<<

. 143
( 145 .)



>>




?
[?(?)]k1 d? + ?2 (C3 ? C1 )k2 ? + C4 ,
2 2
f2 = C3 cosh ?1

?
g1 = C1 sinh ??1 [?(?)]k1 d? ? ?2 (C3 ? C1 )k2 ? + C2 ,
2 2
(4.13)
?
[?(?)]k1 d? + ?2 (C3 ? C1 )k2 ? + C4 ,
2 2
g2 = C3 sinh ?1

? = ?(?) exp[i?(?)],
?(?)
[a? (z) + C6 ]?1/2 dz = ? + C7 , [?(?)]?1/2 d? + C8 ,
?(?) = C5

where
µ2 2(k1 +1)
+ µ2 (C3 ? C1 )2k2 z 2 +
1
?2 2 2
a? (z) = z
k1 + 1
µ1 µ2
?
(C 2 ? C1 )k2 z k1 +2 + 2C5 z;
2 2
+4
k1 + 2 3
(3) A(?) = 0, B(?) = 1:

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

?
[?(?)]k1 d? + ?2 (C1 ? C3 )k2 ? + C4 ,
2 2
f2 = C3 cos ?1

?
[?(?)]k1 d? + ?2 (C1 ? C3 )k2 ? + C2 ,
2 2
g1 = C1 cos ?1
(4.14)
?
[?(?)]k1 d? + ?2 (C1 ? C3 )k2 ? + C4 ,
2 2
g2 = C3 sin ?1

? = ?(?) exp[i?(?)],
?(?)
[a+ (z) + C6 ]?1/2 dz = ? + C7 , [?(?)]?1/2 d? + C8 ,
?(?) = C5

where
µ2
a+ (z) = ? z 2(k1 +1) ? µ2 (C1 ? C3 )2k2 z 2 ?
1
?2 2 2
k1 + 1
µ1 µ2
?
?4 (C1 ? C3 )k2 z k1 +2 + 2C5 z
2 2 2
k1 + 2
(in the above formulae C1 , . . . , C8 are arbitrary constants);
(4) A(?) = m? ?1 , B(?) = 1, m = 2, 3:
616 W.I. Fushchych, R.Z. Zhdanov

(a) k1 > 1/(m ? 1), k2 > 1/2m:

fn (?) = Cn ? ?1/2k2 , gn (?) = ?(?1)n (2k2 m ? 1)1/2 Cn ? ?1/2k2 ,
(4.15)
?(?) = E? ?1/k1 ,
n = 1, 2,

where C1 , C2 and E are constants satisfying the following conditions:
?2
[(1 ? m)k1 + 1]k1 + {µ1 |E|k1 + µ2 [2mk2 (C1 ? C2 )]k2 }2 = 0,
2 2
?
(4.16)
?
±(2k2 m ? 1)1/2 ? 2k2 {?1 |E|k1 + ?2 [2mk2 (C 2 ? C 2 )]k2 } = 0;
1 2

(b) k1 = 2(m ? 1)?1 , k2 > m?1 :

gn (?) = Cn (1 + ?2 ? 2 )?(m+1)/2 ,
fn (?) = (?1)n ??gn (?),
(4.17)
?(?) = E(1 + ?2 ? 2 )(1?m)/2 ,
n = 1, 2,

where the constants C1 , C2 and E satisfy the conditions

?2 (m2 ? 1) = [µ1 |E|2/(m?1) + µ2 (C1 ? C2 )1/m ]2 ,
2 2
?
(4.18)
?
(m + 1)? = [?1 |E|2/(m?1) + ?2 (C 2 ? C 2 )1/m ].
1 2

To obtain the exact solutions of the initial system (4.1) one has to substitute
formulae (4.6)–(4.10), (4.11)–(4.17) into the ansatz (4.2), (4.4). The obtained expres-
sions are very cumbersome and will not be given here.
Let us make some remarks.
? ?
Note 1. If one interprets the nonlinearities ?1 |u|k1 + ?2 (??)k2 , µ1 |u|k1 + µ2 (??)k2 ,
as the masses of a spinor field (M? ) and of a scalar field (Mu ) created because of the
nonlinear interaction of these fields, then for solutions (4.11), (4.15) and (4.17) the
following remarkable relations hold:
2
4k2 [1 + (1 ? m)k1 ]
2
Mu
= , m = 1, 2,
k1 (1 ? 2mk2 )
2
M?
2
4k2 [(m ? 1)k1 ? 1]
2
Mu
(4.19)
= , m = 2, 3,
k1 (2mk2 ? 1)
2
M?
2
m?1
Mu
= , m = 2, 3.
M? m+1
These relations can be interpreted as formulae for the mass spectrum of spinor and
scalar particles. What is more, the discrete variable m arises as the compatibility
condition of the over-determined system (4.3) (compare ref. [50]). So the mass spect-
rum is determined by the geometry of the solutions of the form (4.2), (4.4).
Note 2. If one puts in (4.2) g2 ? f2 ? 0, ?(x) = x · x, then the ansatz suggested by
Heisenberg [2, 14] is obtained,

?(x) = [ig1 (x · x) + ? · xf1 (x · x)]?.

Note 3. If one chooses ?1 = µ2 = 0 then formulae (4.2), (4.4), (4.6)–(4.18) give
exact solutions of the nonlinear Dirac–Heisenberg equation and of the d’Alembert
equations.
Symmetry and exact solutions of nonlinear spinor equations 617

Note 4. Ans?tze (4.2), (4.4) can be used to reduce nonlinear systems of PDE of more
a
general form than (4.1), namely
pµ pµ = F2 (??, u, u? ),
? ?
?µ pµ = F1 (??, |u|)?, (4.20)
where F1 , F2 are arbitrary continuous functions. In particular, the solutions of a
system of equations of the form (4.20) constructed in refs. [12, 13, 51] can be obtained
via ans?tze (4.2), (4.4).
a
5. Exact solutions of the nonlinear Maxwell–Dirac equations
There is a vast literature devoted to the system of equations of classical electrody-
namics (Maxwell–Dirac equations)
[?µ (pµ + eAµ ) + m]?(x) = 0,
(5.1)
?
p? p? Aµ ? pµ p? A? = e??µ ?, µ, ? = 0, 1, 2, 3,
where Aµ = Aµ (x) is the vector potential of the electromagnetic field; m and e
are the mass and the charge of the electron. A number of existence theorems have
been proved (in particular, in ref. [52] the solubility of the Cauchy problem has been
investigated). However, as far as we know there are no publications containing exact
solutions of this system in explicit form.
We look for solutions of eqs. (5.1) in the form
?(x) = (? · ?)?(?), (5.2)
Aµ (x) = ?µ ?(?), µ = 0, 1, 2, 3,
where ? = {?0 , ?1 , ?2 } ? {? · x, b · x, c · x}, ?µ = aµ + dµ , ?(?) and ?(?) are
unknown functions. Substitution of (5.2) into (5.1) gives rise to the following system
of two-dimensional PDE for ?(?) and ?(?)
(? · b)??1 + (? · c)??2 + im? = 0, (5.3a)
??1 ?1 + ??2 ?2 = 2e?(? · ?)?. (5.3b)
?
Let us note that in (5.3) there is no differentiation with respect to ?0 , therefore ? and
? contain ?0 as a parameter.
The general solution of eq. (5.3a) is given by the elliptic analogue of the d’Alembert
formula for the wave equation [38]
?1 +i(?2 ?? )
?2
?
?(?) = F (z, ?0 ) + F (z , ?0 ) ? ie ?(? · ?)?(?, ? )d?d?, (5.4)
?
?1 ?i(?2 ?? )
0

where F is an arbitrary analytical function of z = ?1 + i?2 . So the problem of
constructing particular solutions of the initial system of equations (5.1) is reduced to
that of integrating the linear two-dimensional Dirac equation (5.3a).
Choosing the eigenfunction of the Hermitian operator ?i??1 as a partial solution
of eq. (5.3a) one obtains
? = exp[i??1 + i? · c(m + ?? · b)?2 ]?0 (?0 ), (5.5)
where ?0 is a four-component spinor depending on ?0 in an arbitrary way. Imposing
on (5.5) the additional condition of being periodical with respect to the variable ?1 ,
we come to the following relation:
n ? Z. (5.6)
? = ?n = 2?n,
618 W.I. Fushchych, R.Z. Zhdanov

Substitution of (5.5) into formula (5.4) gives the explicit form of ?(?),
1
?(n) (?) = F (z, ?0 ) + F (z ? , ?0 ) + (m2 + ?2 )?1 ?
n
(5.7)
4
? [?1 cosh 2(m2 + ?2 )1/2 ?2 + ?2 sinh 2(m2 + ?2 )1/2 ?2 ], n ? Z,
n n

where
?1 = 2e?0 (? · ?)?0 ,
z = ?1 + i?2 , ?
?2 = 2ie(m2 + ?2 )?1/2 ?0 (? · ?)(m + ?n ? · b)?0 .
?
n

Substituting (5.5), (5.7) into the ansatz (5.2) one obtains a multi-parameter family of
exact solutions of the Maxwell–Dirac equations depending on three arbitrary complex
functions,
? (n) (x) = (? · a + ? · d) exp[i?n b · x + i? · c(m + ?n ? · b)c · x]?0 (a · x + d · x),
A(n) (x) = (aµ + dµ ) F (z, a · x + d · x) + F (z ? , a · x + d · x) + (5.8)
µ
1
+ (m2 + ?2 )?1 [?1 cosh(2(m2 + ?2 )1/2 c · x) + ?2 sinh(2(m2 + ?2 )1/2 c · x)] .
n n n
4
Analogously if one chooses the following solution of eq. (5.3a):
1 ?1
?(?) = (?1 + ?2 )?1/4 exp ? (? · b)(? · c) arctg ?
2 2
2 ?2
? exp[im(? · c)(?1 + ?2 )1/2 ]?0 (?0 )
2 2


as ?(?), then formulae (5.2), (5.4) give rise to the following family of exact solutions:
b·x
1
?(x) = (? · a + ? · d)|z|?1/2 exp ? (? · b)(? · c) arctg ?
c·x
2
? exp[im? · c|z|]?0 (a · x + d · x),
(5.9)
Aµ (x) = (aµ + dµ ) F (z, a · x + d · x) + F (z ? , a · x + d · x) +
|z|
(?1 sinh 2m? + ?2 cosh 2m?)??1 d? ,
+

where F is an arbitrary analytical function of z = b · x + ic · x,
|z| = (z ? z)1/2 = [(b · x)2 + (c · x)2 ]1/2 , ?1 = 2e?0 [? · a + ? · d)?0 ,
?
?2 = 2ie?0 (? · a + ? · d)(? · c)?0 .
?
Let us consider in more detail the solution of the Maxwell–Dirac equations (5.8)
putting
F ? 0, ?0 = exp[??2 (a · x + d · x)2 ]?,
where ? is an arbitrary constant spinor, ? = const. By direct verification one can

<<

. 143
( 145 .)



>>