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2?? ? w?1 ?2 ? w?1 (1 + w2 ) = 0; (8)
?

2w?? + ?2 ? w?2 ? ? 2 (w + ?)?2 = 1; (9)
?

w2 ?2 + ? + 1 = 0; (10)
? ?

(w ? w2 )?2 + 2w?? ? ?2 ? ? = 0; (11)
? ?

(1 + ?2 )?2 + 2?? + ?2 ? ? = 0; (12)
? ?

(1 + ?2 )?2 + 2(1 ? ?)? + ?2 ? ? = 0; (13)
? ?

?2 + ?? ? ? = 0; (14–16)
? ?

w3 ?2 + w? + ?w ? 1 = 0; (17)
? ?

144(ew ? 1)?2 ? 96?? ? 16?2 = 1; (18, 19)
? ?

(? ? w?)2 ? w?2 = ?. (20)
? ?

Редуцированное уравнение мы снабдили номером той системы инвариантов, кото-
рой оно соответствует.
Укажем некоторые точные решения уравнения эйконала, полученные из реше-
ний редуцированных уравнений (1)–(10).
1/2
u = C 2 + 1 (2C)?1 x2 ? x2 + C 2 ? 1 (2C)?1 x2 ,
1) 0 1
1/2
u = ± x2 ? x2 ? x2 ;
0 1 2

1/2
u = C 2 + 1 (2C)?1 x2 ? x2 ? x2 + C 2 ? 1 (2C)?1 x3 ,
2) 0 1 2
1,2
u = ± x2 ? x2 ? x2 ? x2 ;
0 1 2 3


u = ±x0 ;
3)

1/2
u = ± C(x0 ? x2 ) + x2 ? x2 ? x2 ? ?x2
4) (? = 0, 1);
0 1 2 3
1/2
u = ± C 2 (x0 ? x2 )2 ? 2Cx3 (x0 ? x2 ) + x2 ? x2 ? x2 ;
0 1 2
1/2
u = ± x2 [1 ? C(x0 ? x2 )]?1 + x2 ? x2 ? x2 ? x2 ;
3 0 1 2 3

1/2
u = ± x2 ? x2
5) ;
0 3
1/2
x2 + x2 [1 ? C(x0 ? x3 )]?1 + x2 ? x2 ? x2 ? x2
u=± ;
1 2 0 1 2 3
1/2
u = ± C(x0 ? x3 ) + x2 ? x2 ;
0 3
1/2
u = ± C(x0 ? x3 ) + x2 ? x2 ? x2 ? x2 ;
0 1 2 3
306 Л.Ф. Баранник, В.И. Фущич

1/2
u = ± C(x0 ? x1 ) + x2 ? x2 ? x2
6) ;
0 1 2
1/2
u = ± x2 [1 ? C(x0 ? x1 )]?1 + x2 ? x2 ? x2 ;
2 0 1 2

1/2
u = ± C(x0 + x3 ) + x2 ? x2 ? x2 ? x2
7) ;
0 1 2 3

1/2
u = (x0 ? x1 )?1 x2 + x2 ? x2 ? x2 ?
8) 0 1 2
1/2
? (x0 ? x1 )2 + C(x0 ? x1 ) ? 1 ;

1/2
(x0 ? x3 )3 + (C + ?)(x0 ? x3 )2 + (C? ? ? 2 ? 1)(x0 ? x3 ) ? ?
u=±
9)
(x0 ? x3 )2 (x0 ? x3 + ?)
1/2
x0 ? x3 ?x2 x1
? ? ? ? x2 ? x2
x2 x2 + + ;
x0 ? x3 + ? x0 ? x3 + ? x0 ? x3
0 1 2 3


1/2
u = ± (x0 ? x3 )(t?? + 2? arctg t + 2 arctg x2 x?1 + x0 + x3 + C)
10) ,
1
1/2
t = (2(x0 ? x3 ))?1 x2 + x2 ? 4(x0 ? x3 )2 ? x2 + x2 , ? = ±1.
1 2 1 2




1. Фущич В.И., Никитин А.Г., Симметрия уравнений Максвелла, Киев, Наук. думка, 1983, 200 c.
2. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental
groups of physics. III. The de Sitter groups, J. Math. Phys., 1977, 18, № 12, 2259–2288.
3. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of physics.
II. The similitude group, J. Math. Phys., 1975, 16, № 8, 1615–1624.
4. Burdet G., Patera J., Perrin M., Winternitz P., The optical group and its subgroups, J. Math. Phys.,
1978, 19, № 8, 1758–1780.
5. Фущич В.И., Баранник А.Ф., Баранник Л.Ф., Непрерывные подгруппы обобщенной группы
Евклида, Укр. мат. журн., 1986, 38, № 1, 67–72.
?
6. Баранник Л.Ф., Фущич В.И., Подалгебры алгебры Ли расширенной группы Пуанкаре P (1, n),
Препринт № 85.90, Киев, Ин-т математики АН УССР, 1985, 52 c.
7. Баранник Л.Ф., Лагно В.И., Фущич В.,И., Подалгебры обобщенной алгебры Пуанкаре AP (2, n),
Препринт № 85.89, Киев, Ин-т математики АН УССР, 1985, 50 c.
8. Barannik L.F., Fushchych W.I., On subalgebras of the Lie algebra of the extended Poincar? group
e
?
P (1, n), J. Math. Phys., 1987, 28, № 7, 1445–1458.
9. Баранник Л.Ф., Фущич В.И., О непрерывных подгруппах обобщенных групп Шредингера, Пре-
принт № 87.16, Киев, Ин-т математики АН УССР, 1987, 48 c.
?
10. Chen Su-Shing, On subgroups of the noncompact real exceptional Lie group F4 , Math. Ann., 1973,
204, № 4, 271–284.
11. Тауфик М.С., О полупростых подалгебрах псевдоунитарных алгебр Ли, в сб. Геометрические
методы в задачах алгебры и анализа, Ярославль, Яросл. гос. ун-т, 1980, 86–115.
12. Fushchych W.I., Shtelen W.M., The symmetry and some exact solutions of the relativistic eikonal
equation, Lett. Nuovo Cim., 1982, 34, 498–502.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 307–318.

Exact solutions of multidimensional nonlinear
?
Dirac’s and Schrodinger’s equations
W.I. FUSHCHYCH
A class of nonlinear spinor equations invariant under the extended Poincar? group e
and conformal group is described. New Ans?tze for spinor fields are suggested. Multi-
a
parameter families of exact solutions for the multidimensional families of exact solutions
for the multidimensional nonlinear Dirac and Schr?dinger equations are obtained.
o

In this paper I present some new results, obtained in Intitute of Mathematics,
Academy of Sciences of the Ukranian SSR in Kiev by R. Zhdanov, W. Shtelen,
N. Serov and me on multiparameter families of exact solutions of nonlinear Dirac and
Schr?dinger equations
o
?µ pµ ? + F1 (x, ?? , ?)? = 0, (1)

1
pa pa u + F2 (x, u, u? ) = 0, (2)
p0 +
2m

where ? ? ?(x) = (?0 , ?1 , ?2 , ?3 ) is 4-component spinor, x = (x0 , x1 , x2 , x3 ), ?? is
complex conjugated spinor, ?µ are 4 ? 4 Dirac matrices, u ? u(x0 , x1 , x2 , x3 ), x0 ? t,
u? is complex conjugated wave function,
? ?
pj = ?i
p0 = i , , µ, ? = 0, 3, j = 1, 2, 3,
?x0 ?xj
F1 , F2 are arbitrary smooth function, m is the particle mass.
Fifty years ago D. lvanenko (1938) considered the simplest equation of the type
(1), the case in which
? (3)
F1 = ?(??),
where ? ? ?† ?0 is Dirac-conjugated spinor, ? is arbitrary parameter.
?
W. Heisenberg and his collaborators (1954–1959) have analysed the equation (1)
from a different point of view with the nonlinearity
?
F1 = ???µ ?4 ?? µ ?4 , (4)
?4 = ?0 ?1 ?2 ?3 .
The main efforts of W. Heinseberg directed into the construction of unified quan-
tum field theory based on eq. (1) with the nonlinearities (3), (4). In the works by
R. Finkelstein and his collaborators (1951–1956) eq. (1) has been studied from the
classical point of view, i.e. they studied the exact and approximate solutions of spinor
systems of the type (1).
Some exact solutions of the Dirac equation were obtained by F. Kortel (1956), D.
Kurgeleidze (1957), K.G. Akdezin., A. Smailogic (1984), A.O. Barut, B.W. Xu (1982),
K. Takahashi (1979).
Preprint № 467, Institute for Mathematics and its Applications, University of Minnesota, 1988, 14 p.
308 W.I. Fushchych

The classical Lie’s method for finding exact solutions of multidimensional nonli-
near Schr?dinger equation and d’Allembert equation was applied by Fushchych (1981,
o
1983), Fushchych and Serov (1983, 1987), Gagnon and Winternitz (1988), Grundland,
Harmad and Winternitz (1984), Tajiri (1983).
Evidently if we do not specify the functions F1 , F2 in eqs. (1), (2) there is no hope
to get any profound information about exact solutions of these equations. To specify
the functions F1 and F2 we shall study the symmetry properties of equations (1), and
(2). In what follows, I shall essentially use the classical ideas of S. Lie in application
to nonlinear wave equations.
The wide symmetry of equations (1), (2) makes it possible to reduce the multidi-
mensional partial differential equation (PDE) a set of systems of ordinary differential
equations (ODE). Many of these ODEs can be solved exactly. In this way we are able
to construct many parameter families of exact solutions of the multidimensional wave
equations (1), (2).
1. The symmetry of the nonlinear spinor equation. In this section we will
present the theorems concerning the symmetry properties of equation (1).
Theorem 1. Equation (1) is invariant under the Poincar? group P (1, 3) iff
e

F1 (x, ?? , ?) = F11 (s) + F12 (s)?4 + F13 (s)? µ (??4 ?µ ?) + F14 (s)S µ? (??4 Sµ? ?),
? ?

1
? ? (?µ ?? ? ?? ?µ ), (1.1)
s = (??4 ?, ??), Sµ? = [?µ , ?4 ] =
4
where F11 , F12 , F13 , F14 are arbitrary smooth scalar functions of the invariant
variable s.
?
Theorem 2. Equation (1) is invariant under the extend Poincar? group P (1, 3), i.e.
e
the P (1, 3) group expanded by the one-parameter group of scale transformations of
the type

xµ = xµ exp(?), ?x = ?(x) exp(k?),

iff

F1i = (??)?1/2k F1i ,
? ? (1.2)
i = 1, 2,
(1+2k)
F1j = (??)?
? ? (1.3)
F1j , j = 3, 4,
2k


?
(??)
? ?
where F1i , F1j are arbitrary functions of .
?
(??4 ?)
Theorem 3. Equation (1) is invariant under the conformal group C(1, 3) = P (1, 3),
D, Kµ

xµ = {xµ ? cµ (x · x)}? ?1 (x), ?(x) = 1 ? 2cµ xµ + c2 x2 ,
? (x) = ?(x){1 ? (? · c)(? · x)}?
iff
? ? k = ?3/2,
F1i = (??)1/3 F1i , (1.4)
i = 1, 2,

F1j = (??)?2/3 F1j ,

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