ñòð. 134 |

where gµ? = diag (1, ?1, ?1, ?1) is the metric tensor of Minkowski space.

Theorem 1. Equation (1.5) is Poincar? invariant iff

e

? (2.3)

F (?, ?) = F1 + F2 ?4 ,

? ?

where ? = ? + ?0 , ?4 = ?0 ?1 ?2 ?3 , F1 and F2 are arbitrary scalar functions of ??

?

and ??4 ?.

We give only a sketch of the proof, which is based on the infinitesimal Lie method

?

[32–34]. Expanding the matrix F (?, ?) in a linear combination of ?-matrices, the

?

coefficients of the expansion depending ? and ?,

1

i(?µ ?? ? ?? ?µ ),

F = aI + bµ ? µ + cµ? Sµ? + dµ ?4 ?µ + e?4 , (2.4)

Sµ? =

4

and using the invariance criterion, one obtains the following necessary and sufficient

conditions for the Poincar? invariance of eq. (1.5):

e

Q0k a = 0, Q0k e = 0,

Q0k bµ + b? (g?0 gµk ? g?k gµ0 ) = 0, Q0k dµ + d? (g?0 gµk ? g?k gµ0 ) = 0, (2.5)

µ? µ? µ? µ?

Q0k cµ? + c?? (g?k ??0 + g?0 ??k ? g?0 ??k ? g?k ??0 ) = 0,

where

? ?

iQ0k = ?(S0k ?)? ?/?? ? + (?S0k )? ?/? ? ? , k = 1, 2, 3,

µ? µ?

??? = ?? ?? ? ?? ?? , ?, ?, µ, ? = 0, 1, 2, 3,

µ?

?

and ?? is the Kronecker symbol.

After some cumbersome calculations we obtain the following general solution of

the system of PDE (2.5):

? ?

a = A, e = E, bµ = B1 ??µ ? + B2 ??µ ?4 ?,

(2.6)

? ? ? ?

cµ? = C1 ?Sµ? ? + C2 ??4 Sµ? ?, dµ = D1 ??µ ? + D2 ??4 ?µ ?,

? ?

where A, B1 , . . . , E are arbitrary smooth functions of ?? and ??4 ?.

Symmetry and exact solutions of nonlinear spinor equations 581

Substituting the above formulae into (2.4) one obtains the following expression for

?

the nonlinear term F (?, ?)?:

? ? ? ? ?

F (?, ?)? = {AI + [B1 ??µ ? + B2 ??4 ?µ ?]? µ + [C1 ?Sµ? ? + C2 ??4 Sµ? ?]S µ? +

? ?

+ [D1 ??µ ? + D2 ??4 ?µ ?]?4 ? µ + E?4 }?.

This formula can be essentially simplified with the help of the identity [35]

? ? ?

(?1 ?µ ?2 )? µ ?2 = (?1 ?2 )?2 + (?1 ?4 ?2 )?4 ?2 ,

where ?1 and ?2 are arbitrary four-component spinors, and as a result the nonlinearity

?

F (?, ?) takes the form (2.3). This completes the proof.

Note. In the same way one can prove that the second-order spinor equation

?

pµ pµ ? = F (?, ?)? (2.7)

?

is invariant under the Poincar? group iff F (?, ?) has the form (2.3).

e

?

Theorem 2 [29]. Equation (1.5) is invariant under the Weyl group P (1, 3) iff

?

F (?, ?) has the form (2.3), Fi being determined by the formulae

? ?

Fi = (??)1/2k Fi , (2.8)

i = 1, 2,

??

??

where F1 , F2 are arbitrary functions of ??/??4 ?.

Theorem 3 [29]. Equation (1.5) is invariant under the conformal group C(1, 3) iff

?

F (?, ?) has the form (2.3), (2.8) with k = 3/2.

The proof of the last two statements is obtained with the help of the Lie method

[32–34]; it is omitted here. Let us note that the sufficiency in theorem 3 can be

established by direct verification. To do this we denote by G the following expression:

? ?

? ?

G(?, ?) = ?µ pµ ? + (F1 + F2 ?4 )(??)1/3 ?.

One can verify that the following identities hold:

G(? , ? ) = e?5?/2 G(?, ?),

? ?

?

if ? , ? have the form (2.1) with k = 3/2,

? ?

G(? , ? ) = ? 2 (x)[1 ? (? · ?)(? · x)]G(?, ?),

?

if ? , ? have the form (2.2). Consequently, the equation G = 0 is invariant under the

groups of transformations (2.1), (2.2).

Note 1. Unlike eq. (1.5), the class of equations (2.7) does not include conformally

invariant ones. Therefore it seems reasonable to consider as an equation of motion for

a spinor field the following second-order equation:

? ?

pµ pµ ? = ?(?, ?, ?1 , ?1 ),

(2.9)

? ?

?1 = {??/?xµ , µ = 0, 1, 2, 3}, ?1 = {? ?/?xµ , µ = 0, 1, 2, 3}.

The problem of a complete group-theoretical classification of eqs. (2.9) will be

considered in a future paper. Here we restrict ourselves to an example of a conformally

invariant equation of the form (2.9),

pµ pµ ? ? (3??)?1 ?µ [pµ (??)]?? p? ? = 0.

? ? (2.10)

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

It is worth noting that each solution of the nonlinear Dirac–G?rsey equation [36]

u

satisfies the PDE (2.10).

Note 2. There exist Poincar?-invariant first-order equations which differ principally

e

from the Dirac equation. An example is [29, 37]

?

(?? µ ?)pµ ? = 0. (2.11)

On the set of solutions of the system (2.11) a representation of an infinite-dimensional

Lie algebra is realized. This fact enables us to construct the general solution of eq.

(2.11) in implicit form,

?

f ? (xµ (j · j) ? jµ (j · x), ?, ?) = 0, ? = 0, 1, 2, 3,

?

where jµ = ??µ ?, f ? : R ? C8 > C1 are arbitrary smooth functions.

3. Exact solutions of the nonlinear Dirac equation

According to refs. [23, 24, 37] a solution of eq. (1.5) is looked for as a solution of

the following overdetermined system of PDE:

?

?µ pµ ? + F (?, ?)? = 0,

(3.1)

?

µ

?a ?xµ + ?a (x, ?, ?)? = 0, a = 1, 2, 3,

? ?

where ?a (x, ?, ?) are arbitrary 4 ? 4 matrices, ?a (x, ?, ?) are scalar functions sati-

µ

sfying the condition

?

rank {?a (x, ?, ?)} = 3.

µ

(3.2)

The PDE (3.1) is a system of sixteen equations for four functions ? 0 , ? 1 , ? 2 , ? 3 .

Therefore one has to investigate its compatibility (see also refs. [31, 39, 40]).

Theorem 4. System (3.1) is compatible iff it is invariant under the one-parameter

Lie groups generated by the operators

Qa = ?a ?/?xµ ? (?a ?)? ?/?? ? ,

µ

a = 1, 2, 3.

The main steps of the proof are as follows. Firstly, using condition (3.2) one

reduces the system (3.1) to the equivalent system (to simplify the calculations we

suppose that ??µ /?? ? = ??a /?? µ = 0)

a ??

?

?µ pµ ? + F (?, ?)? = 0,

(3.1 )

?

Qa ? ? (?/?xa + ?? ?/?x0 + ?a )? = 0.

?

It is not difficult to verify that system (3.1 ) admits groups generated by the operators

?

Qa iff the initial system admits groups generated by the operators Qa , while the

following relations hold:

?? (3.3)

[Qa , Qb ] = 0, a, b = 1, 2, 3.

It follows from the general theory of Lie groups that there exists the change of

variables

?(z) = ?(x)?(x), zµ = fµ (x), µ = 0, 1, 2, 3

Symmetry and exact solutions of nonlinear spinor equations 583

?

which reduces the operators Qa satisfying conditions (3.3) to the form

?

? ?

Qa > Qa = ?/?za . (3.4)

System (3.1 ) is rewritten in the following way:

?

??/?z0 = F1 (z, ?, ?)?,

(3.1 )

??/?za = 0.

The necessary and sufficient conditions for the compatibility of the system (3.1 )

are as follows:

? 2 ?/?zµ ?z? = ? 2 ?/?z? ?zµ . (3.5)

Applying these conditions to (3.1 ) one has

(3.6)

?F1 /?za = 0, a = 1, 2, 3,

??

whence the invariance of system (3.1 ) under the operators Qa = ?/?za follows.

The reverse statement is also true — if system (3.1 ) is invariant under the groups

?

?

generated by the operators Qa , then conditions (3.6) hold. Consequently, the initial

system is invariant under the operators Qa . The theorem is proved.

Consequence. Substitution of the ansatz

(3.7)

?(x) = A(x)?(?),

where the 4 ? 4 matrix A(x) and the scalar function ?(x) satisfy the system of PDE

µ

(3.8)

?a ??/?xµ = 0,

µ

(3.9)

Qa A(x) = [?a ?/?xµ + ?a (x)]A(x) = 0,

into eq. (1.5) gives rise to a system of ODE for ? = ?(?).

Proof. Integration of the last three equations of (3.1 ) yields

? = ?(z0 ).

Returning to the original variables x and ?(x), one has

?(x) = [?(x)]?1 ?(z0 ).

Choosing A(x) = [?(x)]?1 , ? = z0 (x), one obtains the statement required, the ODE

for ?(?) having the form

d?/d? = F1 (?, ?, ?)?.

?

Note. If (Q1 , Q2 , Q3 ) is a three-dimensional invariance algebra of PDE (1.5), then the

conditions of theorem 4 are evidently satisfied. Therefore the classical result on the

reduction of PDE to ODE via Qa -invariant solutions [32–34, 41] follows from theorem

4 as a particular case. If Qa are not the symmetry operators then the reduction is

done via conditionally Qa -invariant solutions [31, 39, 40, 42].

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

3.1. Ans?tze for the spinor field. In the following we shall consider spinor equati-

a

ons (1.5) with the nonlinearity (2.3), i.e. Poincar?-invariant systems of the form

e

?

?µ pµ ? = ?(?, ?) = (F1 + F2 ?4 )?. (3.10)

On the set of solutions of system (3.10) the following representation of the Poincar?

e

algebra AP (1, 3) is realized:

Jµ? = xµ p? ? x? pµ + Sµ? . (3.11)

P µ = pµ ,

Using theorem 4 and the group-theoretical properties of eq. (3.10) one can for-

mulate the following algorithm for the reduction of the PDE (3.10) to systems of

ODE.

At the first step one has to describe (to classify) all inequivalent three-dimensional

algebras which are subalgebras of the Poincar? algebra (3.11). As a result we obtain

e

a set of triplets of operators (Q1 , Q2 , Q3 ), each of which determines an ansatz of the

form (3.7).

At the second step the system of equations (3.8), (3.9) is integrated. According

to the consequence of theorem 4 substitution of the obtained ansatze into the initial

equation yields systems of ODE for the unknown function ? = ?(?).

The efficiency of group-theoretical methods is ensured, first of all, by the fact

that intermediate problems to be solved are linear. At the first step linear systems of

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