стр. 30 |

2 2

1 1

(y1 ± y0 ) = arcth (x1 ± x0 ),

2 2

1 1

(y1 + y0 ) = arctg (x1 + x0 ),

2 2

1 1

(y1 ? y0 ) = arcth (x1 ? x0 ).

2 2

Note 2. Equation (1) with the potential V = m exp Cx1 is reduced to the Klein–

Gordon–Fock equation 2u + mu = 0 with the change of variables

1 C

(y1 ± y0 ) = C ?1 exp (x1 ± x0 ).

2 2

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

It is evident that the equation (1) admits VS in Cartesian coordinates ?1 = x0 ,

?2 = x1 under arbitrary function V (x1 ). That is why the most interesting potentials

are such that there exist new coordinate systems providing VS. From the Theorem 1

and Notes 1, 2 it follows that equations (1) admitting VS in, at least, two inequivalent

coordinate systems, are locally equivalent to one of the following wave equations

2u + mu = 0,

1.

2u + mx1 u = 0,

2.

2u + mx?2 u = 0,

3. 1

2u + (m1 + m2 x?2 )u = 0,

4. 1

(17)

2u + (m1 + m2 sin x1 ) cos?2 x1 u = 0,

5.

2u + (m1 + m2 sh x1 ) ch?2 x1 u = 0,

6.

2u + (m1 + m2 ch x1 ) sh?2 x1 u = 0,

7.

2u + (m1 + m2 ex1 )ex1 u = 0.

8.

A detailed analysis of the coordinate systems providing VS in equation (17) will

be carried out in our future work.

In conclusion, we note that the equation (1) is intimately connected with the wave

equation

vtt ? C 2 (x)vxx = 0. (18)

This connection is given by the formula

dx

(19)

v(t, x) = C(x)u t, .

C(x)

Applying the Theorem 1 and the formula (19) it is not difficult to carry out VS in

partial differential equation (18).

Besides, Lorentz-invariant wave equation

uy0 y0 ? uy1 y1 + U (y0 ? y1 )u = 0

2 2

(20)

1

can also be reduced to the form (1), where U (? ) = (ln ? ) by the change of

4? V

variables

y0 = ex1 /2 ch x0 , y1 = ex1 /2 sh x0 .

That is why, one can at once, point out all potentials U = U (? ), ? = x2 ? x2 providing

0 1

VS in the wave equation (20):

= m? ?1 ln ?, U = m? ?1 (ln ? )?2 , U = m1 ? ?1 + m2 ? ?1 (ln ? )?2 ,

U

= m? ?1 , U = ? ?1 (m1 + m2 sin ln ? )(cos ln ? )?2 ,

U

= ? ?1 (m1 + m2 sh ln ? )(ch ln ? )?2 ,

U

= ? ?1 (m1 + m2 ch ln ? )(sh ln ? )?2 , U = m1 + m2 ?.

U

New approach to variable separation in the wave equation with potential 125

1. Miller W., Symmetry and separation of variables, London, Addison Wesley, 1977, 280 p.

2. Шаповалов В.Н., Симметрия и разделение переменных в линейных дифференциальных уравне-

ниях второго порядка, Дифференциальные уравнения, 1980, 16, № 10, 1864–1874.

3. Багров В.Г., Гитман Д.М., Тернов Д.М., О точных, решениях релятивистских волновых урав-

нениях, Новосибирск, Наука, 1982, 143 с.

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

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

5. 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.

6. 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, № 2, 352–360.

7. Koornwinder T.H., A priecise definition of separation of variables, Lecture Notes in Math., 1980,

810, 240–263.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 126–138.

Orthogonal and non-orthogonal separation

of variables in the wave equation

utt ? uxx + V (x)u = 0

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

We develop a direct approach to the separation of variables in partial differential equa-

tions. Within the framework of this approach, the problem of the separation of vari-

ables in the wave equation with time-independent potential reduces to solving an over-

determined system of nonlinear differential equations. We have succeeded in constructing

its general solution and, as a result, all potentials V (x) permitting variable separation

have been found. For each of them we have constructed all inequivalent coordinate

systems providing separability of the equation under study. It should be noted that the

above approach yields both orthogonal and non-orthogonal systems of coordinates.

1. Introduction

Separation of variables (SV) in two- and three-dimensional Laplace, Helmholtz,

d’Alembert and Klein–Gordon–Fock equations has been carried out in classical works

by Bocher [1], Darboux [2], Eisenhart [3], Stepvanov [4], Olevsky [5], and Kalnins

and Miller (see [6] and references therein). Nevertheless, a complete solution to the

problem of sv in a two-dimensional wave equation with time-independent potential

(2 + V (x))u ? utt ? uxx + V (x)u = 0 (1)

has not been obtained yet. In (1) u = u(t, x) ? C 2 (R2 , R1 ), V (x) ? C(R1 , R1 ).

Equations belonging to the class (1) are widely used in modern mathematical

physics and can be related to other important linear and nonlinear partial differential

equations (PDE). First, we mention the Lorentz-invariant wave equation

uy0 y0 ? uy1 y1 + U (y0 ? y1 )u = 0.

2 2

(2)

The above equation can be reduced to the form (1) with the change of variables [7]

t = exp(y1 /2) cosh y0 , x = exp(y1 /2) sinh y0

and what is more, potentials V (? ), U (? ) are connected by the following relation:

U (? ) = (4? )?1 V (? ).

Another related equation is the hyperbolic type equation

vx0 x0 ? c2 (x1 )vx1 x1 = 0 (3)

that is widely used in various areas of mathematical physics.

Equation (3) is reduced to the form (1) by the change of variables

u(t, x) = [c(x0 )]?1/2 v(x0 , x1 ), [c(x1 )]?1 dx1

t = x0 , x=

J. Phys. A: Math. Gen., 1993, 26, P. 5959–5972.

Orthogonal and non-orthogonal separation of variables in the wave equation 127

and what is more

V (x) = ?c3/2 (x1 )[c1/2 (x1 )], (4)

where x = [c(x1 )]?1 dx1 .

The third related equation is the nonlinear wave equation

Wtt ? [c?2 (W )Wx ]x = 0. (5)

By substitution W = Rx , equation (5) is reduced to the form

Rtt ? c?2 (Rx )Rxx = 0.

Applying to the above equation the Legendre transformation

v = tRt + xRx ? R,

x0 = Rt , x1 = Rx , vx0 = t, vx1 = x,

we obtain (3). Consequently, the method of SV in the linear equation (1) makes it

possible to construct exact solutions of the nonlinear wave equation (5).

Let us also mention the Euler–Poisson–Darboux equation

vtt ? vxx ? x?1 vx + m2 x?2 v = 0 (6)

that is reduced to an equation of the form (1)

utt ? uxx + (m2 ? 1/4)x?2 u = 0

by the change of dependent variable v(t, x) = x?1/2 u(t, x).

For the solution of (1) with separated variables ?1 (t, x), ?2 (t, x), we use the ansatz

(7)

u(t, x) = Q(t, x)?1 (?1 )?2 (?2 )

which reduces PDE (1) to two ordinary differential equations (ODE) for functions

?1 , ?2 .

There exist three possibilities for SV in (1). The first is to separate it into two

second-order ODE. The second possibility is to separate (1) into first-order and second-

order ODE, and the third possibility is to separate (1) into two first-order ODE. In

the present paper we shall investigate in detail the first two possibilities. The third

possibility requires special separate consideration and will be the topic of future

publications.

Consider the following ODE:

(8)

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

? ? i = 1, 2,

where Ai , Bi ? C 2 (R1 ? ?, R1 ) are some unknown functions, ? ? ? ? R1 is a real

parameter (separation constant).

Definition 1 [7, 8]. Equation (1) separates into two ODE if substitution of the

ansatz (7) into (1) with subsequent exclusion of the second derivatives ?1 , ?2 ??

according to (8) yields an identity with respect to the variables ?i , ?i , ? (considered

?

as independent).

On the basis of the above definition one can formulate a constructive procedure of

SV in (1), suggested for the first time in [7]. At the first step, one has to substitute

expression (7) into (1) and to express the second derivatives ?1 , ?2 via functions ?i , ?i

?? ?

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

according to (8). At the second step, the equality obtained is split with respect to the

independent variables ?i , ?i , ?. As a result, one obtains an over-determinated system

?

of partial differential equations for functions Q, ?1 and ?2 with undefined coefficients.

The general solution of this system gives rise to all systems of coordinates providing

separability of (1).

Definition 2. Equation (1) separates into first- and second-order ODE

?1 = A(?1 , ?)?1 ,

?

(9)

?2 = B1 (?2 , ?)?2 + B2 (?2 , ?)?2

? ?

if substitution of the ansatz (7) into (1) with subsequent exclusion of derivatives

?1 , ?2 according to (9) yields an identity with respect to the variables ?1 , ?2 , ?2 ,

?? ?

? (considered as independent).

Let us emphasize that the above approach to SV in (1) has much in common with

the non-Lie method of reduction of nonlinear PDE suggested in [9–11]. It is also

important to note that the idea to represent solutions of linear differential equations in

the “separated” form (7) goes as far as the classical works by Fourier and Euler (for

a modern exposition of the problem of SV, see Miller [12] and Koornwinder [13]).

2. Orthogonal separation of variables in equation (1)

It is evident that (1) admits SV in Cartesian coordinates ?1 = t, ?2 = x under

arbitrary V = V (x).

Definition 3. Equation (1) admits non-trivial SV if there exist at least one coordi-

nate system ?1 (t, x), ?2 (t, x) different from the Cartesian system providing its

separability.

Next, if one makes in (1) the following transformations:

t > C1 t, x > C1 x,

(10)

t > t, x > x + C2 , Ci ? R1

then the class of equations (1) transforms into itself and what is more

V (x) > V (x) = C1 V (C1 x),

2

(10a)

V (x) > V (x) = V (x + C2 ).

That is why potentials V (x) and V (x), connected by one of the above relations,

are considered as equivalent ones.

When separating variables in (1) one has to solve an intermediate problem of

description of all inequivalent potentials such that the equation admits non-trivial

SV (classification problem). The next step is to obtain a complete description of the

coordinate systems providing SV in (1) with these potentials.

First, we adduce the principal results on separation of (1) into two second-order

ODE and then give an outline of the proof of the corresponding theorems.

Theorem 1. Equation (1) admits non-trivial SV in the sense of Definition 1 iff the

function V (x) is given, up to equivalence relations (10a), by one of the following

formulae:

(1) V = mx;

V = mx?2 ;

(2)

Orthogonal and non-orthogonal separation of variables in the wave equation 129

V = m sin?2 x;

(3)

V = m sinh?2 x;

(4)

V = m cosh?2 x;

(5)

(6) V = m exp x;

V = cos?2 x(m1 + m2 sin x);

(7)

(11)

V = cosh?2 x(m1 + m2 sinh x);

(8)

= sinh?2 x(m1 + m2 cosh x);

(9) V

(10) V = m1 exp x + m2 exp 2x;

= m1 + m2 x?2 ;

(11) V

(12) V = m.

Here m, m1 , m2 are arbitrary real parameters, m2 = 0.

Note 1. Equation (1) having the potential (6) from (11) is transformed with the change

of variables [7]

x = exp(x/2) cosh t, t = exp(x/2) sinh t

into (1) with V (x) = m.

Note 2. Equations (1) having the potentials (3), (4), (5) from (11) are transformed

into (1) with V (x) = mx?2 by means of changes of variables [7]

x = tan ? + tan ?, t = tan ? ? tan ?,

x = tanh ? + tanh ?, t = tanh ? ? tanh ?,

x = coth ? + tanh ?, t = coth ? ? tanh ?.

Hereafter ? = 1 (x + t), ? = 1 (x ? t) are cone variables.

2 2

By virtue of the above remarks, the validity of the assertion follows from Theo-

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