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i?µ ?/?xµ ? ?(?(x)?(x))k ?(x) = 0, (2)

where ? are 4 ? 4 Dirac matrices (see, for example, Bjorken and Drell [1]), µ, ? =
0, 1, 2, 3, m = 0, k, ? are arbitrary real constants. We use the summation convention
for repeated indices.
It is worthwhile to distinguish equations (1) with m = 0 from (2) because of their
considerable different symmetry properties.
In order to find the exact solutions, we exploit the fact that equation (1) is invariant
under the Poincar? group P (1, 3), equation (2) is invariant under the Weyl group
e
W (1, 3) = {P (1, 3), D} when k = 1 and under the conformal group C(1, 3) when
3
k = 1 . We also show how to draw new families of solutions from known ones.
3
Fushchych [3] has obtained multiparametrical exact solutions of many-dimensional
nonlinear scalar sine-Gordon, Liouville, Hamilton–Jacobi, eikonal, Born–Infeld (Fu-
shchych and Serow [5]), Schr?dinger (Fushchych and Moskaliuk [4]) equations by the
o
method recently proposed (Fushchych [3]). Here we slightly generalise this method
to fit it for the system of partial differential equations.
2. The method
Let Q be an infinitesimal operator of local transformations admitted by equations
(1) or (2). The general form of such an operator is

Q = ? µ (x)?µ + ?(x), (3)

where ? µ (x) are scalar functions of x, ?(x) denotes the 4 ? 4 matrix depending on x
and ?µ ? ?/?xµ .
The operator Q gives the possibility to find exact solutions of equations (1) or (2)
in such a manner.
J. Phys. A: Math. Gen., 1983, 16, ¹ 2, P. 271–277.
On some exact solutions of the nonlinear Dirac equation 325

For the solutions to be found we adopt the ansatz suggested by Fushchych [3]
(4)
?(x) = A(x)?(?),
where the nonsingular 4 ? 4 matrix A(x) can be defined from the equation
QA(x) ? (? µ (x)?µ + ?(x))A(x) = 0, (5)
? are invariants of the differential part of the operatorQ, i.e. functions satisfying
? µ (??/?xµ ) = 0 (6)
or the equivalent Lagrange–Euler system
dx2
dx0 dx1 dx3 def
(7)
=1 =2 =3 = d?.
0 (x)
? ? (x) ? (x) ? (x)
?(?) is the new unknown four-component spinor field depending on new variables ?,
the number of which is one less than the number of variables x.
When A(x) and ? are known, then the substitution of expression A(x)?(?) in
place of ?(x) in equations (1) and (2) leads to a system of differential equations for
?(?) which is often rather easy to solve.
Another procedure for determining the ansatz (4) explicitly is to solve, besides
equation (7), the following system of ordinary differential equations
d?/d? = ??(x(? ))?. (8)
If we insert in the general solution of this system, the value ? defined from (7), and
consider constants of integration as functions of ? then we shall obtain the ansatz
(4) possessing the properties (5) and (6). Let us discuss the procedure of generating
new solutions from known ones.
The general form of transformations generated by operator Q (3) is
x > x = f (x, ?), ?(x) > ? (x ) = R(x, ?)?(x), (9)
where R(x, ?) is a 4 ? 4 matrix, ? is a parameter of transformations. Formula (9)
implies that
?new (x) = R?1 (x, ?)?old (x ) (10)
will be a solution of the equation which admits operator Q as well as ?old (R?1 (x, ?)
denotes the inverse matrix).
Remark. Equation (5) is the consequence of the following obvious condition: the
solutions having the form (4) do not produce new solutions by virtue of the procedure
stated above when transformations (9) are generated by the same operator Q (3).
Indeed, we have according to (4) and (10)
R?1 (x, ?)A(x )?(? ) = A(x)?(?). (11)
? = ? because ? are invariants of the operator Q:
A(x ) = A(x) + ?? µ (x)(?A(x)/?xµ ) + · · · , (12)

R(x, ?) = I ? ??(x) + · · · . (13)
326 W.I. Fushchych, W.M. Shtelen

If we substitute (12) and (13) into (11) and retain terms linear in ? then we obtain (5).
It is clear now that it is the form of transformations (9) that leads to the
ansatz (4).
3. The solutions
First of all we give an example of a conformally invariant solution to the Di-
rac equation (2) with Gursey [6] nonlinearity k = 1 which ensures the conformal
3
invariance of the equation. This solution has the form
?x
exp[i??(??)?)]? ?
?(x) =
(x? x? )2
(14)
?x ??
? cos(????) + i sin(????) ?,
(x? x? )2 ?

where ? = ?x/x? x? , ? ? are arbitrary real constants, ? ? (?? ? ? )1/2 > 0. ? denotes a
space-time independent spinor,

? = a1/3 /? ? ?? , ?x ? ? ? x? ,
?? = a = constant, etc.
?

The solution (14) was sought for in the form

?(x) = ?x/(x? x? )2 ?(?), ? = ?x/x? x? (15)

obtained with the help of the conformal transformation operator

Qconf = cµ kµ = 2(cx)x? ? x2 c? + (?c?x + 2cx),
Qconf ?x/(x? x? )2 ? 0, 2(cx)x? ? x2 c? ? = 0, (16)
? = ? x /x? x? , ?c = 0,

where cµ are arbitrary real constants, x2 ? x? x? , x? = x? (?/?x? ), cx = c? x? . After
the substitution of the expression (15) into equation (2) with k = 1 it implies that
3
?(?) must satisfy the following system of nonlinear ordinary differential equations

d?/d? = i(?/?? ? ? )(??)1/3 (??)?
?

for which it is easy to obtain the general solution

? = exp[i??(??)?]? ? [cos(????) + i(??/?) sin(????)]?

and then (14).
It will be noted that the solution (14) is analytic in the coupling constant ?,
in contrast to the solution obtained by Merwe [8] with the help of the Heisenberg
ansatz [7]. Besides that
?
?(x)?(x) = a/(x? x? )3 ,
?
i.e. ?? dies off very fast when x? x? > ?. It is also noteworthy that such a solution is
easy to generalise to the case of n spatial variables, the conformally invariant equation
being

?
i?? ? ?(?(x)?(x))1/n ?(x) = 0,
On some exact solutions of the nonlinear Dirac equation 327

and the solution takes the form
?x
?(x) = exp[i??(??)?]?, ? = 0, 1, . . . , n
(x? x? )(n+1)/2
(here ?-matrices have appropriate structure (see e.g. Boerner [2]). Using straight-
forward calculations one can make sure that the functions (18), stated below, satisfies
equation (1) as well as equation (2) if m = 0. This solution has been obtained by
virtue of operator QL which is a linear combination of the Lorentz rotation generators
QL = ?a J0a , a = 1, 2, 3,
(17)
1
= x0 ?a + xa ?0 ? ?0 ?a ,
J0a
2
? = A(x)?(?),
? ?
?3 ?? ?3 1 ?? 1
? s? ?
? ? s+ ?
? ? s+ ? s?
? ?
? ?
? ?+ ?
?3 ?+ 1 ?3 1
? ?
? ?
s s?
??+ ?
? ? s+ ? s?
A(x) = ? ?,
? ?
?s ?
1
? ?
0 0
+
? ?
s+ (18)
? ?
? ?
1
0 s? 0
s?
? ?
? ?1/2 [F0 cos(? + ?0 ) + iG0 sin(? + ?0 )]
? ? ?1/2 [F cos(? + ? ) + iG sin(? + ? )] ?
? ?
1 1 1 1
?(?) = ? ?,
? ?[G0 cos(? + ?0 ) + iF0 sin(? + ?0 )] ?
?[G1 cos(? + ?1 ) + iF1 sin(? + ?1 )]
where ? = {?1 , ?2 , ?3 }, ?0 , ?1 , F0 , F1 , G0 , G1 , c = 4(F0 G0 + F1 G1 ) > 0 are arbitrary
real constants:
? = (?x0 )2 ? (? · x)2 , s± = (?x0 ± ? · x)1/2 ,
1/2
?± = ?1 ± i?2 , 2 2 2
? = ?1 + ?2 + ?3 ,
mv
?ck (19)
? (1?k)/2 ?
?= ?, k = 1,
?(k ? 1) ?
mv
?c
? = ? ln ? ? ?, k = 1.
2? ?
This solution is also analytic in the coupling constant ? and in the mass term, and
c 4(F0 G0 + F1 G1 )
?(x)?(x) = v =
? (20)
,
1/2
[(?x0 )2 ? (? · x)2 ]
?
?
i.e. ?? dies off when x? x? > ?.
The next solution has been obtained by means of the operator
QLD = QL + ?D, ? = constant,
(21)
D = x? ?? ? 1/2k
which is admitted only by equation (2) and not by equation (1). We found an explicit
solution in the case k = 1 in such a form
328 W.I. Fushchych, W.M. Shtelen

?(x) = A(x)?(?),
? ?
?3 ?? ?3 ?+ s ?? ?+ s
? e?? s ? e?? s e e
? ?
? ? ? ?
? ?
? ?+ ?? s ?
?3 ?? s ?+ ?+ s ?3
? ?
A(x) = ? ? ? e ? e?+ s
e e ?,
? ? ?
? ?
? ?
e?? s e?+ s
? ?
0 0
e?? s e?+ s
0 0
(22)
? ?
?i?
i?
G0 ? + F0 ?
? ?
G1 ? i? + F1 ? ?i?
? ?
? ?
? ?
? ?
1/2
?+?
?(?) = ? ? ?1/2 ?,
G0 ? i? ? F0 ? ?i?
? ?
???
? ?
? ?
? ?
1/2
?+?
? ?1/2 G1 ? i? ? F1 ? ?i?
???

where ? = {?1 , ?2 , ?3 }, ? are arbitrary real constants and F0 , F1 , G0 , G1 are complex
ones:
1/2
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