стр. 29 |

b = ?2 = 0).

In the above formulae ?1 , ?2 , ?3 , ?4 are arbitrary real constants, ?1 = 0.

Substitution of the expression for F

(12)

F = ?1 ? ln ? + ?1 ? + ?3

into Eq. 3 from the system (10) yields

?ixµ ?ixµ = Qi (?i ), i = 1, 2,

Ci = ?1 Q?1 (?i )?i ln ?i , i = 1, 2,

i

2a = ?1 a ln a + ?2 a, ?3 = 0.

Since in Eq. (12) ?1 = 0, we can rescale the function ? > k? in such a way that

F (?) takes the form F = ?1 ? ln ?. The theorem is proved.

Note. A classical example of the anti-reduction of mathematical physics equations is

the procedure of separation of variables. But the method of separation of variables can

Anti-reduction of the nonlinear wave equation 119

be effectively applied to linear second-order PDEs only, whereas the anti-reduction

procedure is evidently applicable to nonlinear differential equations.

Thus each solution of the system (7) after being substituted into ansatz (6) reduces

the nonlinear PDE (5) to two second-order QDEs

Qi (?i )?i = ??i ln ?i ,

? i = 1, 2.

Let us write down some particular solutions of Eqs. (7) under a = 1.

= ln(x2 ? x2 ), ?2 = ln(x2 + x2 );

1. ?1 0 3 1 2

= ln(x0 ? x3 ), ?2 = x1 ;

2 2

2. ?1

= x0 , ?2 = ln(x2 + x2 );

3. ?1 1 2

2 2

4. ?1 = ln(x1 + x2 ), ?2 = x3 ;

5. ?1 = x0 , ?2 = x1 ;

?1 = (x2 ? x2 ? x2 )?1/2 ,

6. ? = x3 ;

0 1 2

?1 = x0 , ?2 = (x2 + x2 + x2 )?1/2 ;

7. 1 2 3

?1 = x1 cos ?1 + x2 sin ?1 + ?2 , ?2 = x1 sin ?1 ? x2 cos ?1 + ?3 .

8.

In the above formulae ?1 , ?2 , ?3 are arbitrary smooth functions on x0 + x3 .

Let us emphasize that the above ansatzes can not be obtained within the frame-

work of the classical Lie approach (see, e.g. [5, 6]), because the maximal symmetry

group admitted by Eq. (5) is the Poincar? group P (1, 3) [2] and the general form of

e

Poincar?-invariant ansatz is given by the formula (2).

e

1. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных

уравнений математической физики, Киев, Наук, думка, 1989, 336 с.

2. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., Subgroups of the Poincar? group and their

e

invariants, J. Math. Phys., 1976, 17, 977–985.

3. Fushchych W.I., Zhdanov R.Z., Yegorchenko I.A., On the reduction of the nonlinear multi-dimen-

sional wave equations and compatibility of the d’Alembert–Hamilton system, J. Math. Anal. Appl.,

1991, 161, 352–360.

4. Фущич В.И., Условная симметрия уравнений нелинейной математической физики, Укр. мат.

журн., 1991, 43, № 11, 1456–1470.

5. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.

6. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986, 497 p.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 120–125.

On the new approach to variable separation

in the wave equation with potential

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

Пропонується конструктивний пiдхiд до розв’язання проблем роздiлення змiнних

для двомiрного хвильового рiвняння utt ? uxx = V (x)u. У рамках цього пiдходу

описанi yci потенцiали, що допускають, роздiлення змiнних i вказанi вiдповiднi

системи координат.

A problem of variable separation (VS) in the wave equation

ux0 x0 ? ux1 x1 + V (x1 )u = 0 (1)

as considered in [1–3], consists of two problems. The first one is to describe all

functions V (x1 ) providing VS in (1) in, at least, two inequivalent coordinate systems.

The second one is to describe all coordinate-systems such that equation (1) admits

VS for a given potential V (x1 ). Surprisingly enough, the both problems are not

completely solved yet.

Our approach to the problem of VS in the wave equation (1) is based on the idea

of its reduction to two ordinary differential equations

(2)

?i = Ai (?i , ?)?i + Bi (?i , ?)?i ,

? i = 1, 2

with the use of ansatz of special structure [4–6]

(3)

u = A(x0 , x1 )?1 (?1 (x0 , x1 ))?2 (?2 (x0 , x1 )).

In the formulas (2), (3) A1 , A2 , B1 , B2 , A, ?1 , ?2 are sufficiently smooth real

functions, ? ? R1 is some parameter, no summation over i is carried out.

The formulas of the form (3) can be found in the classical works Euler, d’Alembert,

Batemen and by some other contemporary mathematicians (see, for example, the

review by Koornwinder [7]).

Definition. We say, that equation (1) admits VS in the coordinates ?1 , ?2 if substi-

tution of the ansatz (3) into (1) with subsequent exclusion of the second derivatives,

?1 , ?2 according to formulas (2) turns it into zero identically with respect to the

??

variables ?1 , ?2 , ?1 , ?2 , ?.

??

Substituting ansatz (3) into differential equation (1), expressing functions ?i in

?

terms of ?i , ?i , i = 1, 2 and splitting the obtained expression with respect to the

?

independent variables ?1 ?2 , ?1 ?2 , ?1 ?2 , ?1 ?2 we get the following system of nonli-

?? ? ?

near partial differential equations:

1) A2?1 + 2Axµ ?1xµ + AA1 ?1xµ ?1xµ = 0,

2) A2?2 + 2Axµ ?2xµ + AA2 ?2xµ ?2xµ = 0,

(4)

2A + A(B1 ?1xµ ?1xµ + B2 ?2xµ ?2xµ ) + AV (x1 ) = 0,

3)

4) ?1xµ ?2xµ = 0.

Доповiдi АН України, 1993, № 1, С. 27–32.

New approach to variable separation in the wave equation with potential 121

Hereafter, the summation over the repeated Greek indices is under-stood in the

Minkovski space M (1, 1) with a metric tensor gµ? = diag (1, ?1).

Thus to describe all potentials V (x1 ) and coordinate systems ?1 , ?2 providing VS

in (1) one has to solve nonlinear system (4). At first glance such an approach seems

to have poor prospects: to solve linear equation (1) it is necessary to integrate rather

complicated system of nonlinear partial differential equations (4). But system (4) is

overdetermined one. This fact has enabled us to construct its general solution in

explicit form. Let us emphasize that the same is true when reducing nonlinear wave

equation to the ordinary differential equation [5, 6] .

It is not difficult to show that from the forth equation of system (4) it follows that

(5)

(?1xµ ?1xµ )(?2xµ ?2xµ ) = 0.

Differentiating equations 1), 2) from (4) and using (5) we have

A1? = A2? = 0.

Consequently, the relation B1? B2? = 0 holds. Differentiating equation (3) with

respect to ?, we get

B1? ?1xµ ?1xµ + B2? ?2xµ ?2xµ = 0

or

?2xµ ?2xµ

B1?

=? .

B2? ?1xµ ?1xµ

Differentiating the above equality with respect to ?, we obtain

B1?? B2??

(6)

= .

B1? B2?

Since function Bi depends on the variable ?i and the functions ?1 , ?2 are inde-

pendent, it follows from (6) that

Bi?? = ?(?)Bi? , i = 1, 2.

Integration of the above ordinary differential equations yields

Bi = ?(?)fi (?i ) + gi (?i ), i = 1, 2.

After redefining the parameter ?, we have

(7)

Bi = ?fi (?i ) + gi (?i ), i = 1, 2.

Substituting (7) into equation (3) and splitting the obtained equality with respect

to ?, we come to the following partial differential equations:

2A + A(g1 ?1xµ ?1xµ + g2 ?2xµ ?2xµ ) + V (x1 )A = 0,

3a)

(8)

3b) f1 ?1xµ ?1xµ + f2 ?2xµ ?2xµ = 0.

Before integrating overdetermined system of nonlinear equations (4), (8), make an

important remark. It is evident, that the ansatz structure does not change with the

transformation of the form

A > Ah1 (?1 )h2 (?2 ), ?i > ?1 (?i ), (9)

i = 1, 2.

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

That is why, solutions of system (4), (8) connected by relations (9) are considered

as equivalent.

Making the change (9) in equations 1), 2), 3b) by the appropriate, choice of

functions hi , ?i one can obtain f1 = f2 = 1, A1 = A2 = 0. Consequently, functions

?1 , ?2 satisfy the equations

?1xµ ?2xµ = 0, ?1xµ ?1xµ + ?2xµ ?2xµ = 0.

whence

(?1 ± ?2 )xµ (?1 ± ?2 )xµ = 0.

Integrating the above equations we get

?2 = f (?) ? g(?), (10)

?1 = f (?) + g(?),

where f , g are arbitrary smooth functions, ? = 1 (x1 + x0 ), ? = 1 (x1 ? x0 ).

2 2

Substitution of the formulas (10) into equations 1), 2) from (4) yields the following

equations for a function A(x0 , x1 )

(ln A)x0 = 0, (ln A)x1 = 0,

whence A = 1.

At last, substituting the obtained results into the equation 3b) from (8) we come to

a conclusion that the problem of integration of system (4), (8) is reduced to solution

of the functional-differential equation

df dg

V (x1 ) = [g1 (f + g) ? g2 (f ? g)] (11)

.

d? d?

And what is more, solution with separated variables (3) reads

u = ?1 (f (?) + g(?))?2 (f (?) ? g(?)). (12)

To integrate (11) it is convenient to make the hodograph transformation

(13)

? = P (f ), ? = R(g)

? ?

with P ? 0, R ? 0, equation (11) taking the form

? ? ?

g1 (f + g) ? g2 (f ? g) = P (f )R(g)R(g)V (P + R). (14)

Evidently, equality (14) is equivalent to the following relation:

? ?

(?f ? ?g )[P (f )R(g)V (P + R)] = 0

2 2

or

... ...

? ? ?? ? ? ? ?? ? ??

(P R ? R P )V + 3P R(P ? R)V + P R(P 2 ? R2 )V = 0. (15)

Without going into details of integration of equation (15) we give the final results.

Theorem 1. The general solution of (15) is given by one of the following formulas:

? ?

is an arbitrary function, R = P = ?;

1. V = V (x1 )

?

V = m(x1 + C)p2 = ?P + ?, R2 = ?R + ?;

2. ?

New approach to variable separation in the wave equation with potential 123

V = m(x1 + C)?2 , P = F (f ), R = G(g),

3.

?

F 2 = ?F 4 + ?F 3 + ?F 2 + ?F + ?, (16)

?

G2 = ?G4 ? ?G3 + ?G2 ? ?G + ?;

V = m sin?2 (x1 + C),

4. P = arctg F (f ), R = arctg G(g),

where F , G are determined by (16);

V = m sh?2 (x1 + C),

5. P = arth F (f ), R = arth G(g),

where F , G are determined by (16);

V = m ch?2 (x1 + C),

6. P = arctg F (f ), R = arth G(g),

where F , G are determined by (16);

? ?

V = m exp(??x1 ), P 2 = ?e2P + ?eP + ?, R2 = ?e2R + ?eR + ?;

7.

V = cos?2 (x1 + C)[m1 + m2 sin(x1 + C)],

8.

? ?

P 2 = ?2 sin 2P + ? 2 , R2 = ?2 sin 2R + ? 2 ;

V = ch?2 (x1 + C)[m1 + m2 sh(x1 + C)],

9.

? ?

P 2 = ? sh 2P + ? ch 2P ? ? 2 , R2 = ? sh 2R ? ? ch 2R ? ? 2 ;

V = sh?2 (x1 + C)[m1 + m2 ch(x1 + C)],

10.

? ?

P 2 = ? sh 2P + ? ch 2P ? ? 2 , R2 = ?? sh 2R + ? ch 2R ? ? 2 ;

? ? ? ?

V = m1 exp C1 x1 + m2 exp 2Cx1 , P = ?P 2 + ?, R = ?R2 + ?;

11.

V = m1 + m2 (x1 + C)?2 , P 2 = ?P 2 + ?P + ?, R2 = ?R2 ? ?R + ?;

? ?

12.

? ?

V = m, P 2 = ?P 2 + ?1 P + ?1 , R2 = ?R2 + ?2 R + ?2 .

13.

Here ?, ?i , ?i , ?, ?, m, m1 , m2 , C are arbitrary real constants.

Thus, Theorem 1 gives the complete solution of the problem of VS in wave equa-

tion (1).

Note 1. Equation (1) with potentials V = m sin?2 x1 , V = m ch2 x1 , V = m sh?2 x1

is reduced to equation (1) with the potential V = mx?2 by the changes of variables

1

1 1

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