ñòð. 74 |

e

to construct new multiparameter families of exact solutions of the equation starting

from those obtained above. The two following examples illustrate the procedure of

generating solutions.

?

V. Case k = 1/2, nonlinearity ?(??),

a·z

1

? = [(a · z)2 + (b · z)2 ]?1/4 exp ? (? · a)(? · b) tg?1 ?

b·z

2

??

? (4.5)

? exp ?i? (? · b + ?? · a) ?

2(1 + ?2 )

a·z

? ln (a · z)2 + (b · z)2 + 2? tg?1 ?,

b·z

Exact solutions of nonlinear Dirac’s and Schr?dinger’s equations

o 313

zµ = xµ + ?µ , aµ , bµ , ?, ?µ are arbitrary parameters satisfying conditions

a · a = aµ aµ = ?1, b · b = bµ bµ = ?1, a · b = 0,

? · a ? ?0 a0 ? ?1 a1 ? ?2 a2 ? ?3 a3 , ? = (?0 ? ?1 ? ?2 ? ?3 )1/2 .

2 2 2 2

?

VI. Case k = 1/2, nonlinearity ?(??)1/2k ,

a·z

1

?1/4

exp ? (? · a)(? · b) tg?1

?(x) = (a · z)2 + (b · z)2 ?

b·z

2

(4.6)

1?2k

2?k

? exp ?i (? · b)(??)1/2 (a · z)2 + (b · z)2 4k

? ?.

1 ? 2k

Formulas (4.5), (4.6) give multiparameter families of exact solutions of the Dirac

?

equations. These families are nongenerating with respect to the group P (1, 3) in the

sense that solutions (4.5), (4.6) have the same symmetrys as the equation (2.1).

5. Conformally invariant solutions. Conformally invariant Dirac–G?rsey equa-

u

tion has the form

?

?µ pµ + (??)1/3 ? = 0. (5.1)

With the help of a conformally invariant ansatz we can construct the following

solutions

?·x

exp{i?k(? · ?)?}?, k = 1/3,

?(x) =

(x · x)2

(5.2)

?µ xµ ? µ

?= , x? x = 0, ?µ ? > 0;

x? x?

?(x) = ? 2 (x)[1 ? (? · x)(? · ?)] ?

?

(? · a)(? · b)[b · x ? (b · ?)(x · x)]? ?1 (x) ?

? exp

2

i

? exp ? ?(??)1/3 (? · a)[2(a · x) ? (a · ?)(x · x)]?(x) +

?

(5.3)

2

+ ?(b · x ? (b · ?)(x · x))2 ? ?2 (x) ?,

a · b = b · b = 0, a · a = a2 ? a2 ? a2 ? a2 = ?1,

0 1 2 3

c2 = c? c? , x2 = xµ xµ .

Formulae (5.2), (5.3) give multiparameter families of exact solutions of the equati-

on (5.1). The family (5.3) is nongenerating with respect to the group C(1, 3).

If ?1 (x) is a solution of equation (5.1) then

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

(5.4)

xµ ? cµ x2

x= xµ =

?(x)

will also be a solution (5.4) is the formula for multiplication of solutions of Dirac

equation.

314 W.I. Fushchych

6. How to construct solutions of the nonlinear d’Alambert equation via

solutions of the Dirac equation? Complex scalar field can be represented as

? (6.1)

u(x) = ?? exp{i?(x)},

where ?(x) is a solution of the Dirac equation, ?(x) is a phase. In the simplest case,

when

? ?(x) = ?µ xµ

?? = c = const, (6.2)

formula (6.1) gives a plane-wave solution of the linear d’Alambert equation. In most

cases solutions of the nonlinear Dirac equation generate a scalar field (6.1) which

satisfies the nonlinear d’Alambert equation

pµ pµ u = k|u|r u, (6.3)

k, r are constants.

Let us exhibit some exact solutions of the equation (6.3) obtained this way

u(x) = c x2 + x2 exp{i?0 (x0 + x3 )}, (6.4)

r = 2,

1 2

?1/2

?

u(x) = c (x1 + ?1 (x0 + x3 ))2 + (x2 + ?2 (x0 + x3 ))2 x

(6.5)

? exp{i?0 (x0 + x3 )}, r = 2,

?2

u(x) = c x2 ? x2 ? x2 ? x2 (6.6)

, r = 1/2,

0 1 2 3

?1/2

u(x) = c x2 + x2 (6.7)

exp{i?(x0 + x1 )}, r = 2.

2 3

In formulae (6.4)–(6.7) ?0 , ?1 , ?2 , ? are arbitrary smooth functions.

So, solutions of the nonlinear Dirac spinor equation give a possibility to construct

solutions to the nonlinear d’Alambert equation.

All these ideas and results were considered in more detail by Fushchych (1981,

1987), Fushchych and Shtelen (1983, 1987), Fushchych, Shtelen and Zhdanov (1985),

Fushchych and Zhdanov (1987, 1988, 1989), Fushchych and Nikitin (1987) (see Ap-

pendix).

7. The solutions of the multidimensional Schr?dinger equation. Let us consi-

o

der the following nonlinear equation

? 1

? u + F (x, u, u? ) = 0,

? u ? u(x0 ? t, x1 , x2 , x3 ). (7.1)

i

?x0 2m

It is well known that if F = 0, then linear Schr?dinger equation (see Sofus Lie

o

(1881), Hagen (1972), Niederere (1972), Kalnins and Miller (1987)) is invariant under

the generalised Galilei group, which will be denoted by the symbols G2 (1, 3). The basis

elements of the Lie algebra AG2 (1, 3) = P0 , Pa , Jab , Ga , D, A, I have the following

form:

? ? ?

Pa = ?i Jab = xa Pb ? xb Pa ,

P0 = i , , a = 1, 2, 3, I=u ,

?x0 ?xa ?u

(7.2)

3 m

D = 2x0 P0 ? xa Pa + i, A = x0 D + x2 .

Ga = x0 Pa + mxa ,

2a

2

Exact solutions of nonlinear Dirac’s and Schr?dinger’s equations

o 315

Symbols AG1 (1, 3), AG(1, 3) denote the following Lie algebras

AG1 (1, 3) = P0 , Pa , Jab , Ga , D, I ,

AG(1, 3) = P0 , Pa , Jab , Ga , I .

To construct families of exact solutions of (7.1) in explicit form we have to know

the symmetries of (7.1) which obviously depends on the structure of the function F .

By Lie’s algorithm (as given by Ovsyannikov (1978), Olver (1986)), the following

statement can be proved.

Theorem 5. Equation (7.1) is invariant under the following algebras:

(7.3)

iff

AG(1, 3) F = ?(|u|)u,

where ? is arbitrary smooth function, and

F = ?|u|k u, (7.4)

iff

AG1 (1, 3)

where ?, k are arbitrary parameters, the operator of scale transformations D having

the form D = x0 P0 ? xa Pa + 2i/k, k = 0, and

F = ?|u|4/n u. (7.5)

iff

AG2 (1, 3)

Later on we shall construct the exact solutions of the equation (7.1) with nonli-

nearity (7.5), i.e.

p2

p0 ? a u + ?|u|4/3 u = 0. (7.6)

2m

Following Fushchych (1981) we seek solutions of (7.6) with the help of the ansatz

(7.7)

u = f (x)?(?1 , ?2 , ?3 ),

where ? is the function to be calculated. To construct solutions of (7.6) using ansatz

(7.7) it is necessary to have the explicit form of the function f (x) and the new

invariant variables ?1 , ?2 and ?3 . Next I shall present two Ans?tze of the type (7.7).

a

im x0 x 2

?3/4 ?1/2

f (x) = 1 ? x2 ?1 = ?x 1 ? x2

1. exp , ,

2 1 ? x2

0 0

0

?x

?1

?2 = x 2 1 ? x2 , ?3 = arctan x0 + arctan ,

0

?x

where ?, ?, ? are constant vectors satisfying the conditions

? 2 = ? 2 = ? 2 = 1, ?? = ?? = ?? = 0.

The Ans?tze (7.6), (7.7) give the following reduced equation

a

?? ??

? 2im + m2 ?2 ? ? 2?m|?|4/3 ? = 0,

L? + 6

??2 ??3

(7.8)

?2? ?2? ?2? ?2?

+ 4?2 2 + (?2 ? ?1 )?1 2 + 4?1

L? ? 2

.

2

??1 ??2 ??3 ??1 ??2

316 W.I. Fushchych

2. The second Ans?tze has the form

a

im 2 ?1

?3/2

exp ? (7.9)

f (x) = x0 x x0 ,

2

?x

?1 = (?x)x?1 , ?2 = x 2 x?2 , ?3 = x?1 + arctan (7.10)

.

0 0 0

?x

The Ans?tze (7.9), (7.10) reduce the equation (7.6) to

a

?? ??

? 2m|?|4/3 ? = 0.

L? + 6 + 2im

??2 ??3

8. Solutions of the equation (7.6). In this paragraph I present some explicit

solutions of the equation (7.6)

im 3

?3/4

x (1 ? x0 )?1 ,

u = 1 ? x2 (8.1)

exp ?= i;

0

2 2

im 2 ?1

u = (c0 x0 ? cx)?3/2 exp ? (8.2)

x x0 ,

2

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