стр. 71 |

4

?D2

z = (C3 cos bt + C4 sin bt) sin?1 b(t + D1 ) + sin?2 b(t + D1 ),

2

2b

Variable separation in the two-dimensional Schr?dinger equation

o 289

The case 2. ? = 0, ? = 0. On making the change of variables in (A.1)

z

y = exp w, v=

y

we get

w ? w2 = k + ? exp 4w, (A.1a)

? ? v + kv = 0.

?

The first ODE from (A.1a) is reduced to the first-order linear ODE

1

p (w) ? p(w) = k + ? exp 4w

2

to by the substitution w = (p(w))1/2 , whence

?

p(w) = ? exp 4w + ? exp 2w ? k, ? ? R1 .

Equation w = p(w) has a singular solution w = C = const such that p(C) = 0. If

?

w = 0 then integrating equation w = p(w) and returning to the initial variable y, we

? ?

get

y(t)

d?

= t + C1 .

? (?? 4 + ?? 2 ? k)1/2

Taking the integral in the left-hand side of the above equality we obtain the general

solution of the first ODE from (A.1). It is given by the following formulae:

under k = ?a2 < 0

y = C2 (? + sh 2a(t + C1 ))?1/2 , y = C2 (? + ch 2a(t + C1 ))?1/2 ,

(A.7)

y = C2 (? + exp[±2a(t + C1 )])?1/2 ,

under k = b2 > 0

y = D2 (? + sin 2b(t + D1 ))?1/2 . (A.8)

Here C1 , C2 , D1 , D2 are arbitrary real constants.

Integrating the second ODE from (A.1a) and returning to the initial variable z we

have

under k = ?a2 < 0

(A.9)

z = y(t)(C3 sh at + C4 ch at)

under k = b2 > 0

z = y(t)(D3 cos bt + D4 sin bt)

where C3 , C4 , D3 , D4 are arbitrary real constants.

Thus, we have constructed the general solution of the system of nonlinear ODE

(A.1) which is given by formulae (A.5)–(A.9).

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

1. Boyer C., Lie theory and separation of variables for the equation iUt + Ux1 x1 + Ux2 x2 ? (?x?2 +

1

?x?2 )U = 0, SIAM J. Math. Anal., 1976, 7, № 3, 230–263.

2

2. Miller W., Symmetry and separation of variables, Massachusetts, Addison-Wesley, 1977, 285 p.

3. Boyer C., Kalnins E., Miller W., Lie the theory and separation of variables, 6: The equation iUt +

Ux1 x1 + Ux2 x2 = 0, J. Math. Phys., 1975, 16, № 3, 499–511.

4. Fushchych W.I., Zhdanov R.Z., Revenko I.V., On the new approach to the variable separation in the

wave equation with potential, Докл. AH Укpaины, 1993, № 1, 27–32.

5. Niederer U., The maximal kinematical group of the free Sch?dinger equation, Helv. Phys. Acta,

o

1972, 45, № 5, 802–810.

6. Boyer C., The maximal kinematical invariance group for an arbitrary potential, Helv. Phys. Acta,

1974, 47, № 5, 589–605.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 291–310.

Separation of variables

in the two-dimensional wave equation

with potential

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

The paper is devoted to solution of a problem of separation of variables in the wave

equation utt ? uxx + V (x)u = 0. We give a complete classification of potentials V (x) for

which this equation admits a nontrivial separation of variables. Furthermore, we obtain

all coordinate systems that provide separability of the equation considered.

Дана стаття присвячена розв’язанню проблеми роздiлення змiнних для хвильового

рiвняння utt ? uxx + V (x)u = 0. Вказанi всi потенцiали V (x), для яких дане рiв-

няння допускає нетривiальне роздiлення змiнних. Крiм того, одержанi всi системи

координат, в яких роздiлюється дослiджуване рiвняння.

1. Introduction. In this paper, we study the two-dimensional wave equation with

potential

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

where u = u(t, x) ? C 2 (R2 , R1 ) and V (x) ? C(R1 , R1 ), by using the method of

separation of variables (SV). Equations belonging to class (1) are widely used in the

modern quantum physics and can be related to other linear and nonlinear equations

of mathematical physics (these relations will be discussed below, at the end of the

article). In particular, class (1) contains the d’Alembert equation (with V (x) = 0) and

the Klein– Gordon–Fock equation (with V (x) = m ? const).

The separation of variables in two- and three-dimensional Laplace, Helmholtz,

d’Alembert, and Klein–Gordon–Fock equations had been carried out in the classical

works by Bocher [1], Darboux [2], Eisenhart [3], Stepanov [4], Olevsky [5], and

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

of the problem of SV in equation (1) is not obtained yet.

When speaking about solution of equation (1) with separated variables ?1 , ?2 , we

mean the ansatz

(2)

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

reducing (1) to two ordinary differential equations for the functions ?i (?i )

(3)

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

? ? i = 1, 2,

In formulas (2) and (3), A, ?1 , ?2 ? C 2 (R2 , R1 ), Ai , Bi ? C 2 (R1 ? ?, R1 ) are some

unknown functions, ? ? ? ? R1 is a real parameter (separation constant).

Definition 1. Equation (1) admits SV in the coordinates ?1 (t, x), ?2 (t, x) if the

substitution of ansatz (2) into (1) with subsequent exclusion of the second derivati-

ves ?1 , ?2 according to (3) yields an identity with respect to the variables ?i , ?i , ?

?? ?

(considered as independent ones).

Укр. мат. журн., 1994, 46, № 10, P. 1343–1361.

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

On the basis of the above definition, one can formulate the procedure of SV in

equation (1). At the first step; one has to substitute expression (2) into (1) and to

express the second derivatives ?1 , ?2 via the functions ?i , ?i according to equati-

?? ?

ons (3). At the second step, the obtained equality is splitted with respect to the

independent variables ?i , ?i . As a result, one gets an overdetermined system of par-

?

tial differential equations for the functions A, ?1 , ?2 with undefined coefficients. The

general solution of this system gives rise to all systems of coordinates that provide

separability of equation (1).

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

common with the non-Lie method of reduction of nonlinear differential equations

suggested in [7–9]. It is also important to note that the idea of representing solutions

of linear differential equations in the “separated” form (2) goes as far as to classical

works of Euler and Fourier (for a modern exposition of the problem of SV, see

Miller [6] and Koornwinder [10]).

The present paper is organized as follows: In the first section, we adduce principal

assertions about SV in equation (1). In the second section, the detailed proof of these

assertions is given. In the last section, we briefly discuss the obtained results.

2. List of principal results. It is evident that equation (1) admits SV in the

Cartesian coordinates ?1 = t, ?2 = x for an arbitrary V = V (x).

Definition 2. Equation (1) admits a nontrivial SV if there exists at least one coordi-

nate system ?1 = (t, x), ?2 (t, x), different from the Cartesian system, that provides

its separability.

Next, if, in equation (1), one makes the transformations

t > C1 t, x > C1 x, t > t, x > x + C2 , Ci ? R1 ,

then the class of equations (1) transforms into itself and, moreover,

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

2

(4)

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

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

are regarded as equivalent ones.

Theorem 1. Equation (1) admits a nontrivial SV iff the function V (x) is given up

to the equivalence relations (4) by one of the result formulas:

(1) V = mx;

V = mx?2 ;

(2)

V = m sin?2 x;

(3)

V = m sh?2 x;

(4)

V = m ch?2 x; (5)

(5)

(6) V = m exp x;

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

(7)

V = ch?2 x(m1 + m2 sh x);

(8)

V = sh?2 x(m1 + m2 ch x);

(9)

Separation of variables in the two-dimensional wave equation with potential 293

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

V = m1 + m2 x?2 ;

(11)

(12) V = m.

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

Note 1. Equation (1) with the potential V (x) = m exp x is transformed by the change

of variables [11]

x x

x = exp ch t, t = exp sh t

2 2

into equation (1) with V (x) = m (i.e., into the Klein–Gordon–Fock equation).

Note 2. Equations (1) with potentials 3, 4, 5 from (5) are transformed into equation (1)

with V (x) = mx?2 by the changes of variables [11]

x = tg ? + tg ?, t = tg ? ? tg ?,

x = th ? + th ?, t = th ? ? th ?,

x = cth ? + th ?, t = cth ? ? th ?.

Here, ? = (x + t)/2, ? = (x ? t)/2 are cone variables.

In virtue of the above remarks, Theorem 1 implies the following assertion:

Theorem 2. Provided that equation (1) admits a nontrivial SV, it is locally equi-

valent to one of the following equations:

2u + mxu = 0;

(1)

2u + mx?2 u = 0;

(2)

2u + cos?2 x(m1 + m2 sin x)u = 0;

(3)

2u + ch?2 x(m1 + m2 sh x) = 0;

(4)

(6)

2u + sh?2 x(m1 + m2 ch x) = 0;

(5)

2u + ex (m1 + m2 ex )u = 0;

(6)

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

(7)

2u + mu = 0.

(8)

Thus, there exist eight inequivalent types of equations of the form (1) that admit

a nontrivial SV.

It is well known that there are eleven coordinate systems that provide separability

of the Klein–Gordon–Fock equation (2 + m)u = 0 (see, e.g., [12]). This is why the

case V (x) = const is not considered here.

As is shown in Section 2, the general form of the solution of equations (6) with

separated variables is as follows:

(7)

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

here, ?1 (?1 ), ?2 (?2 ) are arbitrary solutions of the separated ordinary differential (6)

here, equations

(8)

?i = (? + gi (?i ))?i ,

? i = 1, 2,

and the explicit form of the systems ?i (t, x), gi (?i ) is given below.

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

Theorem 3. The equation 2u + mxu = 0 separated in two coordinate systems

(1) ?1 = t, ?2 = x, g1 = 0, g2 = m?2 ;

?1 = (x + t)1/2 + (x ? t)1/2 , ?2 = (x + t)1/2 ? (x ? t)1/2 ,

(2) (9)

m m4

g1 = ? m?1 , g2 = ? ?2 .

4

4 4

Theorem 4. The equation 2u + sin?2 x(m1 + m2 cos x)u = 0 is separated in four

coordinate systems

?1 = t, ?2 = x; g1 = 0, g2 = sin?2 ?2 (m1 + m2 cos ?2 );

(1)

x

= arctg sh(?1 + ?2 ) ± arctg sh(?1 ? ?2 ),

(2)

t

g1 = (m1 + m2 ) sh?2 ?1 , g2 = ?(m1 ? m2 ) ch?2 ?2 ;

x

= arctg tn (?1 + ?2 ) ± arctg tn (?1 ? ?2 )

(3)

t

g1 = m1 dn2 ?1 cn?2 ?1 sn?2 ?1 + m2 [sn?2 ?1 ? dn2 ?1 cn?2 ?1 ], (10)

g2 = m1 k 4 sn2 ?2 cn2 ?2 dn?2 ?2 + m2 k 2 [cn2 ?2 dn?2 ?2 ? sn2 ?2 ];

1/2 1/2

k k

x

± arctg cn (?1 ? ?2 ) ,

(4) = arctg cn (?1 + ?2 )

t k k

стр. 71 |