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B3 = C6 (C ? C4 ?1 ? C5 ?2 ), (3.87)
B1 = ?1 , B2 = ?2 ,
?1
? C1 C6 )?1 + (C3 C5 ? C2 C6 )?2 ? CC3 ],
B4 = C6 [(C3 C4

where C1 , . . . , C6 are constants, ? = ?(?), ?i = ?i (?) are arbitrary functions;
(c) two other classes of solutions are obtained via the transposition Ak > Bk ,
Bk > Ak in formulae (3.86), (3.87).
The proof is rather formal; therefore it is omitted.
Using formulae (3.79), (3.81), (3.86) and (3.87) we constructed the following
classes of exact solutions of the initial PDE:
? iC1 ?
e
?eiC2 ?(z) ?
? ?
?(x) = exp(?i??0 x0 ) ? iC4 ?, (3.88)
?
?e ?(z ) cos C3 ?
eiC5 ?(z) sin C3

where {C1 , . . . , C5 } ? R1 , ? is an arbitrary analytical function,
? ?
A3 + B1 + i(A4 + B2 )
?A ? B + i(A ? B ) ?
?3 ?
1 4 2
?(x) = exp(i??3 x3 ) ? ?, (3.89)
?A1 + B3 + i(A2 + B4 ) ?
?A1 + B3 + i(?A2 + B4 )

where the real functions Al (x0 + x1 ), B l (x0 ? x1 ) are determined by formulae (3.86),
(3.87) with C = 1/4.
It is worth noting that solutions (3.88), (3.89) are essentially different from ?1 (x)–
?29 (x). They cannot be obtained with the help of the ansatze in tables 1 and 2.
3.6. On scalar fields generated by the solutions of the nonlinear Dirac–Heisen-
berg equation. In this subsection we construct a scalar field with spin s = 0 using
the exact solutions of the nonlinear Dirac–Heisenberg equation for a spinor field. The
solutions obtained in this way prove to satisfy the nonlinear d’Alembert equation.
The scalar field generated by the solutions of PDE (3.34) is looked for in the form
?
u(x) = ??ei?(x) , (3.90)

where ?(x) is the phase of the field u(x). For ?1 –?10 we have the equality
?
?? = const,
612 W.I. Fushchych, R.Z. Zhdanov

whence it follows that u(x) = Cei?(x) . Choosing ?(x) = ? aµ xµ , ? = const, one obtains
the plane-wave solution
µ
u(x) = Cei? aµ x . (3.91)
So the spinors ?1 –?10 generate plane-wave solutions of the form (3.91) sastisfying
the following equation:
|u|2 = u? u.
pµ pµ u(x) = F (|u|)u(x), (3.92)
We did not succeed in establishing a correspondence between the spinor fields ?22 ,
?24 and a scalar field u(x). Spinor ?19 generates a scalar field of the form
u(x) = C[?0 (x0 + x3 )]?1 [(x1 + ?1 )2 + (x2 + ?2 )2 ]?1/2 exp[i?3 (x0 + x3 )],
where ?µ are arbitrary smooth functions of x0 + x3 . It is easy to check that the
above function satisfies the nonlinear wave equation with variable coupling constant
?(x) = ?[?0 (x0 + x3 )]2 , ? = const, i.e.,
?
pµ pµ u(x) = ?[?0 (x0 + x3 )]2 |u|2 u(x). (3.93)
?
The remaining solutions of the nonlinear Dirac–Heisenberg equation (3.34) gene-
rate scalar fields satisfying the nonlinear d’Alembert equation
pµ pµ = ?|u|? u, (3.94)
? = const.
The corresponding results are given in table 3.
Table 3
No. u(x) ?
C(x2 + x2 )?1/2 exp[i?0 (x0 + x3 )]
11 2
1 2
2 2 ?1/2
C[(x1 + ?1 ) + (x2 + ?2 ) ] exp[i?0 (x0 + x3 )]
12 2
C(x2 x2 )?1/2 exp[i?(x0
+ + x1 )]
13 2
2 3

C(x2 x2 ? x2 )?1
?
14 1
0 1 3
?3/2
C(x · x)
15 2/3
C(x2 x2 )?1/2
?
16 2
0 3

C(x2 + x2 )?1/2 exp[i?(x0 + x1 )]
17 2
2 3
?2
C?0 (x0 + x3 ) exp[i(x1 + ?1 )]
18 0
C(x2 ? x2 ? x2 )?2
20 1/2
0 1 2

C(x0 ? x1 ? x2 )?2
2 2
21 1/2
2

C(x1 + x2 + x2 )?2
2 2
23 1/2
3

C(x0 ? x1 ? x2 )?1
2 2
25 1
3

C(x2 x2 )?1/2
+ exp[i?(x0 + x1 )]
26 2
2 3
?3/2
C(x · x)
27 2/3
?k
2
C [x3 + ?(x0 ? x2 )]2 + x1 + 1 (x0 ? x2 )2 1/k, k < 0
28 2
?1
2
C [x3 + ?(x0 ? x2 )]2 + x1 + 1 (x0 ? x2 )2
29 1
2

?0 , ?1 , ?2 are arbitrary smooth functions of x0 + x3 , ? of x0 + x1 ;
C and ? are constants.
Symmetry and exact solutions of nonlinear spinor equations 613

Thus the spinors ?1 –?29 generate complex scalar fields satisfying the nonlinear
d’Alembert equation (3.94). Let us note that (3.94) with ? = 2 admits the conformal
group C(1, 3). Consequently the fields u(x) generated by the spinors ?11 –?13 , ?16 ,
?17 and ?26 satisfy the conformally invariant d’Alembert equation [though the Dirac–
Heisenberg equation may not be invariant under the group C(1, 3)].
Another interesting feature inherent to the fields u(x) is that u(x) > 0 as x0 =
const, |x| > +? (the only exception is ?28 ). What is more, all the functions u(x)
have a nonintegrable singularity.
4. Exact solutions of the system
of nonlinear Klein–Gordon–Dirac equations
In this section we construct multi-parameter families of exact solutions of the
system of PDE describing the interaction of the spinor field ?(x) and the complex
scalar field u(x),
? ?
?µ pµ ? = [?1 |u|k1 + ?2 (??)k2 ]?, pµ pµ u = [µ1 |u|k1 + µ2 (??)k2 ]2 u, (4.1)
where x = (x0 , x1 , x2 , x3 ), |u| = (uu? )1/2 , ?1 , ?2 , µ1 , µ2 , k1 , k2 are constants.
Let us note that for ?1 = µ2 = 0, k1 = k2 = 0 the system of equations (4.1)
decomposes into the Dirac equation with mass ?2 and the Klein–Gordon equation
with mass µ1 . For ?1 = µ2 = 0, k1 = 1, k2 = 1/3 one obtains the nonlinear
conformally invariant Dirac–G?rsey [36] and d’Alembert [49] equations.
u
With the help of the Lie method one can prove that the system of equations (4.1)
for arbitrary, non-null k1 , k2 is invariant under the extended Poincar? group. For e
k1 = 1, k2 = 1/3 then (4.1) is invariant under the conformal group C(1, 3). The above
facts make it possible to apply the technique of group-theoretical reduction (as was
done in the previous section). But we use another approach which essentially uses the
connection between spinor and scalar fields established earlier and the ansatz
?(x) = {ig1 (?) + g2 (?)?4 ? [if1 (?) + f2 (?)?4 ]?µ pµ ?}?, (4.2)
where g1 , g2 , f1 , f2 are unknown real functions, ? = ?(x) is a scalar function
satisfying the system of PDE
A, B : R1 > R1 .
pµ pµ ? + A(?) = 0, (pµ ?)(pµ ?) + B(?) = 0, (4.3)
Ansatz (4.2) was suggested in refs. [23, 24] for the purpose of constructing exact
solutions of the nonlinear Dirac equation. As shown in ref. [28] it can be used to
obtain solutions of the system (4.1). The scalar field u(x) is looked for in the form
?
u(x) ? C(??), ? ? C 2 (R1 , C2 ).
C = const or (4.4)
u(x) = ?(x),
Substitution of expressions (4.2), (4.4) into (4.1), ? = ?(x) satisfying (4.3), gives
rise to the following system of ODE for gi , fi and ?:
? ?
B ? + A? = ?{µ1 |?|k1 + µ2 [g1 ? g2 + B(f1 ? f2 )]k2 }2 ?,
?2 2 2 2

B f?1 + Af1 = {?1 |?|k1 + ?2 [g 2 ? g 2 + B(f 2 ? f 2 )]k2 }g1 ,
?
1 2 1 2
?
g1 = ?{?1 |?| + ?2 [g1 ? g2 + B(f1 ? f2 )] }f1 ,
k1 2 2 2 2 k2 (4.5)
?
?2
g2 = {?1 |?|k1 + ?2 [g1 ? g2 + B(f1 ? f2 )]k2 }f2 ,
2 2 2
?
B f?2 + Af2 = ?{?1 |?|k1 + ?2 [g1 ? g2 + B(f1 ? f2 )]k2 }g2 ,
?2 2 2 2
614 W.I. Fushchych, R.Z. Zhdanov

?
where ?2 = ?2 (??)k2 , µ2 = µ2 (??)k2 , dot means differentiation with respect to ?.
? ? ?
The system of equations (4.3) is over-determined. Therefore one has to investigate
its compatibility. The compatibility of three-dimensional systems of the form (4.3)
was investigated in detail by C. Collins [50]. He has proved that the system (4.3) is
compatible iff

B(?) ? 0, A(?) ? 0;
(1)
B(?) = ±1, A(?) = N (? + ?)?1 , N = ?1, 0, 1, 2.
(2)

In each case the general solution was constructed.
Generalizing Collins’ results to the four-dimensional case we obtain the following
classes of particular solutions of the system of equations (4.3):
(1) A(?) = ?m? ?1 , B(?) = ?1, m = 1, 2:

? = [(b · y)2 + (c · y)2 + (d · y)2 ]1/2 , (4.6)
m = 2,

? = [(b · y + ?1 )2 + (c · y + ?2 )2 ]1/2 , (4.7)
m = 1;

(2) A(?) = 0, B(?) = ?1:

? = (b · y) cos ?1 + (c · y) sin ?1 + ?2 , a · y = (b · y) cos ?3 + (c · y) sin ?3 + ?4 ;(4.8)

(3) A(?) = 0, B(?) = 1:

? = a · y; (4.9)

(4) A(?) = m? ?1 , B(?) = 1, m = 1, 3:

? = [(a · y)2 ? (b · y)2 ]1/2 , m = 1,
? = [(a · y)2 ? (b · y)2 ? (c · y)2 ]1/2 , (4.10)
m = 2,
? = (y · y)1/2 , m = 3,

In (4.6)–(4.10) yµ = xµ + ?µ , ?µ = const; ?1 , ?2 are arbitrary smooth functions of
a · y + d · y, ?3 , ?4 of ? + d · y; aµ , bµ , cµ , dµ are arbitrary real parameters satisfying
(3.76).
We have succeeded in obtaining the general solution of the system of ODE (4.5)
for A(?) = 0, while in the remaining cases partial solutions are obtained. Let us give
the final result:
(1) A(?) = ?m? ?1 , B(?) = ?1, m = 1, 2:

fn (?) = Cn ? ?1/2k2 , gn (?) = ?(?1)n (1 ? 2k2 m)1/2 Cn ? ?1/2k2 ,
(4.11)
?(?) = E? ?1/k1 ,
n = 1, 2,

the constants k1 , k2 , C1 , C2 and E satisfying the conditions
?2
[(m ? 1)k1 ? 1]k1 + {µ1 |E|k1 + µ2 [2mk2 (C1 ? C2 )]k2 }2 = 0,
2 2
?
?
±(1 ? 2k2 m)1/2 ? 2k2 {?1 |E|k1 + ?2 [2mk2 (C 2 ? C 2 )]k2 } = 0, (4.12)
1 2
k1 < 1/(m ? 1);
k2 < 1/2m,
Symmetry and exact solutions of nonlinear spinor equations 615

(2) A(?) = 0, B(?) = ?1:

?
f1 = C1 cosh ??1 [?(?)]k1 d? ? ?2 (C3 ? C1 )k2 ? + C2 ,
2 2

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