ñòð. 96 |

1 2?2 t x2

?(x, t) = exp e arctan + g(t, ?) +

? x1

mx2

?2 2?2 t x2

+i ? 3

e arctan + + g(t, ?) ,

??1 x1 2t

x2 + x2 , reduces equations (1) to the system

where ? = 1 2

h2 2?1 1

+ 2g2 h2 ? 2 e4?2 t 2 ?

g1 + h22 +

2m ? ? ?2 r

g2 2 1

? d g22 + + 2g2 + 2 e4?2 t 2 ? 2?2 g = 0,

2

? ? r

?2

g2 1 1

h2 ? g22 ? ? g2 ? 2 1 ? 1 e4?2 t 2

2 2

h1 + + 2?1 g = 0,

2

2m ? ? ?2 r

3. A + ?H, J12 + ?P3 , ? ? R=0 , ? ? R. The ansatz

1 2?2 t ?1 2?2 t

+ g(?1 , ?2 ) + i ?

?(x, t) = exp e e + h(?1 , ?2 ) ,

2??2

2??2 2

? x3 , reduces equations (1) to the system

x1

x2 + x2 , ?2 = ? arctan

where ?1 = 1 2 x2

?2 ?2

h1

1+ 2 ?

+ 2g1 h1 + h22 1 + 2 + h2

h11 + 2

?1 ?1 ?1

2

?2

2md g1 ? 4m?2

? ?

2 2

g11 + + 2g1 + g22 1 + + g2 1 + 2 g = 0,

2

?1 ?1 ?1 m

?2 ?2

g1

+ g1 ? h1 + g22 1 + 2 + g2 1+ 2 ?

2 2 2

g11 +

?1 ?1 ?1

2

? 4m?1

? h2 1 + 2 ? g = 0.

2

?1 m

4. A + ?P3 , J12 , ? ? R=0 . The ansatz

1 2?2 t ?1 2?2 t

x3 + g(t, ?) + i ?

?(x, t) = exp e e x3 + h(t, ?) ,

? ??2

400 W.I. Fushchych, V. Chopyk, P. Nattermann, W. Scherer

x2 + x2 , reduces equations (1) to the system

where ? = 1 2

h2 ?1

+ 2g2 h2 ? 2 e4?2 t ?

g1 + h22 +

2m ? ? ?2

g2 2

? d g22 + + 2g2 + 2 e4?2 t ? 2?2 g = 0,

2

? ?

?2

g2 1

h2 ? g22 ? ? g2 ? 2 1 ? 1 e4?2 t

2

h1 + + 2?1 g = 0.

2

?2

2m ? ? 2

4. Conclusions

We have determined the maximal Lie symmetries of equation (1) with an F of

?

the form F [?, ?] := f (?), and have found six different algebras containing among

others the centrally extended Galilei algebra, the Galilei similitude algebra, and the

Schr?dinger algebra. Reduction and ans?tze for these algebras have been studied

o a

previously.

New maximal symmetry algebras, due to the nonlinear character of the equation,

appear in the case f (?) = (?1 + i?2 ) ln(?) (see cases 5 and 6 in Section 2.1). For

these cases we have obtained reduced equations for various subalgebras. The ans?tze a

resulting from these reductions lead to differential equations which we have solved

explicitly in some cases and thus we have obtained explicit solutions of (1). Those

reduced equations, which we have not been able to solve explicitly, are still much more

suitable to numerical treatments than the original equation (1). The list of subalgebras

which we have used for reduction in the case of the new algebras is by no means

complete. In view of the successes of the reduction technique it seems warranted

to obtain a classification of their subalgebras. The non-Lie anz?tze for the nonlinear

a

Schr?dinger equation were constructed by Fushchych and Chopyk [21].

o

1. Doebner H.D., Goldin G.A., Phys. Lett. A, 1992, 162, 397.

2. Doebner H.D., Goldin G.A., Annales de Fisica, Monografias, Vol. II, CIEMAT, 1993, 442–445.

3. Doebner H.D., Goldin G.A., Manifolds, general symmetries, quantization and nonlinear quantum

mechanics, in Proceedings of the First German-Polish Symposium on Paricles and Fields (Rydzyna

Castle, 1992), Singapore, World Scientific, 1993, 115.

4. Doebner H.D. and Goldin G.A., J. Phys. A, 1994, 27, 1771.

5. Fushchych W., Cherniha R., Ukr. Math. J., 1989, 41, ¹ 10, 1349; ¹ 12, 1487.

6. Malomed B.A., Stenflo L., J. Phys. A, 1991, 24, L1149.

7. Nattermann P., Maximal Lie symmetry of the free general Doebner–Goldin equation in 1 + 1

dimensions, Clausthal-preprint ASI-TPA/9/94.

8. Biatynicki-Birula I., Mycielski J., Ann. Phys., 1976, 100, 62.

9. Malomed B.A. and Kivshar Y.S., Rev. Mod. Phys., 61, 1989, 763.

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

nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

11. Fushchych W., in Algebraic-Theoretical Studies in Mathematical Physics, Kiev, Inst. Math. Ukrai-

nian Acad. Sci., 1981, 6–44 (in Russian).

12. Fushchych W., Chopyk V., Symmetry analysis and ansatzes for Schr?dinger equation with the

o

logarithmic nonlinearity, Preprint, Link?ping University, 1993.

o

13. Ovsiannikov L.V., Group analysis of differential equations, Academic Press, 1982.

14. Olver J., Appplications of Lie groups to differential equations, New York, Springer, 1986.

Symmetries and reductions of nonlinear Schr?dinger equations

o 401

15. Baumann G., Lie symmetries of differential equations: A Mathematica program to determine Lie

symmetries, Wolfram Research Inc., Champaign, Ilinois, MathSource 0202-622, 1992.

16. Rideau G., Winternitz P., J. Math. Phys., 1993, 34, 558.

17. Niederer U., Helvet. Phys. Acta, 1972, 45, 802.

18. Fushchych W., Barannik L., Barannik A., Subgroup analysis of the Galilei and Poincar? Groups,

e

and reduction of nonlinear equations, Kiev, Naukova Dumka, 1991 (in Russian).

19. Fushchych W., Chopyk V., Cherkasenko V., Dokl. Ukr. Acad. Sci., 1993, ¹ 2, 32.

20. Chopyk V.I., Symmetry and reduction of multi-dimensional Schr?dinger equation with the logari-

o

thmic nonlinearity, in Symmetry Analysis of Equations of Mathematical Physics, Kiev, Inst. Math.

Ukrainian Acad. Sci., 1992.

21. Fuschchych W., Chopyk V., Ukr. Math. J., 1993, 45, 581.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 402–414.

Symmetry reduction and exact solutions

of nonlinear biwave equations

W.I. FUSHCHYCH, O.V. ROMAN, R.Z. ZHDANOV

Symmetry analysis of a class of the biwave equations 22 u = F (u) and of a system of

wave equations which is equivalent to it is performed. Reduction of the nonlinear biwave

equations by means of the Ans?tze invariant under non-conjugated subalgebras of the

a

extended Poincar? algebra AP (1, 1) and the conformal algebra AC(1, 1) is carried out.

e

Some exact solutions of these equations are obtained.

1 Introduction

It was customary for the classical mathematical physics to use as the mathemati-

cal models for describing real physical processes linear partial differential equations

(PDE) of the order not higher than two. All fundamental equations of mathematical

physics such as the Laplace, heat, Klein–Gordon–Fock, Maxwell, Dirac, Schr?dingero

equations are the first- or the second-order linear partial differential equations. But

now there are strong evidences that linear description is not satisfactory (especial-

ly it is the case in the quantum field theory [1]). That is why, it was attempted to

generalize the classical equations in a non-linear way in order to get more satisfactory

models. There exist different principles of the choice of such generalizations but up

to our mind the most natural and systematic is the symmetry selection principle.

A classical illustration is a group classification of nonlinear wave equations

2u = F (u). (1)

Here and further 2 = ? 2 /?x0 ? ? 2 /?x1 ? · · · ? ? 2 /?xn is the d’Alembertian

in the (n + 1)-dimensional pseudo-Euclidean space R(1, n) with the metric tensor

gµ? = diag (1, ?1, . . . , ?1), µ, ? = 0, n; xµ = x? gµ? ; F (u) is an arbitrary smooth

function; u = u(x) is a real function; the summation over the repeated indices from 0

to n is understood.

With an arbitrary F (u) equation (1) is invariant under the (n+1)(n+2) -parameter

2

Poincar? group P (1, n) having the following generators:

e

? ? ?

? x?

Jµ? = xµ (2)

Pµ = , , µ, ? = 0, n.

?xµ ?x? ?xµ

But equation (1) taken with an arbitrary nonlinearity F (u) is too “general” to be

a reasonable mathematical model for describing a specific physical phenomena. To

specify a form of F (u) symmetry properties of the linear wave equation are utilized.

It is well-known that PDE (1) with F (u) = 0 in addition to the Poincar? groupe

admits a one-parameter scale transformation group and a (n + 1)-parameter group of

special conformal transformations (see, e.g. [2]). Therefore, it is not but natural to

Preprint ASI-TPA/13/95, Arnold-Sommerfeld-Institute for Mathematical Physics, Germany, 1995, 14 p.

Reports on Math. Phys., 1996, 37, ¹ 2, P. 267–281.

Symmetry reduction and exact solutions of nonlinear biwave equations 403

postulate that those nonlinearities are admissible which preserve a symmetry of the

linear equation. It has been proved in [3] that there are only two functions F (u),

namely

F (u) = ?(u + C)k , (3)

F (u) = ? exp Cu,

where C, k = 0 are arbitrary constants, such that Poincar?-invariant equation (1)

e

admits a one-parameter scale transformation group. Furthermore, it was known long

ago that the only equation of the form (1) admitting the conformal group C(1, n)

n+3

is the one with F (u) = ?(u + C) n?1 . Consequently, choosing from the whole set of

PDE (1) equations having the highest symmetry we get the ones with very specific

nonlinearities.

A procedure described above is called group or symmetry classification of PDE (1).

A method used is the classical infinitesimal Lie’s method. Given a representation of

a Lie transformation group (which is fixed by a requirement that this group should

be admitted by the linear wave equation), the problem of symmetry classification of

equations (1) is reduced to solving some linear over-determined system of PDE. This

system is called determining equations (for more detail, see [2, 5]).

But what is most important, the Lie’s method can be applied not only to classify

invariant equations but also to construct their explicit solutions by means of symmetry

reduction procedure. And one more important remark is that equations having broad

Lie symmetry often admit non-trivial conditional symmetry, which can be also used

to obtain their particular solutions [2].

In [6] the description has been suggested of different physical processes with the

help of nonlinear partial differential equations of high order, namely

?u ?u

2l u = F u, (4)

.

?xµ ?xµ

where 2l = 2(2l?1 ), l ? N; F (· , · ) is an arbitrary smooth function.

The equations (4) were considered from different points of view in [2, 7, 8],

where the pseudo-differential equations of type (4) were also studied (in this case l is

fractional or negative).

Assuming l = 1 and F = F (u) in (4) we obtain the standard nonlinear wave

equation (1), which describes a scalar spin-less uncharged particle in the quantum

field theory. Symmetry properties of the equation (1) were studied in [2, 3, 4] and

wide classes of its exact solutions with certain concrete values of the function F (u)

were obtained in [2, 3, 9, 10, 11].

In this paper we restrict ourselves to symmetry analysis of the biwave equation

22 u = F (u), (5)

which is one of the simplest equations of type (4) of the order higher than two (l = 2,

F = F (u)).

2 Symmetry classification of biwave equations

In order to carry out a symmetry classification of the equation (5) we shall establish

at first the maximal transformation group admitted by the equation (5), provided F (u)

404 W.I. Fushchych, O.V. Roman, R.Z. Zhdanov

is an arbitrary function. Next, we shall determine all the functions F (u) such that

the equation (5) admits a more extended symmetry.

Results of symmetry classification of the equation (5) are presented below.

Lemma 1 The maximal invariance group of the equation (5) with an arbitrary

function F (u) is the Poincar? group P (1, n) generated by the operators (2).

e

Theorem 1 Any equation of type (5) admitting a more extended invariance algebra

than the Poincar? algebra AP (1, n) is equivalent to one of the following PDE:

e

1. 22 u = ?1 uk , (6)

?1 = 0, k = 0, 1;

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