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1 u4 2 1 u4 3

u4 u3 u2 u3 u4

? ? 2+

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

b b 2b b b 3b

u3 u2 u4

u1 u2 u4

? 2+ ? 4;

4

?3 = 3 8b4

b b 2b

2 2

2u1 = (F1 + F2 u2 ) exp ? u2 , 2u2 = bF2 exp ? u2 ,

18.

b b

2 2

2u3 = (F3 + F4 u4 ) exp ? u4 , 2u4 = cF4 exp ? u4 ,

c c

?1 = 2bu1 ? u2 , ?2 = 2cu3 ? u4 , ?3 = bu3 + cu1 ? u2 u4 ;

2 2

a?2

b b

2u1 = F1 cos

19. u4 + F2 sin u4 exp u4 ,

c c c

a?2

b b

2u2 = F2 cos u4 ? F1 sin u4 exp u4 ,

c c c

2

2u3 = (F3 + F4 u4 ) exp ? u4 ,

c

2

2u4 = cF4 exp ? u4 , b = 0, c = 0,

c

u4 u1 u4

?1 = ln(u2 + u2 ) ? 2a , ?2 = arctan ? b , ?3 = 2cu3 ? u2 ;

1 2 4

c u2 c

2uj = 0, j = 1, . . . , 4,

20.

where F1 , F2 , F3 , F4 are arbitrary smooth functions and a, b, c are arbitrary constants.

Furthermore, the basis generators Pµ , Jµ? are given by formulae (2) and generators

of corresponding groups of scale transformations are given by the following formulae:

1. D = xµ ?µ + ?1 u1 ?u1 + ?2 u2 ?u2 + ?3 u3 ?u3 + ?4 u4 ?u4 , ?1 = 0;

2. D = xµ ?µ + b?u1 + ?2 u2 ?u2 + ?3 u3 ?u3 + ?4 u4 ?u4 ;

3. D = xµ ?µ + ?1 (u1 ?u1 + u2 ?u2 ) + u2 ?u1 + ?2 u3 ?u3 + ?3 u4 ?u4 ;

4. D = xµ ?µ + u2 ?u1 + b?u2 + ?1 u3 ?u3 + ?2 u4 ?u4 ;

(9)

5. D = xµ ?µ + ?1 (u1 ?u1 + u2 ?u2 ) + u2 ?u1 + b?u3 + ?2 u4 ?u4 ;

6. D = xµ ?µ + ?1 (u1 ?u1 + u2 ?u2 + u3 ?u3 ) + u2 ?u1 + u3 ?u2 + ?2 u4 ?u4 ;

7. D = xµ ?µ + u2 ?u1 + u3 ?u2 + b?u3 + ?u4 ?u4 ;

= xµ ?µ + a1 (u1 ?u1 + u2 ?u2 ) + b1 (u2 ?u1 ? u1 ?u2 ) + ?1 u3 ?u3 + ?2 u4 ?u4 ;

8. D

Symmetry classi?cation of multi-component scale-invariant wave equations 47

= xµ ?µ + a1 (u1 ?u1 + u2 ?u2 ) + b1 (u2 ?u1 ? u1 ?u2 ) + c?u3 + ?u4 ?u4 ;

9. D

10. D = xµ ?µ + ?1 (u1 ?u1 + u2 ?u2 ) + u2 ?u1 + ?2 (u3 ?u3 + u4 ?u4 ) + u4 ?u3 ;

11. D = xµ ?µ + ?(u1 ?u1 + u2 ?u2 + u3 ?u3 + u4 ?u4 ) + u2 ?u1 + u3 ?u2 + u4 ?u3 ;

= xµ ?µ + a(u1 ?u1 + u2 ?u2 ) + b(u2 ?u1 ? u1 ?u2 ) +

12. D

+ ? (u3 ?u3 + u4 ?u4 ) + u4 ?u3 ;

= xµ ?µ a1 (u1 ?u1 + u2 ?u2 ) + b1 (u2 ?u1 ? u1 ?u2 ) +

13. D

+ a2 (u3 ?u3 + u4 ?u4 ) + b2 (u4 ?u3 ? u3 ?u4 );

= xµ ?µ + a1 (u1 ?u1 + u2 ?u2 ) + b1 (u2 ?u1 ? u1 ?u2 ) +

14. D

+ a2 (u3 ?u3 + u4 ?u4 ) + b2 (u4 ?u3 ? u3 ?u4 ) + u3 ?u1 + u4 ?u2 ;

15. D = xµ ?µ + ?(u1 ?u1 + u2 ?u2 + u3 ?u3 ) + u2 ?u1 + u3 ?u2 + b?u4 ;

16. D = xµ ?µ + u4 ?u3 + b?u4 + ?(u1 ?u1 + u2 ?u2 ) + u2 ?u1 , b = 0;

17. D = xµ ?µ + u2 ?u1 + u3 ?u2 + u4 ?u3 + b?u4 ;

18. D = xµ ?µ + u2 ?u1 + b?u2 + u4 ?u3 + c?u4 , b = 0, c = 0;

= xµ ?µ + a(u1 ?u1 + u2 ?u2 ) + b(u2 ?u1 ? u1 ?u2 ) + u4 ?u3 + c?u4 ;

19. D

20. D = xµ ?µ .

Theorem 2. System of PDEs (7) is invariant under the conformal group C(1, 3) i?

it is equivalent to the following system:

u1 u1 u1

2uj = u3 Fj ,, , j = 1, 2, 3, 4.

1

u2 u3 u4

Proofs of Theorems 1, 2 are carried out with the help of the in?nitesimal Lie

algorithm (see, e.g. [2, 5, 6]). Here we present the scheme of the proof of Theorem 1

only.

Within the framework of the Lie method, a symmetry operator for system of PDEs

(7) is looked for in the form

(10)

X = ?µ (x, u)?µ + ?j (x, u)?uj , j = 1, . . . , 4,

where ?µ (x, u), ?j (x, u) are some smooth functions.

The necessary and su?cient condition for system of PDEs (7) to be invariant under

the group having the in?nitesimal operator (10) reads

(11)

X(2uj + Fj ) = 0, j = 1, . . . , 4,

2ui ?Fi =0, i=1,...,4

where X stands for the second prolongation of the operator X.

Splitting relations (11) by independent variables, we get the Killing-type system

of PDEs for ?µ , ?k . Integrating it, we have:

?µ = 2xµ g?? x? k? ? kµ g?? x? x? + cµ? g?? x? + dxµ + eµ , µ = 0, . . . , 3,

4

(12)

akj uj + bk (x) ? 2g?? k? x? uk ,

?k = k = 1, . . . , 4.

j=1

48 W.I. Fushchych, P.V. Marko, R.Z. Zhdanov

Here k? , cµ? = ?c?µ , d, eµ , akj are arbitrary constants, bk (x) are arbitrary functi-

ons satisfying the following relations:

4 4

akl ul + bk (x) ? 2g?? k? x? uk Fjuk + 2bj (x) +

k=1 l=1

(13)

4

+ 2(d + 3g?? k? x? )Fj ? ajl Fl = 0, j = 1, . . . , 4.

l=1

From (12), (13) it follows that system of PDEs (7) is invariant under the Poincar? e

group P (1, 3) having the generators (2) with arbitrary F1 , F2 . To describe all functions

F1 , F2 such that system (7) admits the extended Poincar? group P (1, 3), one has to

e

solve two problems:

1) to describe all operators D of the form (10), (12) which together with opera-

tors (2) satisfy the commutation relations of the Lie algebra of the group P (1, 3) (see,

e.g. [2])

[P? , P? ] = 0, [P? , J?? ] = g?? P? ? g?? P? ,

[J?? , Jµ? ] = g?? J?µ + g?µ J?? ? g?µ J?? ? g?? J?µ ,

[D, J?? ] = 0, [P? , D] = P? , ?, ?, ?, µ, ? = 0, . . . , 3;

2) to solve system of PDEs (13) for each operator D obtained.

On solving the ?rst problem, we establish that the operator D has the form

? ?

4 4

? Aij uj + Bi ? ?ui , (14)

D = xµ ?µ +

i=1 j=1

where Aij , Bi are arbitrary constants.

As noted above, two operators D and D connected by the transformation (8)

(which does not alter the form of the operators Pµ , Jµ? ) are considered as equivalent.

Using this fact we can simplify substantially the form of the operator (14).

On making in (14) the change of variables (8) with ?j = 0, we have

? ?

4 4

? Aij uj + Bi ? ?ui ,

D = xµ ?µ +

i=1 j=1

where

?1

Aij = ?ij Aij ?ij ,

4

(15)

Bi = ?ik Bk , i = 1, 2, 3, 4.

k=1

As an arbitrary (4 ? 4)-matrix can be reduced to a Jordan form by transformation

(15), we may assume without loss of generality that the matrix Aij is in the Jordan

form. The further simpli?cation of the form of operator (14) is achieved at the expense

of transformation (8) with ?ik = 0.

As a result, the set of operators (14) is split into twenty equivalence classes, whose

representatives are adduced in (9).

Symmetry classi?cation of multi-component scale-invariant wave equations 49

Next, integrating corresponding system of PDEs (13), we get P (1, 3)-invariant

systems of equations given above.

Note that when proving Theorem 1, we solve a standard problem of the repre-

sentation theory, namely, we describe inequivalent representations of the extended

Poincar? group which are realized on the set of solutions of system of PDEs (7). But

e

the representation space (i.e., the set of solutions of system (7)) is not a linear vector

space, whereas in the standard representation theory it is always the case. This fact

makes impossible a direct application of the methods of the classical theory of linear

group representations [7].

1. Fushchych W.I., Serov N.I., J. Phys. A, 1983, 16, 3645.

2. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations

of nonlinear mathematical physics, Dordrecht, Kluwer Academic Publ., 1993.

3. Fushchych W.I., Yehorchenko I.A., J. Phys. A, 1989, 22, 2643.

4. Zhdanov R.Z., Fushchych W.I., Marko P.V., Physica D, 1996, 95, 158–162.

5. Olver P., Applications of Lie groups to di?erential equations, New York, Springer, 1986.

6. Ovsjannikov L.V., Group analysis of di?erential equations, Moscow, Nauka, 1978.

7. Barut A.O., Raczka R., Theory of group representations and applications, Warszawa, Polish

Scienti?c Publ., 1980.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 50–55.

New scale-invariant nonlinear di?erential

equations for a complex scalar ?eld

R.Z. ZHDANOV, W.I. FUSHCHYCH, P.V. MARKO

We describe all complex wave equations of the form 2u = F (u, u? ) invariant under

the extended Poincar? group. As a result, we have obtained the ?ve new classes of

e

P (1, 3)-invariant nonlinear partial di?erential equations for the complex scalar ?eld.

It is well-known that the maximal symmetry group admitted by the nonlinear

wave equation

2u ? ux0 x0 ? (1)

3u = F (u)

with an arbitrary smooth function F (u) is the 10-parameter Poincar? group P (1, 3)

e

having the following generators:

Jµ? = gµ? x? ?? ? g?? x? ?µ , (2)

P µ = ?µ ,

where ?µ = ?/?xµ , gµ? = diag(1, ?1, ?1, ?1), µ, ?, ? = 0, 1, 2, 3. Hereafter, the

summation over the repeated indices from 0 to 3 is understood.

As established in [1] Eq. (1) admits a wider symmetry group only in the two cases:

(1) F (u) = ?uk , (3)

k = 1,

(2) F (u) = ?eku , (4)

k = 0.

where ?, k are arbitrary constants.

Eqs. (1) with nonlinearities (3) and (4) admit the one-parameter groups of scale

transformations D(1) having the following generators:

2

(1) D = xµ ?µ + u?u ,

1?k (5)

2

(2) D = xµ ?µ ? ?u .

k

The 11-parameter transformation group with generators (2) and (5) is called the

extended Poincar? group P (1, 3).

e

The above result admits the following group-theoretical interpretation: on the set

of solutions of the nonlinear wave equation (1) two inequivalent representations of

the extended Poincar? group are realized. Each representation gives rise to a P (1, 3)-

e

nonlinear wave equation with a very speci?c nonlinearity.

Surprisingly enough, there is no an analogous result for the complex nonlinear

wave equation

2u = F (u, u? ) (6)

Physica D, 1996, 95, P. 158–162.

New scale-invariant nonlinear di?erential equations 51

which is a more realistic model for describing a charged meson ?eld in the modern

quantum ?eld theory. Eq. (6) admits the Poincar? group with generators (2) under

e

?

arbitrary F (u, u ). It is natural to formulate the following problem: to describe all

functions F such that the said equation admits wider symmetry groups. We are

interested in those equations of the form (6) which are invariant under the natural

extensions of the Poincar? group — the extended Poincar? and the conformal groups.

e e

A usual approach to the description of partial di?erential equations admitting

some Lie transformation group is to ?x a representation of the group and then use

the in?nitesimal Lie method (see, e.g. [2, 3]) to obtain an explicit form of the unknown

function F . In this way in the paper [4] two classes of P (1, 3)-invariant equations of the

form (6) were constructed. But this approach may result in loosing some subclasses

of invariant equations (which is the case for the paper mentioned). It means that one

should not ?x a priori a representation of the group. The only thing to be ?xed is the

commutational relations of the corresponding Lie algebra. This approach guarantees

that all equations admitting a given group will be obtained.

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