стр. 91 |

Отметим, что в трехмерном случае x ? R(1, 2) общее решение системы (3), (4)

было построено Коллинзом [7]. Однако используемый геометрический метод, как

отмечал сам автор, не обобщается на случай четырех независимых переменных.

Полученные им решения ее держатся в приведенных выше классах решений при

N = 0, 1, 2.

Таким образом, теоремы 1–3 дают полное решение задачи исследования совме-

стности и построения общего решения системы нелинейных волновых уравнений

(3), (4).

382 В.И. Фущич, Р.З. Жданов, И.В. Ревенко

1. Фущич В.И., Симметрия в задачах математической физики, в сб. Теоретико-алгебраические

исследования в математической физике, Киев, Ин-т матем. АН УССР, 1981.

2. Фущич В.И., Жданов Р.З., Ревенко И.В., Совместимость и решения нелинейные уравнений

д’Аламбера и Гамильтона, Препринт N 90.39, Киев, Ин-т математики АН УССР, 1990, 65 с.

3. Fushchych W.I., Zhdanov R.Z., On some new exact solutions of nonlinear d’Alembert and Hamilton

equations, Preprint N 468, Minneapolis, Institute for Mathematics and its Applications, 1988, 5 p.

4. Fushchych W.I., Zhdanov R.Z., On some new exact solutions of the nonlinear d’Alembert–Hamilton

system, Phys. Lett. A, 1989, 141, № 3–4, 113–115.

5. Соболев С.Л., Функционально-инвариантные решения волнового уравнения, Труды физико-

математического ин-та им. В. А. Стеклова, Л., Изд-во АН СССР, 1934, 5, 259–264.

6. Бейтмен Г., Математическая теория распространения электромагнитных волн, М., Физматгиз,

1958, 179 с.

7. Collins С.В., Complex potential equations I. A technique for solutions, Proc. Camb. Ph. Soc., 1976,

80, № 1, 165–171.

W.I. Fushchych, Scientific Works 2002, Vol. 4, 383–389.

On the reduction of the nonlinear

multi-dimensional wave equations

and compatibility of the d’Alembert–

Hamilton system

W.I. FUSHCHYCH, R.Z. ZHDANOV, I.A. YEGORCHENKO

The necessary conditions of the compatibility of the d’Alembert–Hamilton system in

Minkowsky space R(1, n) are established. The problem of reduction of P (1, n)-invariant

wave equations to ordinary differential equations is discussed.

1. Since Euler the method of reduction of partial differential equations (PDE) to

ordinary differential equations (ODE) is one of the most effective ways to construct

the exact solutions of PDE.

The papers [1–5] contain the symmetry reduction to ODE of the d’Alembert equati-

on

2u = G(u), 2 ? ?x0 ? ?x1 ? · · · ? ?xn

2 2 2

(1)

(where G(u) is an arbitrary smooth function). So the many-dimensional PDE [1] with

the ansatz

(2)

u = ?(?),

where ? ? C 2 (R1 , R1 ); ? = ?(x) ? C 2 (Rn+1 , R1 ), the new variable, is reduced to the

ODE of the form

(3)

?µ ?µ ?(?) + (2?)?(?) = G(?),

? ?

where ?µ ? ??/?xµ , µ = 0, . . . , n. Hereafter summation over repeated indices is

understood in the Minkowsky space R(1, n) with the metric gµ? = diag(1, ?1, . . . , ?1).

In [3–5] using the symmetry properties of Eq. (1) and the subgroup structure of

the P (1, n) group the new variables ? = ?(x) for Eq. (3) had been constructed.

Equation (3) depends on ? and does not depend on “old” variables x. ?(x) are

invariants of the corresponding subgroups of the Poincar? group P (1, n).

e

In the present paper we suggest the approach to the problem of reduction of PDE

to ODE more general than one based on the employment of the symmetry properties

of PDE [1–5].

Definition. We say that the ansatz (2) reduces PDE (1) to ODE (3) when the new

variable ? = ?(x) satisfies both

2? = F1 (?), (4)

?µ ?µ = F2 (?),

where F1 (?), F2 (?) are arbitrary smooth functions. Further we call Eqs. (4) the

d’Alembert–Hamilton system.

J. Math. Anal. and Appl., 1991, 161, № 2, 352–360.

384 W.I. Fushchych, R.Z. Zhdanov, I.A. Yegorchenko

Evidently for every ?(x) satisfying the system (4) ODE (1) depends on ? only.

Thus the problem of finding of the ansatze (2) reducing PDE (1) to ODE leads to the

construction of solutions of the d’Alembert–Hamilton system (4).

Before solving the system (4) it is necessary to clear the matter of its compatibility,

i.e., to describe all functions F1 , F2 for the system (4) to possess nontrivial solutions.

In the three-dimensional case (n = 2) the compatibility of the system (4) was

investigated by Collins [6] with the geometry methods. The compatibility of the

d’Alembert–Hamilton system in the four-dimensional space R(1, 3) was investigated

in detail in [7]. We had generalized the results of [7] for the case of (1+n)-dimensional

system of PDE (4) using the classical Hamilton–Cayley theorem.

2. The system (4) with the change of dependent variable z = z(?) transforms to

the following system of PDE

2? = F (?), (5)

?µ ?µ = ?, ? = const,

Eq. (3) having the form

(6)

?? + F (?)? = G(?).

? ?

Before formulating the main result we adduce some preliminary statements.

Lemma 1. The solutions of the system (5) satisfy the equalities

1 2?

?

?µ?1 ??1 µ = ??F (?), ?µ?1 ??1 ?2 ??2 µ = ? F (?), ...,

2!

(7)

(??)N (N )

N ? 1,

?µ?1 ??1 ?2 . . . ??N µ = F (?),

N!

where ?µ? ? ? 2 ?/?xµ ?x? , µ, ? = 0, . . . , n.

Proof. We prove the lemma with the method of mathematical induction by N .

Having differentiated twice the second equation of the system (5) with respect to

x? , x? we obtain the relation

(8)

?µ?? ?µ + ?µ? ?µ? = 0.

Convoluting (8) with the metric tensor g ?? we come to the equality

?µ? ?µ? + ?µ 2?µ = 0.

?

Since 2?µ = (?/?xµ )F (?) = ?µ F (?), on the solutions of the system (5) the last

expression can be rewritten in the form

?

?µ? ?µ? + ?F (?) = 0.

Thus the basic statement of induction is proved.

Let us suppose that the lemma holds for N = k. We prove that whence its

statement follows for N = k + 1.

Convoluting (8) with the tensor

???2 ??2 ?3 · · · ??k ?

we get the equality

?µ? ???2 · · · ??k ? ??µ + ?µ ???µ ???2 · · · ??k ? = 0. (9)

On the reduction of the nonlinear multi-dimensional wave equations 385

Since

1

?µ (??? ???2 · · · ??k ? )µ =

?µ ???µ ???2 . . . ??k ? =

k+1

(??)k (k) (??)k+1 (k+1)

1

=?

= ?µ F (?) F (?)

k+1 k! (k + 1)!

µ

(we used the assumption of induction) then it follows from (9) that

(??)k+1 (k+1)

?µ?1 ??1 ?2 · · · ??k+1 µ = F (?)

(k + 1)!

The Lemma is proved.

Lemma 2. On the solutions of the system (5) the equality

(10)

det ?µ? = 0

holds.

The proof follows from the fact that (10) is the criterium of functional dependence

of ?0 , . . . , ?n .

Theorem 1. For the system (5) to be compatible it is necessary that

F (?) = ?f?(?)f ?1 (?), (11)

f satisfying the condition

dn+1 f (?)

(?) ?

(n+1)

(12)

f = 0.

d? n+1

Proof. Let us first consider the case ? = 0. For an arbitrary (n + 1) ? (n + 1)-matrix

W = wµ? by virtue of the Hamilton–Cayley theorem the equality

n?1

(?1)k Mk tr (W n?k ) + (?1)n n det W = 0 (13)

k=0

is true. Mk in (13) is the sum of basic minors of the order k of the matrix W ,

which is calculated with the recurrent formula

k?1

Ml tr (W k?l ) (?1)k?l k ?1 , k ? 1;

(?1)l

Mk =

(14)

l=0

M0 = 1.

We take the matrix elements of W as

n

wµ? = g?? ?µ? ,

?=0

then from Lemmas 1, 2

(??)k?1 (k?1)

tr (W k ) = (15)

F (?), det W = 0.

(k ? 1)!

386 W.I. Fushchych, R.Z. Zhdanov, I.A. Yegorchenko

The substitution of formula (15) into (14) gives the ODE for determination of the

function F = F (?). Let us show that this ODE reduces using the nonlocal change of

variable (11) to the form (12).

Let

N

Mk tr (W N ?k+1 )

(?1)k

YN =

k=0

Mk = ((?1)k?1 /k)Yk?1 ; whence

then

(N ?k)

N

(?1)N ?k+1 N +1?k f?

YN = ? Yk?1 .

k(N ? k)! f

k=0

Using the method of mathematical induction we prove that

(?1)N +1 N +1 f (N +1)

(16)

YN = ? .

N! f

For N = 1, 2, 3 this equality follows from the results of [7]. Let us assume that (16)

holds for every m ? N, m ? N ? 1. We show that whence it follows that (16) is true

for m = N .

Indeed

(N ?k)

N

(?1)k+1 k f (k) (?1)N ?k N +1?k f?

YN = ? ? =

k(k ? 1)! f (N ? k)! f

k=0

(N ?k)

N

f?

(?1)N +1 ?N +1 f (k) (?1)N +1 ?N +1 f (N +1)

k

= CN = ,

N! f f N! f

k=0

the same as what was to be proved.

From the equality (10) Yn = (?1)n+1 n det W = 0 whence by virtue of (15), (16)

we obtain

f (n+1) = 0.

Let us consider now the case ? = 0. Using Lemmas 1, 2 we have

tr (W k ) = 0, k = 2, n; det W = 0.

Taking into account these equalities we can rewrite the Hamilton–Cayley identity

in the form

Yn = 0,

where Yn = (?1)n+1 (F/n!). Whence we conclude that F = 0. The theorem is proved.

Consequence. The system 2u = F (u), uµ uµ = 0 is compatible iff F (u) = 0.

Proof. The necessity of the above statement follows from the Theorem 1. The suffi-

ciency is proved by the fact that the function u(x) = x0 + x1 satisfies both the

d’Alembert (2u = 0) and the Hamilton (uµ uµ = 0) equations.

Let us note that this consequence was proved in [9] by another technique.

On the reduction of the nonlinear multi-dimensional wave equations 387

Theorem 2. The system of PDE (5) is invariant with respect to the conformul group

of transformations of the Minkowskv space R(1, n) iff [7, 8]

F (?) = ?n(? + C)?1 , (17)

c = const, ? > 0.

The proof is carried out by S. Lie’s method.

Let us note that the formula (17) is obtained from (11) when f = (? + c)n . So

Theorem 2 demonstrates the deep connection between the symmetry of overdetermi-

ned system of PDE (5) and its compatibility.

Note. It is well known that PDE (1) is invariant under the conformal group C(1, n) iff

G(u) = cu(n+3)/(n?1) (see, e.g., [3, 10, 11]). Thus the additional condition uµ uµ = ?

picks out the subset of solutions of Eq. (1) which admits a wider symmetry group

than the set of its solutions in a whole. In other words the nonlinear d’Alembert

equation is conditionally invariant under the conformal group if G(u) = ?n(u + c)?1

(the notion of conditional invariance of PDE was introduced in [12–14]; see also [15,

16]).

The sufficient conditions of the compatibility of the d’Alembert–Hamilton sys-

tem (5) are

F (?) = |?|N (? + c)?1 , (18)

where c = const, N = 1 ? n, 2 ? n, . . . , 0, 1, . . . , n.

As shown by Collins [6] the above conditions are the necessary and sufficient

ones for the system (5) to be compatible if n = 1, 2. In the Appendix we list exact

solutions of the d’Alembert Hamilton system under (18) for n = 3 obtained in [3, 5,

7–9, 17]. Let us emphasize that solutions numbered (5)–(7), (9) are not invariants

of the Poincar? group P (1, n). Nevertheless they satisfy the d’Alembert–Hamilton

e

system and, consequently, can be used to reduce Eq. (1) to ODE via ansatz (2).

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