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Vigier [9] and by Guerra and Pusterla [10],

2|u| m2 c2

2u = u ? 2 u. (3)

|u|

This equation arose in the modelling of an equation for de Broglie’s theory of the

double solution [1]. Gu?ret and Vigier were able to show that a solution to this

e

problem, obtained by Mackinnon [11] satis?ed Eq. (3). Guerra and Pusterla obtained

(3) as a relativistic version of a nonlinear Schr?dinger equation they had found by

o

applying stochastic methods to quantum mechanics.

Eq. (3) is from our point of view (namely, the symmetry view) a remarkable

nonlinear equation, since it is invariant under the conformal algebra AC(1, n + 1) in

an unusual representation.

It is well-known (see, for instance, Refs. [3, 7]) that the free wave equation 2u = 0

is invariant under the conformal group AC(1, n) with in?nitesimal operators

?

Jµ? = xµ P? ? x? Pµ , (4)

Pµ = ,

?xµ

n?1

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

D = xµ Pµ ? Kµ = 2xµ ? x2 Pµ , (5)

2

Physics Letters A, 1997, 226, P. 150–154.

On a new conformal symmetry for a complex scalar ?eld 101

with x2 = xµ xµ . The wave equation is also invariant under the operators

I = i(u?u ? u? ?u? ), Q = u?u + u? ?u? ,

L1 = u? ?u + u?u? , L2 = i(u? ?u ? u?u? ),

which are important in reducing the wave equation to the Schr?dinger and heat equa-

o

tions (see Refs. [4, 5, 6]).

The conformal operators Kµ generate the ?nite conformal transformations

xµ ? x2 cµ

xµ > xµ = (6)

,

1 ? 2c? x? + c2 x2

u > u = (1 ? 2c? x? + c2 x2 )(n?1)/2 u, (7)

where cµ are parameters.

All equations of the form (2) invariant under the conformal group with in?nitesimal

generators given in the representation (4), (5) were classi?ed in Ref. [2]. In particular,

it was shown there that when the function F is independent of the derivatives of u,

then the equation is conformally invariant under (4), (5) if and only if

F (u) = ?|u|4/(n?1) , (8)

where n ? 2 and ? is an arbitrary parameter. Thus, Eq. (1), when the right-hand

side does not depend on the derivatives of u, has the same conformal invariance as

the free wave equation if and only if F is given by (8).

An analysis of the proof of this statement shows that two things are ?xed at the

outset: the independence of F of the derivatives; and the representation of the algebra

AC(1, n). One then sees that the following natural question arises: does there exist

a representation of AC(1, n) di?erent from (4), (5)? That is, are there operators Kµ ,

D which are not equivalent to those given in (5)? Our answer to this question is that

there exists such a representation.

To this end, we have calculated the Lie point symmetry algebra of the equation

(see, for instance, Ref. [12, 3])

2|u|

2u = (9)

u + ?u,

|u|

with ? an arbitrary parameter. It is evident that this equation is Poincar? invariant

e

with respect to the operators (4). On the other hand, it is de?nitely not invariant

under the conformal operators given in (5). However, this does not mean that it is

not at all conformally invariant, as we see from the following result.

Theorem 1. Eq. (9) with ? < 0 has maximal point-symmetry algebra AC(1, n+1)?Q

generated by operators

(1)

Pµ , Jµ? , Pn+1 , Jµn+1 , D(1) , Kµ , Kn+1 , Q,

(1)

where

? ?

= i(u?u ? u? ?u? ),

Jµ? = xµ P? ? x? Pµ ,

Pµ = , Pn+1 =

?xµ n+1

?x

102 P. Basarab-Horwath, W.I. Fushchych, O.V. Roman

n

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

Jµn+1 = xµ Pn+1 ? xn+1 Pµ , D(1) = xµ Pµ + xn+1 Pn+1 ?

2

Kµ = 2xµ D(1) ? (xµ xµ + xn+1 xn+1 )Pµ ,

(1)

Q = u?u + u? ?u? ,

(1)

Kn+1 = 2xn+1 D(1) ? (xµ xµ + xn+1 xn+1 )Pn+1 ,

where the additional variable xn+1 is de?ned as

u?

i

=v

= ?xn+1

n+1

x ln , ? < 0.

2 ?? u

For ? > 0 the maximal symmetry algebra of (9) is AC(2, n) ? Q generated by the

same operators above, but with the additional variable

u?

i

xn+1 = xn+1 = v ln , ? > 0.

u

2?

Remark 1. In this theorem we have introduced a new metric tensor

gAB = diag (1, ?1, . . . , ?1, gn+1 n+1 )

with gn+1 n+1 = 1 when ? > 0 and gn+1 n+1 = ?1 when ? < 0.

Direct veri?cation shows that the above operators satisfy the commutation relati-

ons of the conformal algebra AC(1, n + 1) ? Q when ? < 0 and AC(2, n) ? Q when

? > 0.

(1) (1)

The meaning of the new operators Pn+1 , Jµn+1 , Kµ , Kn+1 is best understood

when Eq. (9) is rewritten in the amplitude-phase representation, namely, on putting

u = Rei? with R and ? being real functions. Then equation (9) becomes the system

g µ? ?µ ?? = ??, (10)

R2? + 2g µ? Rµ ?? = 0. (11)

The symmetry algebra of Eq. (9) is actually obtained by ?rst calculating the symmetry

algebra of the system (10), (11). Then we have, in the amplitude-phase representation

? ? ?

?? (12)

Pn+1 = , Jµn+1 = xµ ,

xµ

?? ??

? ? n?

?R

D(1) = xµ (13)

+? ,

µ

?x ?? 2 ?R

?

Kµ = 2xµ D(1) ? (xµ xµ + gn+1 n+1 ?2 )

(1)

(14)

,

?xµ

?

(1)

Kn+1 = 2gn+1 n+1 ?D(1) ? (xµ xµ + gn+1 n+1 ?2 ) (15)

.

??

From the expressions (12)–(15), we see that the phase variable ? has been added to

the n + 1-dimensional geometric space of the xµ . This is the same e?ect we see for the

eikonal equation [3], and it is not surprising, since the ?rst equation of system (10),

(11) is indeed the eikonal equation for the phase function ?. What is novel here is that

equation (11), which is the equation of continuity, does not reduce the symmetry of

On a new conformal symmetry for a complex scalar ?eld 103

equation (10). On using an appropriate ansatz (see Ref. [5]) for ? and A one can reduce

system (10), (11) to another system consisting of the Hamilton–Jacobi equation and

the non-relativistic continuity equation. This second system also exhibits surprising

symmetry properties [8]: it is again conformally invariant.

(1) (1)

Let us remark that the operators D(1) , Kµ , Kn+1 are a nonlinear representation

of the dilatation and conformal translation operators. They generate the following

?nite transformations:

xµ > xµ = exp(b)xµ , ? > ? = exp(b)?,

D(1) :

R > R = exp(?bn/2)R;

xµ ? cµ (x? x? + gn+1 n+1 ?2 )

(1)

xµ > xµ =

Kµ : ,

1 ? 2c? x? ? 2cn+1 ? + c2 (x? x? + gn+1 n+1 ?2 )

?

?>? = ,

1 ? 2c? x? ? 2cn+1 ? + c2 (x? x? + gn+1 n+1 ?2 )

n/2

R > R = 1 ? 2c? x? ? 2cn+1 ? + c2 (x? x? + gn+1 n+1 ?2 ) R;

xµ

(1)

xµ > xµ =

Kn+1 : ,

1 ? 2c? x? ? 2cn+1 ? + c2 (x? x? + gn+1 n+1 ?2 )

? ? cn+1 (x? x? + gn+1 n+1 ?2 )

?>?= ,

1 ? 2c? x? ? 2cn+1 ? + c2 (x? x? + gn+1 n+1 ?2 )

n/2

R > R = 1 ? 2c? x? ? 2cn+1 ? + c2 (x? x? + gn+1 n+1 ?2 ) R.

where b, c? , cn+1 are the group parameters and c2 = c? c? + cn+1 cn+1 with the

usual lowering and raising of indices using the metric gAB used in Theorem 1. The

expressions for these ?nite transformations can be compared with those given in (6),

(7). The form is exactly the same, but the new feature is that ? is considered as

a geometrical variable on the same footing as the xµ , and it is the amplitude R which

transforms as the dependent variable, just as u does in (7).

It should be added that Eq. (9) is the only equation of type (1) which is invariant

under AC(1, n + 1) ? Q in the representation given in Theorem 1. This is not the

standard representation. However, if we keep the standard representation (4), (5) of

the conformal algebra but allow dependence of the nonlinearity in (1) on the deri-

vatives, then we ?nd that there are other equations of this type which are invariant

under the conformal algebra:

2u = |u|4/(n?1) F |u|(3+n)/(1?n) 2|u| u, n = 1,

2|u|

2u = 2|u|F , |u| u, n = 1,

(?|u|)2

2|u| (2|u|)n

42u = +? u, n arbitrary,

|u| |u|n+4

2|u|

2u = (1 + ?) u,

|u|

2|u| ?

2u = 1 + 4 u,

|u| |u|

2|u| ?

2u = 1+ u.

|u| 1 + ?|u|4

104 P. Basarab-Horwath, W.I. Fushchych, O.V. Roman

Thus, we see that wave equations which have a nonlinear quantum potential term

2|u|/|u| have an unusually wide symmetry. This is in sharp contrast with nonli-

nearities not containing derivatives. Moreover, we see that the representation of

a given algebra plays a fundamental role in picking out certain equations which

are invariant. This remark leads us to asking how one can construct all possible

representations, linear and nonlinear. Linear representation theory is well-developed,

but nonlinear representations are not at all well understood. Certainly, the equation

dictates the symmetry and the representation of the symmetry, and both equation

and representation are intimately tied together. From the symmetry point of view,

we cannot truly distinguish between them as phenomena.

Finally, we remark that given an equation, its symmetry algebra can be exploi-

ted to construct ansatzes (see, for example, [3]) for the equation, which reduce the

problem of solving the equation to one of solving an equation of lower order, even

ordinary di?erential equations. We examine this question for some of the equations

we have given above in a future article, and we hope that some of them will ?nd

some application in nonlinear quantum mechanics or optics, not least because of their

beautiful symmetry properties and relation to nonlinear Schr?dinger equations.

o

Acknowledgements. This work was supported in part by NFR grant R-RA

09423-315, and INTAS and DKNT of Ukraine. W.I. Fushchych thanks the Mathema-

tics Department of Link?ping University for its hospitality and for support for travel.

o

P. Basarab-Horwath thanks the Wallenberg Fund of Link?ping University the Tornby

o

Fund and the Magnusson Fund of the Swedish Academy of Science for travel grants,

and the Mathematics Institute of the Ukrainian Academy of Sciences in Kiev for its

hospitality. Olena Roman thanks the International Soros Science Education Program

for ?nancial support through grant number PSU061088.

1. de Broglie L., Non-linear wave mechanics, Elsevier Publishing Company, 1960.

2. Fushchych W.I., Yehorchenko I.A., The symmetry and exact solutions of the nonlinear d’Alem-

bert equation for complex ?elds, J. Phys. A, 1989, 22, 2643–2652.

3. Fushchych W., Shtelen W., Serov M., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

4. Basarab-Horwath P., Fushchych W., Barannyk L., Exact solutions of the wave equation by

reduction to the heat equation, J. Phys. A, 1995, 28, 5291–5304.

5. Basarab-Horwath P., Fushchych W., Barannyk L., Solutions of the relativistic nonlinear wave

equation by solutions of the nonlinear Schr?dinger equation, Preprint, Link?ping University,

o o

1996.

6. Basarab-Horwath P., A symmetry connection between hyperbolic and parabolic equations,

J. Nonlinear Math. Phys., 1996, 3, 311–318.

7. Fushchych W., Nikitin A., Symmetries of the equations of quantum mechanics, New York,

Allerton Press, 1994.

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