ñòð. 127 |

assertions giving general solutions of the system of PDE (1) 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 sufficiently smooth real function on n real variables x0 , . . . ,

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

following formula:

(3)

Cµ (u)xµ + Cn (u) = 0,

where Cµ (u), Cn (u) are arbitrary real functions that satisfy the condition

(4)

Cµ (u)Cµ (u) = 0

(the condition (4) means that n-vector (C0 , C1 , . . . , Cn?1 ) is an isotropic one).

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

the formulae (3), (4) when n = 3 [1, 2]. That is why it is natural to call (3), (4)

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

JSSF up to the case n = 4 [3]. Recently, Collins [4] proved that JSSF give the

general solution of system (1) under arbitrary n ? N. He applied rather complicated

in Symmetry Analysis of Equations of Mathematical Physics, Kyiv, Institute Mathematics, 1992, P. 68–

90.

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

differential geometry technique. Below we show that to integrate Eqs. (1) it is quite

enough to apply only classical methods of mathematical physics.

Theorem 2. The general solution of the system of nonlinear PDE (1) in the class of

functions u = u(x0 , x1 , x2 , x3 ) ? C 2 (C4 , C1 ) is given by the following formula:

(5)

F (Aµ (u)xµ , B? (u)x? , u) = 0,

where F ? C 2 (C3 , C1 ) is an arbitrary function, Aµ , Bµ ? C 2 (C1 , C1 ) are arbitrary

smooth functions satisfying the conditions

(6)

Aµ Aµ = Aµ Bµ = Bµ Bµ = 0.

Theorem 3. The general solution of the system of nonlinear PDE (1) in the class

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

formulae:

(7)

1) Aµ (?, u)xµ + C1 (?, u) = 0,

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

(8)

Bµ (?, u)xµ + C2 (?, u) = 0,

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

?Aµ ?Bµ

(9)

Aµ Aµ = Aµ Bµ = Bµ Bµ = 0, Bµ = Aµ =0

?? ??

and what is more

?Aµ ?C1 ?Aµ ?C1

xµ + xµ +

?? ?? ?? ??

(10)

? = det = 0.

?Bµ ?C2 ?Bµ ?C2

xµ + xµ +

?? ?? ?? ??

(11)

2) Aµ (x)xµ + C1 (u) = 0,

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

(12)

Aµ Aµ = 0

(in the formulae (7)–(12) the index µ takes the values 0, 1, 2, 3, 4).

Note 2. In 1915 Bateman [5] investigating particular solutions of the Maxwell equa-

tions came to the problem of integrating the d’Alembert equation 24 u = 0 with

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

following class of exact solutions of the above system:

(13)

u(x) = Cµ (? )xµ + C4 (? ),

where ? = ? (x) is a smooth function determined from the equation

? ? (14)

Cµ (? )xµ + C4 (? ) = 0,

cµ (? ), C4 (? ) are arbitrary smooth functions satisfying the conditions

?? (15)

Cµ Cµ = Cµ Cµ = 0.

On the general solution of the d’Alembert equation 541

It is not difficult to show that the solutions (13)–(15) are complex (see Lemma 1

below). Another class of complex solutions of the system (1) with n = 4 was construc-

ted by Yerugin [3]. But neither Bateman formulae (13)–(15) not Yerugin’s results give

the general solution of the system (1) with n = 4.

3. Proof of Theorems 1–3. It is well-known that the system of PDE (1) admits

an infinite-dimensional Lie algebra [6]. It is this very fact that enables us to construct

its general solution.

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

a = 1, n ? 1, (16)

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

Evidently, the transformation (16) is defined for all functions u(x), such that ux0 ? 0.

But the system (1) with ux0 = 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 (16) is defined on the whole set of solutions

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

Being rewritten in the new variables z, w(z) the system (1) takes the form

n?1 n?1

2

(17)

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 we have

n?1

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

a,b=1

whence, by force of (17),

n?1

2

(18)

wza zb = 0.

a,b=1

Since w(z) is a real valued function from (18) it follows that wza zb = 0, a, b =

1, n ? 1, whence

n?1

(19)

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

a=1

In (19) ?a , ? ? C 2 (R1 , R1 ) are arbitrary functions.

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

Substituting (19) into the second equation of system (17), we have

n?1

2

(20)

?a (z0 ) = 1.

a=1

Thus, the formulae (19), (20) give the general solution of the system of nonlinear

PDE (17). Rewriting (19), (20) in the initial variables, we get

n?1 n?1

2

(21)

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

a=1 a=1

To represent the formula (21) in the manifestly covariant form (3) 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 (21) we come to the formulae (7).

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

the formulae (7) under Aµ ? 0, µ = 0, n ? 1, B(u) = u + const, JSSF (7) give the

general solution of the system of the PDE (1) with an arbitrary n ? N. The theorem

is proved.

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

real-valued function u(x). It a solution of the system (1) is looked for in the class of

complex-valued functions u(x). JSSF (7) do not give its general solution with n > 3.

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

Further we shall adduce the proof of Theorem 3 (Theorem 2 is proved in the same

way).

Case 1. ux0 = 0. In this case the hodograph transformation (16) reducing the

system (1) with n = 5 to the form

4 4

2

(22)

wza za = 0, wza = 1, wz0 = 0

a=1 a=1

is defined.

The general solution of nonlinear complex Eqs. (22) was constructed by the authors

in [7]. It is given by the following formulae:

4

(23)

1) w(z) = ?a (?, z0 )za + ?1 (?, z0 ),

a=1

where ? = ? (z0 , . . . , z4 ) is the function determined from the equation

4

(24)

?a (?, z0 )za + ?2 (?, z0 ) = 0

a=1

and ?a , ?a , ?1 , ?2 ? C 2 (C2 , C1 ) are arbitrary functions satisfying the relations

4 4 4 4

??a

2 2

(25)

?a = 1, ?a ?a = ?a = 0, ?a = 0.

??

a=1 a=1 a=1 a=1

On the general solution of the d’Alembert equation 543

4

(26)

2) w(z) = ?a (z0 )za + ?1 (z0 ),

a=1

where ?a , ?1 ? C 2 (C2 , C1 ) are arbitrary functions satisfying the relation

4

2

(27)

?a = 1.

a=1

Rewriting the formulae (24), (25) in the initial variables x, u(x), we have

4

(28)

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

a=1

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

4

(29)

?a (?, u)xa + ?2 (?, u) = 0

a=1

and the relations (25) hold.

Evidently, the formulae (7) under

C1 = ??1 ,

A0 = 1, Aa = ?a ,

(30)

C2 = ??1 ,

B0 = 0, Ba = ?a , a = 1, 4.

Further, by force of inequality wza ? 0 we get from (23)

4

(31)

(?az0 + ?a? ?z0 )xa + ?1z0 + ?1? ?z0 = 0.

a=1

Differentiation of (24) with respect to z0 yields the following expression for ?z0 :

?1

4 4

?z0 = ? ?az0 xa + ?2z0 ?a? xa + ?2? .

a=1 a=1

Substitution of the above result, into (31) yields relation of the form

4 4

?az0 xa + ?1z0 ?a? xa + ?1?

?1

4

a=1 a=1

?a? xa + ?2? = 0.

4 4

a=1

?az0 xa + ?2z0 ?a? xa + ?2?

a=1 a=1

As the direct, check shows the above inequality follows from (10) with the condi-

tions (30).

Now we turn to solutions of the system (22) of the form (26). Rewriting the

formulae (26), (27) in the initial variables x, u(x) we get

4 4

2

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

a=1 a=1

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

After making in the obtained equalities the chance ?a = Aa A?1 , a = 1, 4, ?1 =

0

?C1 A?1 , we arrive at the formulae (11), (12).

0

Thus, under ux0 ? 0 the general solution of the system (1) is contained in the

class of functions u(x) given by the formulae (7)–(10) or (11), (12).

Case 2. ux0 ? 0, u ? const. It is well-known that the system of PDE (1) is

invariant under the generalized Poincar? group P (1, n ? 1) (see, e.g. [8])

e

xµ = ?µ? x? + ?µ , u (x ) = u(x),

ñòð. 127 |