ñòð. 118 |

? ? ?

+ Q ( ?0 )?0 ?1 ?2 + ( ?1 )?0 ?1 ?2 + ( ?2 )?0 ?1 ?2 + ?0xa ?0xa U0 ?1 ?2 +

? ? ?

+ ?1xa ?1xa ?0 U1 ?2 + ?2xa ?2xa ?0 ?1 U2 + 2?0xa ?1xa ?0 ?1 ?2 +

??

+ 2?0xa ?2xa ?0 ?1 ?2 + 2?1xa ?2xa ?0 ?1 ?2 = V Q?0 ?1 ?2 .

? ? ??

Splitting the above equality with respect to ?0 ?1 , ?0 ?2 , ?1 ?2 we obtain the

?? ?? ??

equalities:

(A14)

?0xa ?1xa = 0, ?0xa ?2xa = 0, ?1xa ?2xa = 0.

Since the functions ?µ , µ = 0, 2 are real-valued, equalities (A.14) mean that there

are three real two-component vectors which are mutually orthogonal. This is possible

only if one of them is a null-vector. Without loss of generality we may suppose that

(?0x1 , ?0x2 ) = (0, 0), whence ?0 = f (t) ? t.

Consequently, Ansatz (A.12) necessarily takes the form (5). But Ansatz (5) can

not separate Eq. (1) into three second-order ODEs, since it contains no second-order

derivative with respect to t.

Thus, we have proved that the Schr?dinger equation (1) is not separable into three

o

second-order ODEs.

1. Boyer C., SIAM J. Math. Anal., 1976, 7, 230.

2. Boyer C., Kalnins E., Miller W., J. Math. Phys., 1975, 16, 499.

3. Miller W., Symmetry and separation of variables, Massachusetts, Addison-Wesley, 1977.

4. Shapovalov V.N., Sukhomlin N.B., Izvestiya Vuzov, Fizika, 1974, ¹ 12, 268.

5. Nikitin A.G., Onufriychuk S.P., Fushchych W.I., Teoret. Mat. Fiz., 1992, 12, 268.

6. Fushchych W.I., Zhdanov R.Z., Revenko I.V., Proc. Ukrain. Acad. Sci., 1993, ¹ 1, 27.

7. Fushchych W.I., in Symmetry in Mathematical Physics Problems, Kiev, Inst. of Math., 1981, 6.

8. Koornwinder T.H., Lect. Notes in Math., 1980, 810, 240.

9. Zhdanov R.Z., Revenko I.V., Fushchych W.I., J. Phys. A: Math. Gen., 1993, 26, 5959.

10. Zhdanov R.Z., J. Phys. A: Math. Gen., 1994, 27, L291.

11. Boyer C., Helv. Phys. Acta, 1974, 47, 589.

12. Niederer U., Helv. Phys. Acta, 1972, 45, 802.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 497–516.

On the general solution of the d’Alembert

equation with a nonlinear eikonal constraint

and its applications

R.Z. ZHDANOV, I.V. REVENKO, W.I. FUSHCHYCH

We construct the general solutions of the system of nonlinear differential equations

2n u = 0, uµ uµ = 0 in the four- and five-dimensional complex pseudo-Euclidean spaces.

The results obtained are used to reduce the multi-dimensional nonlinear d’Alembert

equation 24 u = F (u) to ordinary differential equations and to construct its new exact

solutions.

1 Introduction

Kaluza [1] was the first who put forward an idea of extension of the four-dimensional

Minkowski space in order to use it as a geometric basis for unification of the

electromagnetic and gravitational fields. Nowadays, Kaluza’s idea is well-known and

there are a lot of papers where further development and various generalizations of

this idea are obtained [2].

In [3–5] it was proposed to apply five-dimensional wave equations to describe

particles (fields) having variable spins and masses. Such physical interpretation of the

five-dimensional equations is based on the fact that the generalized Poincar? group

e

P (1, 4) acting in the five-dimensional de Sitter space contains the Poincar? group

e

P (1, 3) as a subgroup. It means that the mass and spin Casimir operators have conti-

nuous and discrete spectrum, respectively, in the space of irreducible representations

of the group P (1, 4) [3–6].

The simplest P (1, 4)-invariant scalar linear equation has the form

25 u + ?2 u = 0, (1)

? = const,

where 25 is the d’Alembert operator in the five-dimensional Minkowski space with

the signature (+, ?, ?, ?, ?).

The problem of construction of exact solutions of the above equation is, in fact,

completely open. One can obtain some its particular solutions applying the symmetry

reduction procedure or the method of separation of variables (both approaches use

essentially symmetry properties of the whole set of solutions of Eq. (1)). In the

present paper we suggest a method for construction of solutions of partial differential

equation (1) which utilizes implicitly the symmetry of a subset of the set of its

solutions. Namely, a special subset of its exact solutions obtained by imposing an

additional constraint

u2 0 ? u2 1 ? u2 2 ? u2 3 ? u2 4 = 0,

x x x x x

which is the eikonal equation in the five-dimensional space, will be investigated.

As shown in [7, 8], the system obtained is compatible if and only if ? = 0. We

J. Math. Phys., 1995, 36, ¹ 12, P. 7109–7127.

498 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

will construct general solutions of multi-dimensional systems of partial differential

equations (PDE)

2n u = 0, uµ u µ = 0 (2)

in the four- and five-dimensional complex pseudo-Euclidean spaces.

In (2) u = u(x0 , x1 , . . . , xn?1 ) ? C 2 (Cn , C1 ). Hereafter, the summation over the

repeated indices in the pseudo-Euclidean space M (1, n ? 1) with the metric tensor

gµ? = diag (1, ?1, . . . , ?1) is understood, e.g. 2n ? ?µ ? µ = ?0 ??1 ?· · ·??n?1 , ?µ =

2 2 2

n?1

?/?xµ .

It occurs that solutions of system of PDE (2), being very interesting by itself, can

be used to reduce the nonlinear d’Alembert equation

24 u = F (u), F (u) ? C(R1 , R1 ), (3)

to ordinary differential equations, thus giving rise to families of principally new exact

solutions of (3). More precisely, we will establish that there exists a nonlinear map

from the set solutions of the system of PDE (2) into the set of solutions of the

nonlinear d’Alembert equation, such that each solution of (2) corresponds to a family

of exact solutions of Eq. (3) containing two arbitrary functions of one argument. It

will be shown that solutions of the nonlinear d’Alembert equation obtained in this

way can be related to its conditional symmetry.

The paper is organized as follows. In Section 2 we give assertions describing

the general solution of system of PDE (2) in the n-dimensional real and in the

four- and five-dimensional complex pseudo-Euclidean spaces. In Section 3 we prove

these assertions. Section 4 is devoted to discussion of the connection between exact

solutions of system (2) and the problem of reduction of the nonlinear d’Alembert

equation (3). In Section 5 we construct principally new exact solutions of Eq. (3).

2 Integration of the system (2):

the list of principal results

Below we adduce assertions giving general solutions of the system of PDE (2) with

arbitrary n ? N provided u(x) ? C 2 (Rn , R1 ), and with n = 4, 5, provided u(x) ?

C 2 (Cn , C1 ).

Theorem 1. Let u(x) be a sufficiently smooth real function of n real variables

x0 , . . . , xn?1 . Then, the general solution of the system of nonlinear PDE (2) is given

by the following formula:

Aµ (u)xµ + B(u) = 0, (4)

where Aµ (u), B(u) are arbitrary real functions which satisfy the condition

Aµ (u)Aµ (u) = 0. (5)

Note 1. As far as we know, Jacobi, Smirnov and Sobolev were the first who obtained

the formulas (4), (5) with n = 3 [9, 10]. That is why, it is natural to call (4), (5)

the Jacoby–Smirnov–Sobolev formulas (JSSF). Later on, in 1944 Yerugin generalized

General solution of the d’Alembert equation with a nonlinear eikonal constraint 499

JSSF for the case n = 4 [11]. Recently, Collins [12] has proved that JSSF give the

general solution of system (2) for an arbitrary n ? N. He applied rather complicated

differential geometry technique. Below we show that to integrate Eqs. (2) it is quite

enough to make use of the classical methods of mathematical physics only.

Theorem 2. The general solution of the system of nonlinear PDE (2) in the class

of functions u = u(x0 , x1 , x2 , x3 , x4 ) ? C 2 (C5 , C1 ) is given by one of the following

formulas:

(1) Aµ (?, u)xµ + C1 (?, u) = 0, (6)

where ? = ? (u, x) is a complex function determined by the equation

Bµ (?, u)xµ + C2 (?, u) = 0, (7)

and Aµ , Bµ , C1 , C2 ? C 2 (C2 , C1 ) are arbitrary functions satisfying the conditions

?Aµ

Aµ Aµ = Aµ B µ = Bµ B µ = 0, Bµ (8)

= 0,

??

and what is more,

?Aµ ?C1 ?Aµ ?C1

xµ xµ

+ +

?? ?? ?u ?u

(9)

? = det = 0;

?Bµ ?C2 ?Bµ ?C2

xµ xµ

+ +

?? ?? ?u ?u

(2) Aµ (u)xµ + C1 (u) = 0, (10)

where Aµ (u), C1 (u) are arbitrary smooth functions satisfying the relations

Aµ Aµ = 0 (11)

(in the formulas (6)–(11) the index µ takes the values 0, 1, 2, 3, 4).

Theorem 3. The general solution of the system of nonlinear PDE (2) in the class of

functions u = u(x0 , x1 , x2 , x3 ) ? C 2 (C4 , C1 ) is given by the formulas (6)–(11), where

the index µ is supposed to take the values 0, 1, 2, 3.

Note 2. Investigating particular solutions of the Maxwell equations, Bateman [13]

arrived at the problem of integrating the d’Alembert equation 24 u = 0 with an

additional nonlinear condition (the eikonal equation) uxµ uxµ = 0. He has obtained the

following class of exact solutions of the said system of PDE:

u(x) = cµ (? )xµ + c4 (? ), (12)

where ? = ? (x) is a complex-valued function determined in implicit way

cµ (? )xµ + c4 (? ) = 0, (13)

? ?

and cµ (? ), c4 (? ) are arbitrary smooth functions satisfying conditions

cµ cµ = cµ cµ = 0. (14)

??

(hereafter, a dot over a symbol means differentiation with respect to a corresponding

argument).

It is not difficult to check that solutions (12)–(14) are complex (see the Lemma 1

below). An another class of complex solutions of the system (2) with n = 4 was

constructed by Yerugin [11]. But neither the Bateman’s formulas (12)–(14) nor the

Yerugin’s results give the general solution of the system (2) with n = 4.

500 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

3 Proofs of Theorems 1–3

It is well-known that the maximal symmetry group admitted by equation (1) is finite-

dimensional (we neglect a trivial invariance with respect to an infinite-parameter

group u(x) > u(x) + U (x), where U (x) is an arbitrary solution of Eq. (1), which is

due to its linearity). But being restricted to a set of solutions of the eikonal equation

the set solutions of PDE (1) admits an infinite-dimensional symmetry group [14]! It

is this very fact that enables us to construct the general solution of (2).

Proof of the Theorem 1. Let us make in (2) the hodograph transformation

a = 1, n ? 1, (15)

z0 = u(x), za = xa , w(z) = x0 .

Evidently, the transformation (15) is defined for all functions u(x), such that

ux0 ? 0. But the system (2) with ux0 = 0 takes the form

n?1 n?1

u2 a = 0,

uxa xa = 0, x

a=1 a=1

whence uxa ? 0, a = 1, n ? 1 or u(x) = const.

Consequently, the change of variables (9) is defined on the whole set of solutions

of the system (2) with the only exception u(x) = const.

Being rewritten in the new variables z, w(z) the system (2) takes the form

n?1 n?1

2

(16)

wza za = 0, wza = 1.

a=1 a=1

Differentiating the second equation with respect to zb , zc we get

n?1

(wza zb zc wza + wza zb wza zc ) = 0.

a=1

Choosing in the above equality c = b and summing up we have

n?1

(wza zb zb wza + wza zb wza zb ) = 0,

a,b=1

whence, by force of (16),

n?1

2

(17)

wza zb = 0.

a,b=1

Since u(z) is a real-valued function, it follows from (17) that an equality wza zb = 0

holds for all a, b = 1, n ? 1, whence

n?1

(18)

w(z) = ?a (z0 )za + ?(z0 ).

a=1

In (18) ?a , ? ? C 2 (R1 , R1 ) are arbitrary functions.

General solution of the d’Alembert equation with a nonlinear eikonal constraint 501

Substituting (18) into the second equation of system (16), we have

n?1

2

(19)

?a (z0 ) = 1.

a,b=1

Thus, the formulas (18), (19) give the general solution of the system of nonlinear

PDE (16). Rewriting (18), (19) in the initial variables x, u(x), we get

n?1 n?1

2

(20)

x0 = ?a (u)xa + ?(u), ?a (u) = 1.

a=1 a=1

To represent the formulas (20) in a manifestly covariant form (4), (5) we redefine

the functions ?a (u) in the following way:

Aa (u) B(u)

?(u) = ? a = 1, n ? 1.

?a (u) = , ,

A0 (u) A0 (u)

Substituting the above expressions into (20) we arrive at the formulas (4), (5).

Next, as u = const is contained in the class of functions u(x) determined by the

formulas (4), (5) under Aµ ? 0, µ = 0, n ? 1, B(u) = u+const, JSSF (4), (5) give the

general solution of the system of the PDE (2) with an arbitrary n ? N. The theorem

is proved.

Let us emphasize that the reasonings used above can be applied to the case of

a real-valued function u(x) only. If a solution of the system (2) is looked for in a class

of complex-valued functions u(x), then JSSF (4), (5) do not give its general solution

with n > 3. Each case n = 4, 5 . . . requires a special consideration.

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