ñòð. 22 |

v = exp ? = ?F (|?|)?

G2 ?(?1 , ?2 ) ?1 = t, ?2 = y1 i?1 + ?22 +

4t 2?1

2

2

iy1

v = exp

G1 + aP2 (a > 0) ?(?1 , ?2 ) ?1 = t, ?2 = ay1 + ty2 i?1 + (?1 + a2 )?22 +

4t

i

i?2

+ ? = ?F (|?|)?

?2 +

?1 2?1

v = exp(iat)?(?1 , ?2 ) ?1 = y1 , ?2 = y2

T ? 2aM (a ? R) ?11 + ?22 ? a? = ?F (|?|)?

23

at aty1 a?1

+ ? = ?F (|?|)?

T + aG1 (a > 0) ?(?1 , ?2 ) 4?11 + ?22 +

v = exp ? ?1 = at2 ? 2y1 , ?2 = y2

6 2 4

a2

2 2

J12 + aT + 2bM v = exp(?ibt)?(?1 , ?2 ) 4?1 ?11 + ?22 +

?1 = y1 + y2 ,

?1

y1

+t

?2 = a arctan + 4?1 + i?2 + ? = ?F (|?|)?

(a > 0, b ? R, or a = 0, b ? 0

y2

95

96

Table 5. Reduction by one-dimensional subalgebras of AG1 (1, 2). These ansazes and reduced equations are for equations (10) and (11).

Subalgebra Ansatz ? Reduced equation

2

y2

y1

v = t?(ia+1/k) ?(?1 , ?2 ) ?1 = , ?2 = 2

D + 4aM (a ? R) 4?1 ?11 + 4?2 ?22 + (2 ? i?1 )?1 +

t t

1

? = ?|?|k ?

+ (2 ? i?2 )?2 ? i ia +

k

2 2

a2

1 y1 + y2

,

aD + 2abM

J12 + v = t?(ib+1/k) ?(?1 , ?2 ) ?1 = 4?1 ?11 + ?22 + (4 ? i?1 )?1 +

t

2 ?1

i

y1

+t ? = ?|?|k ?

?2 = a arctan

(a ? 0, b ? R) + i?2 + b ?

k

y2 t

Table 6. Reduction by one-dimensional subalgebras of AG2 (1, 2). These ansazes and reduced equations are for equation (11).

Subalgebra Ansatz ? Reduced equation

2

1 y1

v=v ?1 = 4?1 ?11 + 4?2 ?22 + 2?1 + 2?2 +

S + T + 2aM (a ? R) exp i ? a arctan t +

t2 + 1

t2 + 1

2 2 2

? 1 + ?2

t(y1 + y2 ) y2

+ ? = ?|?|2 ?

?(?1 , ?2 ) ?2 + a?

=2

4(t2 + 1) t +1 4

2 2

1 1

y1 + y2

v=v

S + T + aJ12 + 2bM ?1 4?1 ?11 + ?22 + 4?1 +

exp i ? b arctan t + =2

t +1

t2 + 1 ?1

2 2

y1 ?1

t(y1 + y2 )

+ = arctan + a arctan t ? = ?|?|2 ?

?(?1 , ?2 ) ?2

(a > 0, b ? R) + ia?2 + b ?

4(t2 + 1) 4

y2

1 y1 + ty2

i t2 ? 1 2

v= exp

v

S + T + J12 + a(G1 + P2 ) ?1 ?11 + ?22 + i(2?1 ? a)?2 ?

?1 + =2

t

4 t +1

t2 + 1

y2 ty1 ? y2 2

+1 ?(?1 , ?2 ) ?2

(a ? 0) ? a arctan t ? ?1 ? = ?|?|2 ?

=2

t t +1

P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk

Solutions of the relativistic nonlinear wave equation 97

It has interesting symmetry properties, with its symmetry algebra containing both

the Poincar? and Galilei algebras. We intend to return to this equation in future

e

publications.

Finally, let us note that our ansatz relates the Schr?dinger equation with any

o

equation related to the wave equation, such as the Dirac equation. Indeed, the Dirac

equation is

(i? µ ?µ ? m)? = 0,

so that we may represent ? as

? = (i? µ ?µ + m)?, (27)

where ? is a four-component vector of functions satisfying

2? + m2 ? = 0.

Clearly, each of the components can be related (independently) to the Schr?dinger

o

equation by using our ansatz (7). In this way, we can use (27) to construct solutions of

the Dirac equation from the Schr?dinger equation. Similarly, we can use the complex

o

heat equation

?v

= ?v

?t

to construct solutions of the Dirac equation. Instead of ansatz (6), which uses the

operator M , we have the ansatz

?

? = ek(?x) v(?x, ?x, ?x), ? = ek(?x) v (?x, ?x, ?x),

?

which uses the operator L of Theorem 1. Exact solutions of the complex heat equation

in 1+2 space-time dimensions can be obtained from those of the real heat equation

given in [19]. Thus we see that solutions of the Dirac equation can be obtained from

the Schr?dinger and heat equations, or a mixture of both.

o

6 Appendix

In the following tables we give inequivalent ansatzes for equations (9), (10) and (11)

constructed from one- and two-dimensional subalgebras of the corresponding algebras

of invariance. This is organized as follows: we consider subalgebras in the ascending

chain AG(1, 2) ? AG1 (1, 2) ? AG2 (1, 2) (strictly speaking, this is incorrect, since the

dilatation operator D has a di?erent representation in AG1 (1, 2) and AG2 (1, 2), but

but here we treat the inclusions as abstract Lie algebra inclusions up to isomorphism).

In Tables 1, 2 and 3, we give a list of inequivalent two-dimensional subalgebras, with

the corresponding ansatzes and reduced equations (these are ordinary di?erential

equations); in Tables 4, 5 and 6, we do the same for one-dimensional subalgebras of

the chain, the reduced equations being partial di?erential equations. The reductions

have been veri?ed using MAPLE.

In order to avoid repetition in the reduced equations, we shall, in the following,

regard the function F in equation (9) as being arbitrary; in equation (10), k is an

arbitrary real number, so that with this convention equation (10) is a particular case

98 P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk

of equation (9), and equation (11) is a particular case of equation (10). Further, in

performing the symmetry reductions of (9) for arbitrary F , we use the inequivalent

subalgebras (of dimensions 1 and 2) of AG1 (1, 2) the symmetry reduction of (10) is

done using those subalgebras of AG2 (1, 2) which are not equivalent to subalgebras

of AG(1, 2); the reductions of (11) are done with respect to subalgebras of AG2 (1, 2)

which are not equivalent to subalgebras of AG1 (1, 2).

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 Swedish

Institute for ?nancial support and the Mathematics Department of Link?ping Univer-

o

sity for its hospitality. P. Basarab-Horwath thanks the Wallenberg Fund of Link?ping

o

University, the Tornby 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.

It is with great sadness that we announce that Professor W.I. Fushchych died

on April 7th 1997, after a short illness. This is a tremendous loss for his family, his

many students, and for the scienti?c community. His deep contributions to the ?eld of

symmetry analysis of di?erential equations have made the Kyiv school of symmetries

known throughout the world. We take this oportunity to express our deep sense of

loss as well as our gratitude for all the encouragement in research Wilhelm Fushchych

gave during the years we knew him.

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Interscience, 1966.

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7. Fushchych W.I., Serov M.I., J. Phys. A., 1983, 16, 3645.

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9. Grundland A.M., Tuszynski J.A., Winternitz P., Phys. Lett. A, 1987, 119, 340.

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

nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

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14. Niederer, U., Helv. Phys. Acta, 1974, 47, 119.

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Nauka, 1986 (in Russian).

19. Basarab-Horwath P., Fushchych W., Barannyk L., J. Phys. A, 1995, 28, 5291.

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

Allerton Press, 1994.

Solutions of the relativistic nonlinear wave equation 99

21. Fushchych W., Cherniha R., Ukrain. Math. J., 1989, 41, 1161; 1456.

22. Fushchych W., Chopyk V., Ukrain. Math. J., 1993, 45, 539.

23. Clarkson P., Nonlinearity, 1992, 5, 453.

24. Bluman G.W., Kumei S., Symmetries and di?erential equations, New York, Springer, 1989.

25. Fushchych W.I., Barannyk L.F., Barannyk A.F., Subgroup analysis of the Galilei and Poincar?

e

groups and reduction of nonlinear equations, Kiev, Naukova Dumka, 1991.

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

27. Tajiri M., J. Phys. Soc. Japan, 1983, 52, 1908.

28. Borhardt A.A., Karpenko D.Ya., Di?erential Equations, 1984, 20, 239.

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Vol. 2, New York, McGraw-Hill, 1953.

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35. Petiau G., Nuovo Cimento, 1958, Suppl. IX, 542.

36. Basarab-Horwath P., Fushchych W., Roman O., New conformally invariant nonlinear wave

equations for a complex scalar ?eld, in preparation.

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invariant non-linear Schr?dinger equations, in preparation.

o

38. Fanchi J.R., Found. Phys., 1993, 23, 487.

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W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 100–104.

On a new conformal symmetry

for a complex scalar ?eld

P. BASARAB-HORWATH, W.I. FUSHCHYCH, O.V. ROMAN

We exhibit a new nonlinear representation of the conformal algebra which is the

symmetry algebra of a nonlinear hyperbolic wave equation. The equation is the only

one of its type invariant under the conformal algebra in this nonlinear representation.

We also give a list of some nonlinear hyperbolic equations which are invariant under

the conformal algebra in the standard representation.

In this note we examine a nonlinear wave equation for a complex ?eld, having the

following structure

2u = F (u, u? , ?u, ?u? , ?|u|?|u|, 2|u|)u, (1)

where u = u(x) = u(x0 , x1 , . . . , xn ), ?u = (ux0 , . . . , uxn ), ?u? = (u? 0 , . . . , u? n ),

x x

µ? ?|u| ?|u|

?|u|?|u| = |u|µ |u| = g ?xµ ?x? , g = diag(1, ?1, . . . , ?1), and we use the usual

µ µ?

summation convention. Here, F is an arbitrary real-valued function.

Examples of equations such as (1) can be found in the literature, the most common

being the nonlinear Klein–Gordon type [2, 3],

2u = F (|u|, |u|µ |u|µ )u. (2)

Another such equation is that proposed (independently of each other) by Gu?ret and

ñòð. 22 |