B?cklund approach does, but more reliable and easy.
a
In this note we solve the problem of finding finite transformations generated by
non-Lie operators, and show that any such operator leads to a one-parametrical group
of transformations.
Formulae of finite transformations discussed here can be used for generating new
solutions of equations in question by analogy with that done in the local case [4–8].
Any linear arbitrary-order differential operator Q acting in the space of r-compo-
nent ?-function (? = ?(x), x = {x0 , x1 , . . . , xn }) can be written down in the form

Q(x, ?) = ? µ ?µ + ?(x, ?), (1)

where ? µ (x) are scalar functions, µ = 0, 1, . . . , n; ? = {?? = ?/?x? }, ?(x, ?) is a
matrix (r ? r), the differential operator does not contain terms like ? µ (x)?µ .
Definition. A linear system of PDE

(2)
L(x, ?)?(x) = 0

is invariant under the transformations

x > x = f (x, ?), ?(x) > ? (x ) = R(x, ?, ?)?(x), (3)

if

(4)
L(x , ? )? (x ) = 0.

Theorem. Operator Q (1) will be an operator of symmetry of eq.(2), if on the
manifold of solutions of eq.(1) the following condition holds true:

LQ? = 0 or [L, Q]? ? (LQ ? QL)? = 0, (5)
Lettere al Nuovo Cimento, 1985, 44, № 1, P. 40–42.
522 W.I. Fushchych, W.M. Shtelen

the transformations generated by Q having the form
x? = exp[?? · ?]x? exp[????], ? (x ) = exp[?? · ?] exp[??Q]?(x). (6)
Proof. As a result of transformations (6) operator L(x, ?) of eq.(2) will be rewritten
in such a manner
L(x, ?) > L(x , ? ) = exp[?? · ?]L(x, ?) exp[??? · ?]. (7)
Hence, we have
L(x , ? )? (x ) = exp[?? · ?]L(x , ?) exp[??? · ?] exp[???] exp[??Q]?(x) =
= exp[???]L(x, ?) exp[??Q]?(x) = 0,
since eq.(5) takes place. According to (4) it proves our theorem.
Remark 1. If Q is a first-order differential operator (case of Lie symmetry), that
is ?(x, ?) = ?(x), then formulae (6) give the same result as does integration of
corresponding Lie equations.
Remark 2. Transformations (6) form a one-dimensional group. Indeed,
x? = exp[?? · ?]x? exp[??? · ?] = exp[?? · ?] exp[?? · ?]x? ?
? exp[????] exp[??? · ?] = exp[(? + ?)? · ?]x? exp[?(? + ?)? · ?].
As far as the transformation of ?(x) is concerned, let us rewrite it in this way
? (x ) = exp[?? · ?] exp[??Q]?(x) = exp[?R]?(x).
To do it, we have used the Campbell–Baker–Hausdorff formula. R = R(x, ?) is an
operator constructed from ? · ? and Q and their various commutators. So we have
? (x ) = exp[?R]? (x ) = exp[?R] exp[?R]?(x) = exp[(? + ?)R]?(x).
This proves our statement.
In ref. [2, 3] it is shown that Dirac equation is the sense of condition (5), under
the set operators satisfying the commutation relations of the Poincar? algebra
e
Pa = ?i?a ,
P0 = i?0 , a = 1, 2, 3,
i
Jab = xa Pb ? xb Pa ? ?a ?b ,
2
1
J0a = tPa ? (Hxa + xa H), H = ?0 ?a Pa + ?0 m.
2
But now operator J0a does not generate Lorentz transformations. In accordance with
formulae (6) we get
x0 = x0 , xa = xa + va x0 ,
which are the well-known Galilei transformations;
i
? (x ) = exp[ix0 va pa ] exp ?ix0 va pa + (Hxa + xa H) ?(x)
2
and it is a nonlocal law of transformation.
On nonlocal transformations 523

1. Anderson R.L., Ibragimov N.H., Lie–B?cklund transformations in applications, Philadelphia, Pa.,
a
l979, p. 150.
2. Fushchych W.I., in Group-Theoretic Methds in Mathematical Physics, Kiev, Mathematical Institute,
1978, 5–44 (in Russian).
3. Fushchych W.I., Nikitin A.G., The symmetry of Maxwell equations, Kiev, Naukova dumka, 1983
(in Russian).
4. Fushchych W.I., Shtelen W.I., Dokl. Akad. Nauk USSR, 1983, 269, № 1, 88 (in Russian).
5. Fushchych W.I., Shtelen W.I., J. Phys. A, 1983, 16, № 2, 271.
6. Fushchych W.I., Shtelen W.I., Lett. Nuovo Cimento, 1982, 34, № 16, 498.
7. Fushchych W.I., Shtelen W.I., Lett. Nuovo Cimento, 1983, 38, № 2, 37.
8. Fushchych W.I., Shtelen W.I., Phys. Lett. B, 1983, 128, № 3/4, 215.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 524–527.

On the new conformally invariant equations
for spinor fields and their exact solutions
W.I. FUSHCHYCH, W.M. SHTELEN, R.Z. ZHDANOV
The Poincar? and conformally invariant nonlinear generalizations of the Dirac equati-
e
on are discussed and, in particular, the conformally invariant version of the Dirac–
Heisenberg equation is obtained. For the latter equation some exact solutions are found
and among them there is a family which is invariant under the full 15-parameter
conformal group.


Consider the following Poincar? invariant nonlinear generalization of the Dirac
e
equation
? ? ? ?
? µ [i?µ + F1 ??µ ? + F2 ??4 ?µ ? + F3 (??µ ?)?4 + F4 (??4 ?µ ?)?4 ]?+
(1)
? ?
+F5 (??µ? ?)? µ? ? + F6 (??µ? ?)?4 ? µ? ? = (F7 + F8 ?4 )?,
? ?
where F1 , . . . , F8 are arbitrary functions of ?? and ??4 ?,
1
i(?µ ?? ? ?? ?µ ).
?4 = i?0 ?1 ?2 ?3 , ?µ? =
4
The well-known Dirac–Heisenberg [1] and Dirac–G?rsey [2] equations belong to this
u
class.
We shall choose from (1) such equations which are invariant under the scale
transformation

xµ = e? xµ , ? (x ) = ek? ?(x), k, ? = const. (2)

and under the conformal ones (see e.g. ref. [3])

xµ ? cµ x2
? (x ) = ?(x)(1 ? ?c?x)?(x),
xµ = ,
?(x) (3)
?(x) = 1 ? 2cx + c2 x2 , cx ? c? x? , c2 ? c? c? , ? = 0, 1, 2, 3.

Theorem 1. Eq.(1) is invariant under the scale transformation (2) if and only if
?(1+2k)/4k ?(1+2k)/4k
? ? ? ?
F1 = ?1 (??µ ?)(?? µ ?) F2 = ?2 (??4 ?µ ?)(??4 ? µ ?)
, ,
?(1+2k)/4k ?(1+2k)/4k
? ? ? ?
F3 = ?3 (??µ ?)(?? µ ?) F4 = ?4 (??4 ?µ ?)??4 ? µ ?
, ,
(4)
?(1+2k)/4k ?(1+2k)/4k
? ? ? ?
F5 = ?5 (??µ? ?)?? µ? ? F6 = ?6 (??µ? ?)?? µ? ?
, ,
F7 = ?7 (??)?1/2k , F8 = ?8 (??)?1/2k ,
? ?

??
where ?1 , . . . , ?8 are arbitrary functions of ??/??4 ?.
Physics Letters B, 1985, 159, № 2/3, P. 189–191.
On the new conformally invariant equations for spinor fields 525

Proof. It is easy to see that transformations (2) leave eq.(1) invariant if
? ? ??
e?(2k+1) FB ??e2k? , ??4 ?e2k? = FB ??, ??4 ? , B = 1, 2, . . . , 6,
(5)
? ? ??
?(k+1)
??e2k? , ??4 ?e2k? = FC ??, ??4 ? ,
e FC C = 7, 8.
Taking into account the well-known identities [4]

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