ñòð. 143 |

?

[?(?)]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

ñòð. 143 |