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the extended Poincar? group P (1, 1) using the approach described above. They have

e

classi?ed second-order P (1, 1)-invariant equations within the change of independent

and dependent variables.

In the present paper we will describe within the a?ne transformations all equa-

tions belonging to the class (6) which are invariant under the 11-parameter extended

Poincar? group.

e

Putting u = u1 + iu2 , u? = u1 ? iu2 we rewrite the complex equation (6) as a

system of two real equations

2uj = Fj (u1 , u2 ), (7)

j = 1, 2.

Before formulating the principal assertions we make a remark. As a direct check

shows, the class of Eqs. (7) is invariant under the linear transformations of dependent

variables

2

uj > uj = (8)

?jk uk + ?j ,

k=1

where ?jk , ?j , j = 1, 2 are arbitrary constants with det ?jk = 0.

That is why we carry out symmetry classi?cation of Eqs. (7) within the equivalence

transformations (8).

Theorem 1. The system of partial di?erential equations (7) is invariant under the

extended Poincar? group P (1, 3) i? it is equivalent to one of the following systems:

e

(a?2)/a

2u1 = u1

(i) F1 (?),

F2 (?), ? = ub u?a ;

(b?2)/a

2u2 = u1 12

u1 u1

2u1 = exp (a ? 2)

(ii) F1 (?) + F2 (?) ,

u2 u2

u1 u1

2u2 = exp (a ? 2) F2 (?), ? = a ? ln u2 ;

u2 u2

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

a?2

2u1 = exp

(iii) u2 F1 (?),

b

2

2u2 = exp ? u2 F2 (?), ? = au2 ? b ln u1 ;

b

a?2 u1

2u1 = (u2 + u2 )?1/2 exp

(iv) arctan u2 F1 (?) + u1 F2 (?) ,

1 2

b u2

a?2 u1

2u2 = (u2 + u2 )?1/2 exp u2 F2 (?) ? u1 F1 (?) ,

arctan

1 2 (9)

b u2

u1

? = b ln(u2 + u2 ) ? 2a arctan ;

1 2

u2

2

2u1 = exp ? u2

(v) F1 (?) + u2 F2 (?) ,

b

2

2u2 = b exp ? u2 F2 (?), ? = 2bu1 ? u2 ; 2

b

2u1 = 0, 2u2 = 0;

(vi)

where F1 , F2 are arbitrary smooth functions, a, b are arbitrary constants.

And what is more, the basis generators Pµ , Jµ? are given by the formulae (2) and

the generators of the corresponding groups of scale transformations are given by the

following formulae:

(i) D = xµ ?µ + au1 ?u1 + bu2 ?u2 , a = 0;

(ii) D = xµ ?µ + a(u1 ?u1 + u2 ?u2 ) + u2 ?u1 ;

(iii) D = xµ ?µ + au1 ?u1 + b?u2 , b = 0;

(10)

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

(iv) D b = 0;

(v) D = xµ ?µ + u2 ?u1 + b?u2 , b = 0;

(vi) D = xµ ?µ .

Theorem 2. The system of PDE (8) is invariant under the conformal group C(1, 3)

i? it is equivalent to the following system:

u1

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

1

u2

where F1 , F2 are arbitrary smooth functions.

Proofs of the Theorems 1, 2 are carried out with the use of in?nitesimal algorithm

by Lie [2, 3]. Here we present the proof of the Theorem 1 only.

Within the framework of the Lie’s approach a symmetry operator for the system

of PDE (7) is looked for in the form

(11)

X = ?µ (x, u)?µ + ?1 (x, u)?u1 + ?2 (x, u)?u2 ,

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

Necessary and su?cient condition for the system of PDE (7) to be invariant under

the group having the in?nitesimal operator (11) reads

X(2uj ? Fj ) (12)

= 0, j = 1, 2,

2u1 ?F1 =0

2u2 ?F2 =0

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

New scale-invariant nonlinear di?erential equations 53

Splitting relations (12) by independent variables we get a Killing type system of

PDE for ?µ , ?k . Integrating it we have:

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

2

(13)

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

?k = k = 1, 2,

j=1

where k? , cµ? = ?c?µ , d, eµ , akj are arbitrary constants, bk (x) are arbitrary functions

satisfying the following relations:

2 2

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

k=1 l=1

(14)

2

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

l=1

From (13) and (14) it follows that the system of PDE (7) is invariant under the

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

e

all functions F1 , F2 such that system (7) admits the extended Poincar? group P (1, 3)

e

one has to solve the following two problems:

• to describe all operators D of the form (11), (13) which together with the

operators (2) satisfy the commutational relations of the Lie algebra of the group

P (1, 3):

[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;

• to solve system of PDE (14) for each operator D obtained.

Substituting the operator D ? X with ?µ , ?k of the form (11) and (13) into

the above commutational relations and computing the coe?cients of the linearly-

independent operators ?xµ we arrive at the following relations:

k? = 0, cµ? = 0, ?, µ, ? = 0, . . . , 3,

?bk (x)

= 0, k = 1, 2, µ = 0, . . . , 3.

?xµ

Consequently, the generator of the one-parameter scale transformation group D

admitted by the PDE (7) necessarily takes the form

? ?

2 2

? Aij uj + Bi ? ?ui , (15)

D = xµ ?µ +

i=1 j=1

where Aij , Bi are some constants.

Before integrating the determining Eqs. (14) we simplify the operator D using

the equivalence relation (8). Making in (15) the change of variables (8) with ?j = 0

(which does not alter the form of the operators Pµ , Jµ? ) we have

? ?

2 2

? Aij uj + Bi ? ?ui ,

D = xµ ?µ +

i=1 j=1

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

where

2 2

?1

(16)

Aij = ?ik Akl ?lj , Bi = ?ik Bk , i = 1, 2.

k,l=1 k=1

?1

Here ?lj are elements of the (2 ? 2)-matrix inverse to the matrix ?ij .

Since an arbitrary (2 ? 2)-matrix can be reduced to the Jordan form by the

transformation (16) we may assume, without loss of generality, that the matrix Aij

is in Jordan form. The further simpli?cation of the form of operator (15) is achieved

at the expense of the transformation (8) with ?ik = 0.

As a result, the set of operators (15) is divided into the six equivalence classes

whose representatives are adduced in (10).

Next, integrating corresponding system of PDE (14) we get P (1, 3)-invariant sys-

tems of equations (9).

Note 1. When proving the Theorem 1 we solve the classical problem of representation

theory: the description of inequivalent representations of the extended Poincar? group

e

which are realized on the set of solutions of the system of nonlinear PDE (7). 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 standard methods of linear representation

theory (for more detail, see [5, 6]).

Note 2. If one put in the formulae (1) and (3) from (6) a = k1 , b = k2 and a = k1 ,

b = 0 respectively, then we get P (1, 3)-invariant systems of PDE constructed in [4].

Further, if we make in (6) the change of variables

1 1

(u + u? ) , (u ? u? ) ,

u1 = u2 =

2 2i

then we get the six classes of inequivalent PDE for complex ?eld invariant under the

extended Poincar? group.

e

Equations of the form (3) are widely used in the quantum ?eld theory to describe

at the classical level spinless charged mesons [7]. But PDE (3) with arbitrary F1 , F2 is

“two general” to be used as a reasonable mathematical model of a real physical process.

The nonlinearities F1 , F2 should be restricted in some way. To our minds the symmetry

selection principle is the most natural way of achieving this target. Furthermore, the

wide symmetry of the equation under study makes it possible to apply the symmetry

reduction procedure to obtain its exact solutions. Since all connected subgroups of

the extended Poincar? group are known [8–10] one can apply the said procedure to

e

reduce and to construct particular solutions of the PDE (9). This problem is now

under consideration and will be a topic of our future paper.

Acknowledgments. One of the authors (R.Zh.) is supported by the Alexander

von Humboldt Foundation. The authors are thankful to the referee for useful sugges-

tions.

New scale-invariant nonlinear di?erential equations 55

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

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

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

4. Fushchych W.I., Yegorchenko I.A., J. Phys. A, 1989, 22, 2643.

5. Rideau G., Winternitz P., J. Math. Phys., 1990, 31, 1095.

6. Fushchych W.I., Zhdanov R.Z., Lagno V.I., J. Nonlinear Math. Phys., 1994, 1, 295.

7. Bogoljubov N.N., Shirkov D.V., Introduction into theory of quantized ?elds, Moscow, Nauka,

1984.

8. Patera J., Winternitz P., Zassenhaus J., J. Math. Phys., 1975, 16, 1597.

9. Patera J., Winternitz P., Zassenhaus J., J. Math. Phys., 1976, 17, 717.

10. Patera J., Sharp R.T., Winternitz P., Zassenhaus J., J. Math. Phys., 1976, 17, 977.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 56–66.

Some exact solutions of a conformally

invariant nonlinear Schr?dinger equation

o

P. BASARAB-HORWATH, L.L. BARANNYK, W.I. FUSHCHYCH

We consider a nonlinear Schr?dinger equation whose symmetry algebra is the confor-

o

mal algebra. Using some of these symmetries, we construct some ansatzes for solutions

of the equation. This equation can be thought of as giving a wave-function description

of a classical particle.

1 Introduction

Many authors have proposed nonlinear generalisations of the linear equation of the

following type [1, 2, 3, 4, 5, 6, 7]:

|u| |u|a |u|a u

(1)

iut + u= ?1 + ?2 + ?0 ln ? u,

|u| |u| u

where ut = ?u , |u|a = ?xa , |u| = uu? , a = 1, . . . , n, ?0 , ?1 , ?2 are constants, and

?u

?t

we sum over repeated indices. These types of equations were introduced to include

e?ects such as dissipation and di?usion.

The symmetry properties and classi?cation of equations of type (1) are studied in

[6, 7]. An important property of all equations of the above type is their admit the

Galilei group G(1, n) as symmetries.

In this article we shall consider the following equation belonging to the class (1):

|u|

(2)

iut + u= u

|u|

which has remarkable symmetry properties. Indeed, it has the largest local symmetry

algebra of all known nonlinear Schr?dinger equations, being invariant under the

o

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