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Now, making use of the Poincar? invariance of the Dirac equation, is not difficult
e
to construct new multiparameter families of exact solutions of the equation starting
from those obtained above. The two following examples illustrate the procedure of
generating solutions.
?
V. Case k = 1/2, nonlinearity ?(??),
a·z
1
? = [(a · z)2 + (b · z)2 ]?1/4 exp ? (? · a)(? · b) tg?1 ?
b·z
2
??
? (4.5)
? exp ?i? (? · b + ?? · a) ?
2(1 + ?2 )
a·z
? ln (a · z)2 + (b · z)2 + 2? tg?1 ?,
b·z
Exact solutions of nonlinear Dirac’s and Schr?dinger’s equations
o 313

zµ = xµ + ?µ , aµ , bµ , ?, ?µ are arbitrary parameters satisfying conditions

a · a = aµ aµ = ?1, b · b = bµ bµ = ?1, a · b = 0,
? · a ? ?0 a0 ? ?1 a1 ? ?2 a2 ? ?3 a3 , ? = (?0 ? ?1 ? ?2 ? ?3 )1/2 .
2 2 2 2

?
VI. Case k = 1/2, nonlinearity ?(??)1/2k ,
a·z
1
?1/4
exp ? (? · a)(? · b) tg?1
?(x) = (a · z)2 + (b · z)2 ?
b·z
2
(4.6)
1?2k
2?k
? exp ?i (? · b)(??)1/2 (a · z)2 + (b · z)2 4k
? ?.
1 ? 2k
Formulas (4.5), (4.6) give multiparameter families of exact solutions of the Dirac
?
equations. These families are nongenerating with respect to the group P (1, 3) in the
sense that solutions (4.5), (4.6) have the same symmetrys as the equation (2.1).
5. Conformally invariant solutions. Conformally invariant Dirac–G?rsey equa-
u
tion has the form
?
?µ pµ + (??)1/3 ? = 0. (5.1)

With the help of a conformally invariant ansatz we can construct the following
solutions
?·x
exp{i?k(? · ?)?}?, k = 1/3,
?(x) =
(x · x)2
(5.2)
?µ xµ ? µ
?= , x? x = 0, ?µ ? > 0;
x? x?

?(x) = ? 2 (x)[1 ? (? · x)(? · ?)] ?
?
(? · a)(? · b)[b · x ? (b · ?)(x · x)]? ?1 (x) ?
? exp
2
i
? exp ? ?(??)1/3 (? · a)[2(a · x) ? (a · ?)(x · x)]?(x) +
?
(5.3)
2
+ ?(b · x ? (b · ?)(x · x))2 ? ?2 (x) ?,

a · b = b · b = 0, a · a = a2 ? a2 ? a2 ? a2 = ?1,
0 1 2 3

c2 = c? c? , x2 = xµ xµ .

Formulae (5.2), (5.3) give multiparameter families of exact solutions of the equati-
on (5.1). The family (5.3) is nongenerating with respect to the group C(1, 3).
If ?1 (x) is a solution of equation (5.1) then
?2 (x) = ? ?2 (x)[1 ? (? · x)(? · c)]?1 (x ),
(5.4)
xµ ? cµ x2
x= xµ =
?(x)
will also be a solution (5.4) is the formula for multiplication of solutions of Dirac
equation.
314 W.I. Fushchych

6. How to construct solutions of the nonlinear d’Alambert equation via
solutions of the Dirac equation? Complex scalar field can be represented as
? (6.1)
u(x) = ?? exp{i?(x)},
where ?(x) is a solution of the Dirac equation, ?(x) is a phase. In the simplest case,
when
? ?(x) = ?µ xµ
?? = c = const, (6.2)
formula (6.1) gives a plane-wave solution of the linear d’Alambert equation. In most
cases solutions of the nonlinear Dirac equation generate a scalar field (6.1) which
satisfies the nonlinear d’Alambert equation
pµ pµ u = k|u|r u, (6.3)
k, r are constants.
Let us exhibit some exact solutions of the equation (6.3) obtained this way
u(x) = c x2 + x2 exp{i?0 (x0 + x3 )}, (6.4)
r = 2,
1 2

?1/2
?
u(x) = c (x1 + ?1 (x0 + x3 ))2 + (x2 + ?2 (x0 + x3 ))2 x
(6.5)
? exp{i?0 (x0 + x3 )}, r = 2,
?2
u(x) = c x2 ? x2 ? x2 ? x2 (6.6)
, r = 1/2,
0 1 2 3

?1/2
u(x) = c x2 + x2 (6.7)
exp{i?(x0 + x1 )}, r = 2.
2 3

In formulae (6.4)–(6.7) ?0 , ?1 , ?2 , ? are arbitrary smooth functions.
So, solutions of the nonlinear Dirac spinor equation give a possibility to construct
solutions to the nonlinear d’Alambert equation.
All these ideas and results were considered in more detail by Fushchych (1981,
1987), Fushchych and Shtelen (1983, 1987), Fushchych, Shtelen and Zhdanov (1985),
Fushchych and Zhdanov (1987, 1988, 1989), Fushchych and Nikitin (1987) (see Ap-
pendix).
7. The solutions of the multidimensional Schr?dinger equation. Let us consi-
o
der the following nonlinear equation
? 1
? u + F (x, u, u? ) = 0,
? u ? u(x0 ? t, x1 , x2 , x3 ). (7.1)
i
?x0 2m
It is well known that if F = 0, then linear Schr?dinger equation (see Sofus Lie
o
(1881), Hagen (1972), Niederere (1972), Kalnins and Miller (1987)) is invariant under
the generalised Galilei group, which will be denoted by the symbols G2 (1, 3). The basis
elements of the Lie algebra AG2 (1, 3) = P0 , Pa , Jab , Ga , D, A, I have the following
form:
? ? ?
Pa = ?i Jab = xa Pb ? xb Pa ,
P0 = i , , a = 1, 2, 3, I=u ,
?x0 ?xa ?u
(7.2)
3 m
D = 2x0 P0 ? xa Pa + i, A = x0 D + x2 .
Ga = x0 Pa + mxa ,
2a
2
Exact solutions of nonlinear Dirac’s and Schr?dinger’s equations
o 315

Symbols AG1 (1, 3), AG(1, 3) denote the following Lie algebras

AG1 (1, 3) = P0 , Pa , Jab , Ga , D, I ,
AG(1, 3) = P0 , Pa , Jab , Ga , I .

To construct families of exact solutions of (7.1) in explicit form we have to know
the symmetries of (7.1) which obviously depends on the structure of the function F .
By Lie’s algorithm (as given by Ovsyannikov (1978), Olver (1986)), the following
statement can be proved.
Theorem 5. Equation (7.1) is invariant under the following algebras:

(7.3)
iff
AG(1, 3) F = ?(|u|)u,

where ? is arbitrary smooth function, and

F = ?|u|k u, (7.4)
iff
AG1 (1, 3)

where ?, k are arbitrary parameters, the operator of scale transformations D having
the form D = x0 P0 ? xa Pa + 2i/k, k = 0, and

F = ?|u|4/n u. (7.5)
iff
AG2 (1, 3)

Later on we shall construct the exact solutions of the equation (7.1) with nonli-
nearity (7.5), i.e.
p2
p0 ? a u + ?|u|4/3 u = 0. (7.6)
2m
Following Fushchych (1981) we seek solutions of (7.6) with the help of the ansatz

(7.7)
u = f (x)?(?1 , ?2 , ?3 ),

where ? is the function to be calculated. To construct solutions of (7.6) using ansatz
(7.7) it is necessary to have the explicit form of the function f (x) and the new
invariant variables ?1 , ?2 and ?3 . Next I shall present two Ans?tze of the type (7.7).
a
im x0 x 2
?3/4 ?1/2
f (x) = 1 ? x2 ?1 = ?x 1 ? x2
1. exp , ,
2 1 ? x2
0 0
0

?x
?1
?2 = x 2 1 ? x2 , ?3 = arctan x0 + arctan ,
0
?x

where ?, ?, ? are constant vectors satisfying the conditions

? 2 = ? 2 = ? 2 = 1, ?? = ?? = ?? = 0.

The Ans?tze (7.6), (7.7) give the following reduced equation
a
?? ??
? 2im + m2 ?2 ? ? 2?m|?|4/3 ? = 0,
L? + 6
??2 ??3
(7.8)
?2? ?2? ?2? ?2?
+ 4?2 2 + (?2 ? ?1 )?1 2 + 4?1
L? ? 2
.
2
??1 ??2 ??3 ??1 ??2
316 W.I. Fushchych

2. The second Ans?tze has the form
a
im 2 ?1
?3/2
exp ? (7.9)
f (x) = x0 x x0 ,
2

?x
?1 = (?x)x?1 , ?2 = x 2 x?2 , ?3 = x?1 + arctan (7.10)
.
0 0 0
?x

The Ans?tze (7.9), (7.10) reduce the equation (7.6) to
a
?? ??
? 2m|?|4/3 ? = 0.
L? + 6 + 2im
??2 ??3
8. Solutions of the equation (7.6). In this paragraph I present some explicit
solutions of the equation (7.6)
im 3
?3/4
x (1 ? x0 )?1 ,
u = 1 ? x2 (8.1)
exp ?= i;
0
2 2

im 2 ?1
u = (c0 x0 ? cx)?3/2 exp ? (8.2)
x x0 ,
2
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