ñòð. 141 |

= i?1 ? i?1 = 0.

The cases k = 2, 3, . . . , l ? 1 are treated in the same way.

Consequence. On the set of solutions of eq. (3.64) the following representation of

the Galilei algebra AG(1, l ? 1) is realized:

Pa = ?i??a , Jab = ?a Pb ? ?b Pa + Sab ,

P0 = i??0 ,

1

Ga = ?0 Pa + (Sal ? S0a ), a, b = 1, . . . , l ? 1.

2

Note 1. In general, the algebra (3.65) is not a maximal invariance algebra of (3.64).

As an example one can take eq. (13) of (3.16), whose symmetry is described by

proposition 1. Other examples are given in refs. [27, 29].

Note 2. Proposition 3 holds true for Poincar?-invariant generalizations of the Bhabha

e

equation of the form

[?µ pµ + F (?? , ?)]?(x) = 0.

This makes it possible to construct exact solutions of the above nonlinear equations

including arbitrary functions with the help of the procedure of generating solutions

[27, 29]. By a special choice of the arbitrary functions one can pick out classes of

solutions possessing some additional properties.

Symmetry and exact solutions of nonlinear spinor equations 607

Choosing in ?18 (x)

?0 ? exp[?2 (x0 + x3 )2 ], ?1 ? ?2 ? ?3 ? 0,

? = const,

one obtains the following solution of the Dirac–Heisenberg equation:

?(x) = exp[??2 (x0 + x3 )2 ][I + ?2 (x0 + x3 )(?1 x1 + ?2 x2 )(?0 + ?3 )] ?

(3.67)

? exp{i?(??)1/2 ?1 x1 exp[??2 (x0 + x3 )2 ]}?.

?

This solution is not localized in R3 but it is localized inside an infinite cylinder with

its axis parallel to the Ox3 axis. Moreover (3.67) decreases exponentially in all points

of R3 as x0 > +?.

Let us mention that for ? = 0 takes the form

?(x) = exp[i?(??)1/2 ?1 x1 ]?. (3.68)

?

Consequently, (3.67) can be considered as a perturbation of the stationary state (3.68).

3.4. Nongenerable families of solutions of the nonlinear Dirac equation. The so-

lutions ?1 (x)–?29 (x) depend on the variables xµ in an asymmetrical way, while in the

Dirac–Heisenberg equation all independent variables have equal rights. Using physical

language one can say that the system (3.34) is solved in some fixed reference system.

To obtain solutions (more precisely families of solutions) which do not depend on the

chosen reference system it is necessary to apply a procedure of generating solutions

by a group of transformations [21, 47]. This procedure is based on the following

statement.

Let eq. (3.34) be invariant under the group of transformations

(3.69)

? (x ) = A(x, ?)?(x), xµ = fµ (x, ?),

where A(x, ?) is an invertible 4 ? 4 matrix, ? = (?1 , . . . , ?r ) are group parameters.

Besides there is some solution ? = ?I (x) of eq. (3.34).

Proposition 4. The spinor ?II (x),

?II (x) = A?1 (x, ?)?I (f (x, ?)), (3.70)

satisfies eq. (3.34) too.

The proof can be found in refs. [21, 32].

We call formula (3.70) the solutions generating formula. Let us mention the solu-

tion generating formulae with transformations of the conformal group C(1, 3).

(1) The group of translations,

(3.71)

?II (x) = ?I (x ), xµ = xµ + ?µ , ?µ = const,

(2) the Lorentz group O(1, 3),

(a) the group of rotations

i

?II (x) = exp ? ?abc ?a Sbc ?I (x ),

2

??x ?(? · x) (3.72a)

x0 = x0 , x = x cos ? ? (1 ? cos ?),

sin ? +

?2

?

1

?k = const, ? = (? · ?)1/2 , Sab = i(?a ?b ? ?b ?a ),

4

608 W.I. Fushchych, R.Z. Zhdanov

(b) the Lorentz transformations,

1

?II (x) = exp ? ??0 ?a ?I (x ),

2

(3.72b)

x0 = x0 cosh ? + xa sinh ?, xa = xa cosh ? + x0 sinh ?,

xb = xb , b = a, a, b = 1, 2, 3, ? = const;

(3) the group of scale transformations,

?II (x) = ek? ?I (x ), xµ = e? xµ , (3.73)

? = const;

(4) the group of special conformal transformations,

?II (x) = ? ?2 (x)[1 ? (? · x)(? · ?)]?I (x ),

(3.74)

xµ = [xµ ? ?µ (x · x)]? ?1 (x), µ = 0, 1, 2, 3,

where ?(x) = 1 ? 2? · x + (? · ?)(x · x), ?µ = const.

As an example we shall consider the procedure of generating the solution ?1 (x),

the remaining cases being treated in an analogous way. Let us apply to ?1 (x) formula

(3.72b) with a = 3,

1

?(x) = exp ? ??0 ?3 exp[i?(??)1/2k (x3 cosh ? + x0 sinh ?)?3 ]?.

?

2

Rewriting the above formula in the equivalent form one obtains

1

?(x) = exp ? ??0 ?3 exp[i?(??)1/2k (x3 cosh ? + x0 sinh ?)?3 ]?

?

2

1 1

? exp ??0 ?3 exp ? ??0 ?3 ?.

2 2

Taking into account the identities

?

??0 cosh ? + ?3 sinh ?, µ = 0,

?

1 1

exp ? ??0 ?3 ?µ exp = ?3 cosh ? + ?0 sinh ?, µ = 3,

??0 ?3

?

2 2 ?

?µ , µ = 1, 2,

one has

?II (x) = exp[i?(??)1/2k (?3 cosh ? + ?0 sinh ?)(x3 cosh ? + x0 sinh ?)]?,

? ?

where ? = exp ? 2 ??0 ?3 ?. Using formula (3.72a) one comes to the following family

1

?

of solutions:

?II (x) = exp[i?(??)1/2k (? · d)(d · x)]?. (3.75)

?

Hereafter aµ , bµ , cµ and dµ are arbitrary real parameters satisfying the relations

?a · a = b · b = c · c = d · d = ?1, a · b = a · c = a · d = b · c = b · d = c · d = 0(3.76)

[in other words, the four-vectors a, b, c, d create an orthonormal basis in the Mi-

nkowski space R(1, 3)]. It is not difficult to verify that the family (3.74) is invariant

under the transformations (3.71), (3.73).

Symmetry and exact solutions of nonlinear spinor equations 609

Solution (3.75) depends on the variables xµ in a symmetrical way and its form

is not changed both under a transition from one inertial reference system to another

and under a change of the scale according to formula (3.73). In other words, we

?

have constructed a P (1, 3)-nongenerable family of solutions of the nonlinear Dirac–

Heisenberg equation (the corresponding definition is given in ref. [48]). The transition

from the solution ?1 (x) to the family of solutions (3.75) seems to be very important

because one obtains a class of exact solutions having the same symmetry as the

equation of motion (3.34).

?

Generating ?2 (x)–?5 (x) we obtain the following P (1, 3)-nongenerable families of

solutions of eq. (3.34):

?2 (x) = exp[?i?(??)1/2k (? · a)(a · x)]?;

?

1

?3 (x) = exp ? ?(? · a + ? · d)(? · b)(a · z + d · z) ?

2

1

? exp i?(??)1/2k (? · b)[2b · z + ?(a · z + d · z)2 ] ?;

?

2

1 i?

?4 (x) = exp ? ?(? · a + ? · d)(? · b)(a · z + d · z) exp (??)1/2k ?

?

2)

2 2(1 + ?

? (? · c ? ?? · b)[2(c · z ? ?b · z) ? ??(a · z + d · z)2 ] ?;

?b · z ? ? ln[?(a · z + d · z)]

(? · a + ? · d)? · b ?

?5 (x) = exp

2?(a · z + d · z)

1

? exp (? · a)(? · d) ln[?(a · z + d · z)] ?

2

? exp{[(? · c)(? · a + ? · d) + i?(??)1/2k ][? · c ? ?(? · a + ? · d)]?

?

? [c · z ? (?/?) ln[?(a · z + d · z)]]}?,

where zµ = xµ + ?µ ; ?, ?, ?, ?µ = const.

If in (3.34) k = 3/2, then the equation is invariant under the conformal group

C(1, 3). Therefore one can generate solutions by the transformations (3.74). Genera-

ting solutions ?1 (x)–?14 (x) (for k = 3/2) one comes to C(1, 3)-nongenerable families

of solutions. The corresponding formulae are omitted because of their cumbersome

character.

3.5. Conditionally invariant solutions of the Dirac–Heisenberg equation. As em-

phasized in refs. [37, 42] additional constraints enlarging the symmetry of the equati-

on are not necessarily differential ones. Let us impose on the solutions of PDE (3.34)

?

an algebraic condition ?? = 1, i.e., we consider the over-determined system

? ?

[?µ pµ ? ?(??)1/2k ]?(x) = 0, ?? = 1,

or

?

(?µ pµ ? ?)?(x) = 0, (3.77)

?? = 1.

Proposition 5. The system (3.77) is conditionally invariant under the operators

Q1 = p0 ? ??0 , Q2 = p3 ? ??3 .

610 W.I. Fushchych, R.Z. Zhdanov

Proof. According to the definition of conditional invariance it is to be proved that the

system

?

(?µ pµ ? ?)?(x) = 0, (3.78)

?? = 1, Q1 ? = 0

is invariant in the Lie sense [32] under the group of transformations generated by Q1 .

?

Acting on the system (3.78) with the extended operator Q1 [32] one obtains

?? ?

Q1 ?? = 0, Q1 (p0 ? ? ??0 ?) = 0,

?

Q1 (?µ pµ ? ? ??) = i??0 (?µ pµ ? ? ??) ? 2i?(p0 ? ? ??0 ?),

whence it follows that the statement holds true. The case of the operator Q2 is treated

in the same way.

Let us perform a reduction of the system (3.37) using the above statement. Integ-

ration of the equation Q1 ? = 0 yields the following ansatz:

(3.79)

?(x) = exp(?i??0 x0 )?(x).

Substituting (3.79) into (3.77) one obtains

(3.80)

?1 ?x1 + ?2 ?x2 + ?3 ?x3 = 0, ?? = 1.

?

Analogously integration of the equation Q2 ? = 0 yields the ansatz

(3.81)

?(x) = exp(i??3 x3 )?(x0 , x1 , x2 ),

?(x0 , x1 , x2 ) satisfying a PDE of the form

(3.82)

?0 ?x0 + ?1 ?x1 + ?2 ?x2 = 0, ?? = 1.

?

If one chooses in (3.80) ? = ?(x1 , x2 ), then the obtained two-dimensional PDE can

be integrated. Its general solution is given by

? = (?0 (z ? ), ?1 (z), ?2 (z ? ), ?3 (z))T ,

where ?1 , ?3 (?0 , ?2 ) are arbitrary analytical (anti-analytical) functions.

Imposing on ? the condition ?? = 1 [we use the form (1.2b) of the ?-matrices]

?

one comes to the following relation for ?µ :

|?) |2 + |?1 |2 ? |?2 |2 ? |?3 |2 = 1, |?µ |2 = ?µ? ?µ . (3.83)

Analogously choosing in (3.82) ? = ?(x0 , x1 ) and integrating the obtained equation

one has

? = (h0 + g0 , h0 ? g0 , h1 + g1 , ?h1 + g1 )T ,

where

hµ = h1 (x0 + x3 ) + ih2 (x0 + x3 ),

µ µ

gµ = gµ (x0 ? x3 ) + igµ (x0 ? x3 ),

1 2

µ = 0, 1,

hi , gµ are arbitrary smooth functions. From ?? = 1 it follows that hi , gµ satisfy the

i i

?

µ µ

equality

1

h1 g0 + h2 g0 + h1 g1 + h2 g1 =

1 2 1 2

(3.84)

.

1 1 0 0

4

Symmetry and exact solutions of nonlinear spinor equations 611

It is easy to convince oneself that (3.83), (3.84) can be written in the form

(3.85)

Al (?)Bl (?) = C

(summation over repeated indices from 1 to 4 is implied).

Lemma. The general solution of the algebraic equation (3.85) is given by formulae

Ak = ?k (?l ?l ) ? ?k (?l ?l ) + C?k /(?l ?l ), (3.86)

(a) Bk = ?k , k = 1, 2, 3, 4,

where ?k = const, ?k = ?k (?) are arbitrary functions;

(b) A1 = C1 ? + C4 , A2 = C2 ? + C5 , A3 = C3 ? + C6 , A4 = ?,

?1

ñòð. 141 |