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следующее представление:

L1 = f L2 + gL? , L? = f ? L? + g ? L2 ,
2 1 2

где f f ? ? gg ? = 0; f , g зависит от u, u? , u? , u? .
?
Теорема 4. Уравнения (2) инвариантны относительно алгебр A, A1 , . . . , A5 то-
лько в том случае, если они эквивалентны следующей системе:

?i? W ?i + ?? (?) = 0,
?i (?)W i + ?(?),

где ? — дифференциальные инварианты нулевого и первого порядка, W i —
инварианты второго порядка вида (12) соответствующих алгебр.
4. Примеры. Приведем пример системы, инвариантной относительно конформ-
ной группы C(1, 3) в четырехмерном пространстве (x0 , x1 , x2 , x3 ):
u? u?
2u? = u?3 ?
2u = u3 ? , , , ,
u? u6 u? u6
Эта система является комплексным обобщением единственного конформно-
инвариантного уравнения 2u + ?u3 = 0 для действительной функции.
Уравнение
n?1
{(R3 ? R5 )r1 + (R4 ? R2 )r2 }(r1 r3 ? r2 )?1 =
2u ? 2
2
= (u2 + u?2 )?1 (f (r1 u + r2 u? ) + i(r2 u ? r1 u? )g) = 0,
О симметрийных свойствах комплекснозначных волновых уравнений 355

где f , g — действительные функции от инвариантов нулевого и первого порядка,
инвариантные относительно алгебры AC(1, n) с нулевой конформной степенью.
Уравнения
n+1 u?
u2u ? r1 = ? , ,
u? u2
2
2u ?2 u?
(u R1 ? 2uu? R2 + u2 R3 ) = r1 + ? ,
u? u2
?
инвариантны относительно AC(1, n) с конформной степенью 1; ri , ?, Rj — обо-
значения, использовавшиеся при записи инвариантов первого и второго порядка.
К приведенным уравнениям, конечно, следует дописать комплексно-сопряженные.

1. Боголюбов Н.Н., Ширков Д.В., Введение в теорию квантованных полей, М., Наука, 1973, 416 с.
2. Эйзенхарт Л.П., Непрерывные группы преобразований, М., ИЛ, 1947, 358 с.
3. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.
4. Фущич В.И.. Серов Н.И., ДАН, 1984, 278, № 4, 847–851.
5. Fushchych W.I., Serov N.I., J. Phys. A, 1983, 16, 3645–3656.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 356–360.

On the reduction and some new exact
solutions of the non-linear Dirac and
Dirac–Klein–Gordon equations
W.I. FUSHCHYCH, R.Z. ZHDANOV
New ans?tze tor spinor fields are suggested. Using these, we construct multiparameter
a
families of exact solutions of the non-linear many-dimensional Dirac and Dirac–Klein–
Gordon equations, some solutions including arbitrary functions.

In this letter we have constructed new families of exact solutions of the following
equations:
?
(?µ pµ ? ?(??)k )?(x) = 0, (1)
µ = 0, 1, 2, 3,
?
?µ pµ ? (?1 |u|k1 + ?2 (??)k2 ) ?(x) = 0,
(2)
?
pµ pµ ? (µ1 |u|k1 + µ2 (??)k2 )2 u(x) = 0,

where ?µ are (4 ? 4)-Dirac matrices, ? = ?(x) is a four-component spinor, u = u(x)
is a complex scalar function, p0 = i?/?x0 , pa = ?i?/?xa , a = 1, 3; ?, k, ?i , µi and
ki are constants. Hereafter we use the summation convention.
Solutions obtained by us differ from those already known in the literature [1–8].
These solutions can be useful in the relativistic quantum field theory.
To construct exact solutions of equation (1) we use the following ans?tze:
a
?(x) = [ig1 (?) + ?4 g2 (?) ? (if1 (?) + ?4 f2 (?))?µ pµ ?]?, (3)
?4 = ?0 ?1 ?2 ?3 ,

?(x) = [G1 (?1 , ?2 ) + i(?µ aµ + ?µ dµ )G2 (?1 , ?2 ) +
(4)
+ i(?µ bµ )F1 (?1 , ?2 ) + (?µ aµ + ?µ dµ )(?? b? )F2 (?1 , ?2 )]?,

where ? = ?(x) are scalar functions satisfying conditions of the form
pµ pµ ? + A(?) = 0, (pµ ?)(pµ ?) + B(?) = 0, (5)
where fi , gi , Fi , Gi , A and B are arbitrary differentiable functions, ?1 = aµ xµ +dµ xµ ,
?2 = b? x? and ? is an arbitrary constant spinor. Hereafter aµ , bµ , cµ and dµ are
arbitrary real parameters satisfying the following conditions:
?aµ aµ = bµ bµ = cµ cµ = dµ dµ = ?1,
aµ bµ = aµ cµ = aµ dµ = bµ cµ = bµ dµ = cµ dµ = 0.
Substitution of ans?tze (3) and (4) into the initial equation (1) leads to the following
a
systems of differential equations for unknown functions fi , gi , Fi , Gi :
B f?1 + Af1 = ?[g1 ? g2 + B(f1 ? f2 )]k g1 ,
?2 2 2 2
(6)
?
g1 = ??[g 2 ? g 2 + B(f 2 ? f 2 )]k f1 ,
? 1 2 1 2

J. Phys. A: Math. Gen., 1988, 21, L5–L9.
On the reduction and some new exact solutions 357

?2
g2 = ?[g1 ? g2 + B(f1 ? f2 )]k f2 ,
2 2 2
?
B f?2 + Af2 = ??[g 2 ? g 2 + B(f 2 ? f 2 )]k g2 ,
?
1 2 1 2
? f?i = dfi /d?,
? = ?(??)k ,
? gi = dgi /d?,
? i = 1, 2,
? ?
F?2 = ??[(G1 )2 ? (F 1 )2 ]k G2 , G1 2 = ??[(G1 )2 ? (F 1 )2 ]k F 1 ,
1
?
(7)
? ?
G1 + F 2 = ??[(G1 )2 ? (F 1 )2 ]k G2 ? G2 + F 1 = ?[(G1 )2 ? (F 1 )2 ]k F 2 .
?1 ?2 ?2 ?1

Not going into details of the integration of systems (5) and (6) we shall write down
exact solutions of the non-linear Dirac equation (1) obtained through the substitution
of expressions for fi , gi into ansatz (3)
(i) k < 1
4

?(x) = ? ?1/2k {?(1 ? 4k)1/2 (?iC1 + ?4 C2 ) + (C1 ? i?4 C2 ) ?
? [(?b)(by) + (?c)(cy) + (?d)(dy)]? ?1 }?, (8)
? = [(by)2 + (cy)2 + (dy)2 ]1/2 , Cj = const
and the condition holds
?
±(1 ? 4k)1/2 ? 2k ?[4k(C1 ? C2 )]k = 0;
2 2

1
(ii) k > 4

?(x) = ? ?1/2k {?(4k ? 1)1/2 (?iC1 + ?4 C2 ) + (C1 ? i?4 C2 ) ?
? [(?a)(ay) ? (?b)(by) ? (?c)(cy)]? ?1 }?, (9)
? = [(ay)2 ? (by)2 ? (cy)2 ]1/2 , Cj = const
and the condition holds
?
±(4k ? 1)1/2 ? 2k ?[4k(C1 ? C2 )]k = 0;
2 2

1
(iii) k > 6

?(x) = ? ?1/2k [?(6k ? 1)1/2 (?iC1 + ?4 C2 ) + (C1 ? i?4 C2 )(?y)? ?1 ]?,
(10)
? = (yy)1/2 , Cj = const
and the condition holds
?
±(6k ? 1)1/2 ? 2k ?[6k(C1 ? C2 )]k = 0;
2 2


(iv) k ? R
?
?(x) = {ig1 (?) + ?4 g2 (?) + (f1 (?) ? i?4 f2 (?))[?b + (?a + ?d)F (ay + dy)]}?,
?2 ?2
f1 = C1 cosh[?(C3 ? C1 )k ? + C2 ], f2 = C3 cosh[?(C3 ? C1 )k ? + C4 ],
2 2
(11)
?2 ?2
g1 = C1 sinh[?(C3 ? C1 )k ? + C2 ], g2 = C3 sinh[?(C3 ? C1 )k ? + C4 ],
2 2

? = by + F (ay + dy), Cj = const,
where F is an arbitrary differentiable function;
?(x) = [ig1 (?) + ?4 g2 (?) + (f1 (?) ? i?4 f2 (?))(?a)]?,
?2 ?2
f1 = C1 sin[?(C1 ? C3 )k ? + C2 ], f2 = C3 cos[?(C1 ? C3 )k ? + C4 ],
2 2
(12)
? ?
g1 = C1 cos[?(C 2 ? C 2 )k ? + C2 ], g2 = C3 sin[?(C 2 ? C 2 )k ? + C4 ],
1 3 1 3
Cj = const, ? = ay;
358 W.I. Fushchych, R.Z. Zhdanov

(v) k = 1/m, m = 2, 3

?(x) = (1 + ?2 ? 2 )?(m+1)/2 [iC1 + ?4 C2 ? ?(C1 + i?4 C2 )] ?
[(?a)(ay) ? (?b)(by) ? (?c)(cy)], m = 2,
?
?y, m = 3, (13)
[(ay)2 ? (by)2 ? (cy)2 ]1/2 , m = 2,
?=
(yy)1/2 , m=3

and the condition holds
?2
(m + 1)? ? ?(C1 ? C2 )1/m = 0.
2


In the formulae (8)–(13) the following notations were used:
ay ? aµ y µ , ?a ? ?µ aµ , ?y ? ?µ y µ , µ = 0, 1, 2, 3,
(14)
?
?µ = const, ? = ?(??)k .
yµ = xµ + ?µ , ?
If g2 ? f2 ? 0, ? = xµ xµ then (3) coincides with the ansatz suggested by
Heisenberg in [1]. That is why exact solutions of the equation (1) obtained with the
help of the Heisenberg ansatz in [2–4] belong to classes (10) and (13).
It was G?rsey who showed that under k = 1 equation (1) is conformally inva-
u 3
riant [9]. This makes it possible to construct new families of exact solutions using the
solution generation technique (see [6]). As is shown in [6] the formula of generating
solutions by final transformations of the four-parameter special conformal group has
the form
?2 (x) = ? ?2 (x)[1 ? (?x)(??)]?1 (x ),
xµ = (xµ ? ?µ xx)? ?1 (x), (15)
?(x) = 1 ? 2?x + (??)(xx), ?µ = const, µ = 0, 3.

Using (9)–(13) under k = 1 as ?1 (x) one can obtain multi-parameter families of
3
solutions of the non-linear Dirac equation (1) which are invariant under the conformal
group C(1, 3).
System (7) proved to be an integrable one. Substituting its general solution into the
ansatz (4) we obtain a multi-parameter family of exact solutions of the equation (1)
depending on four arbitrary functions
?
?(x) = ?1 cosh(?bx) + ?2 sinh(?bx) + i(?a + ?d)[(bx/2?)? ?

? (?2 cosh(?bx) + ?1 sinh(?bx) + ?3 cosh(?bx) + ?4 sinh(?bx))] +
+ i(?b)(? sinh(?bx) + ?2 cosh(?bx)) + (?a + ?d)(?b) ?
? ? ?
?2 ? ?1 ?2 ? ?2 ? ?2 (16)
? ? ? ? ?
bx sinh(?bx) +
?2 ?2
? 2? ?
?
?1 ?
? bx cosh(?bx) + ?3 sinh(?bx) + ?4 cosh(?bx) ?,
2?
d? d?i
? ?
?= , ?i = , i = 1, 2,
d? d?
On the reduction and some new exact solutions 359

where
?a ? ?µ aµ , ?b ? ?µ bµ , bx ? bµ xµ ,
?(?) = ??(??)k [(?1 (?))2 ? (?3 (?))2 ]k ,
?
?1 , . . . , ?4 are arbitrary differentiable functions of ? = aµ xµ + dµ xµ .
We note that it is not difficult to construct an explicit form of the energy-momen-
tum tensor
1? ?
Tµ? = i(??µ ?? ? ?? ?µ ?) + gµ? L,
2
1? ?
? ?
L = i(??µ ?µ ? ?µ ?µ ?) ? (??)k+1
2 k+1
corresponding to obtained solutions. For the solutions (8)–(10) one has
?µ? ? ?(k+1)/k ,
?>+?
? ?µ? = const. (17)
Tµ?

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