стр. 69 |

If k = a, then

E k + iH k + ?a (E a + iH a )xk

E k + iH k > E k + iH k = ,

1 + ?a [x0 (E a + iH a ) ? xa ]

(54)

E k ? iH k + ?a (E a ? iH a )xk

E k ? iH k > E k ? iH k (no sum over a).

=

1 + ?a [x0 (E a ? iH a ) ? xa ]

Note 1. Setting ? = aE + ibH, where a, b are arbitrary functions of the invariants

E 2 , H 2 , E H, we obtain more general form of the equation (44). The equation

??k k

l ??

= F (E H, E 2 , H 2 )?k

+?

?x0 ?xl

is invariant only under some subalgebras of algebra (46) depending on the choice of

function F .

280 W.I. Fushchych, I.M. Tsyfra, V.M. Boyko

Note 2. If we analyse the symmetry of the following equations

? ? ?

+ El + Hl E k = 0,

?x0 ?xl ?xl

(?)

? ? ?

+ El + Hl k

H = 0;

?x0 ?xl ?xl

or

?E k ? ?

= ± El + Hl Hk,

?x0 ?xl ?xl

(??)

k

?H ? ?

= ± El + Hl Ek,

?x0 ?xl ?xl

we obtain concrete examples of nonlinear representations for the Poincar? and Galilei

e

algebras. This problem will be considered in a future paper.

6. Symmetry of the continuity equation and the Poynting vector

Let us consider the continuity equation for the electromagnetic field

??

L(E, H) ? (55)

+ div ?v = 0.

?x0

According to the Poynting definition ? and ?v k have the forms

12

(E + H 2 ), ?v k = ?kln E l H n . (56)

?=

2

Theorem 3. The nonlinear system (55), (56) is not invariant under the Lorentz

algebra, with basis elements:

Jkl = xk ?xl ? xl ?xk + E k ?E l ? E l ?E k + H k ?H l ? H l ?H k ,

(57)

J0k = xk ?x0 + x0 ?xk + ?kln (E l ?H n ? H l ?E n ), k, l, n = 1, 2, 3.

To prove theorem 3 it is necessary to substitute ? and ?v k , from formulas (56),

to equation (55) and to apply Lie’s algorithm, i.e., it is necessary to verify that the

invariance condition

?0 (58)

Jµ? L(E, H)

L=0

1

is not satisfied, where Jµ? is the first prolongation of the operator Jµ? .

1

Theorem 4. The continuity equation (55), (56) is conditionally invariant with

respect to the operators Jµ? , given in (57) if and only if E, H satisfy the Maxwell

equation (4), (5).

Thus the continuity equation, which is the mathematical expression of the conser-

vation law of the electromagnetic field energy and impulse is not Lorentz-invariant if

E, H does not satisfy the Maxwell equation. A more detailed discussion on conditional

symmetries can be found in [1, 2].

The following statement can be proved in the case when

? = F 0 (E, H) and ?v k = F k (E, H), (59)

where F 0 , F k are arbitrary smooth functions F 0 ? 0, F k ? 0.

Nonlinear representations for Poincar? and Galilei algebras

e 281

Theorem 5. The continuity equation (55), (59) is invariant with respect to the

classic geometrical Lorentz transformations if and only if

(60)

r(B) = 4,

where r(B) is the rank of the Jacobi matrix of functions F µ .

In conclusion we present some statements about the symmetry of the following

systems:

?E ?H

= ?rot E + F2 (E, H),

= rot H + F1 (E, H),

(61)

?x0 ?x0

div E = R1 (E, H), div H = R2 (E, H),

?(RE) ?N H

= ?rot (N E),

= rot (RH),

(62)

?x0 ?x0

div (RE) = 0, div (N H) = 0.

Here

W1 = E 2 ? H 2 ,

R = R(W1 , W2 ), N = N (W1 , W2 ), W2 = E H.

(63)

div (RE + N H) = 0.

Theorem 6. The system of equations (61) is invariant under the Lorentz algebra

with the basis elements (57) if and only if

F1 ? F2 ? 0, R1 ? R2 ? 0.

Theorem 7. The system of equations (62) is invariant under the Lorentz algebra

(57) if R and N are arbitrary functions of the invariants W1 = E 2 ? H 2 , W2 = E H.

Theorem 8. The equation (63) is invariant under the Lorentz algebra with the basis

elements (57) if and only if E, H satisfy the system of equations

?(RE + N H)

= rot (RH ? N E).

?x0

Thus it is established that, besides the generally recognized linear representation of

the Lorentz group discovered by Henry Poincar? in 1905 [5], there exists the nonlinear

e

representation constructed by using the nonlinear equations of hydrodynamical ty-

pe [4]. It is obvious that for instance the linear superposition principle does not hold

for a non-Maxwell electrodynamic theory based on the equation (25) or (45).

? ?

The nonlinear representations for the algebras AG(1, 3), AP (1, 2), AP (2, 2),

AC(1, 2), AC(2, 2) for a scalar field have been considered in [6], AP (1, 1) in [7],

and AP (1, 2) in [8].

1. Fushchych W., Nikitin A., Symmetries of Maxwell’s equations, Dordrecht, Reidel Publ. Comp., 1987.

2. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of nonlinear

mathematical physics, Dordrecht, Kluwer Publishers, 1993.

3. Fushchych W., On symmetry and exact solutions of the multidimensional equations of mathemati-

cal physic, in Algebraic-Theoretical Studies in Mathematical Physics Problems, Kyiv, Institute of

Mathematics, 1983, 4–23.

282 W.I. Fushchych, I.M. Tsyfra, V.M. Boyko

4. Fushchych W., New nonlinear equations for electromagnetic field having the velocity different from

c, Dopovidi of the Ukrainian Academy of Sciences, 1982, № 4, 24–28.

5. Poincar? H., On the dynamics of the electron, Comptes Rendus, 1905, 140, 1504–1510.

e

6. Fushchych W., Zhdanov R., Lahno V., On nonlinear representation of the conformal algebra AC(2, 2),

Dopovidi of the Ukrainian Academy of Sciences, 1993, № 9, 44–47.

7. Rideau G., Winternitz P., Nonlinear equations invariant under the Poincar?, similitude and conformal

e

groups in two-dimensional space-time, J. Math. Phys., 1990, 31, 1095–1105.

8. Yehorchenko I., Nonlinear representation of the Poincar? algebra and invariant equations, in Symmetry

e

Analysis of Equations of Mathematical Physics, Kyiv, Inst. of Math., 1992, 62–66.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 283–290.

On the new approach to variable separation

?

in the two-dimensional Schrodinger equation

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

Для двовимiрного рiвняння Шредiнгера з потенцiалом, який не залежить вiд часової

змiнної, повнiстю розв’язано задачу класифiкацiї потенцiалiв, при яких воно допу-

скає роздiлення змiнних. Для кожного з потенцiалiв описано всi системи координат,

в яких роздiлюється вiдповiдне рiвняння Шредiнгера.

There is a lot of papers devoted to separation of variables (SV) in the two-

dimensional Schr?dinger equation

o

(1)

iu1 + ux1 x2 + ux2 x2 = V (x1 , x2 )u

with some specific V (x1 , x2 ) (see, e.g. [1–3] and references therein). Saying about

the problem of SV in the Eq. (1), we imply two mutually connected problems. The

first one is to describe all functions V (x1 , x2 ) such that the equation (1) admits

separation of variables (classification problem). The second problem is to construct

for each function V (x1 , x2 ) all coordinate systems making it possible to separate

corresponding Schr?dinger equation.

o

As far as we know, the first problem has been solved provided V = 0 [3] and

V = ?x?2 + ?x?2 [1] and the second one has not been considered in the literature at

1 2

all. We guess that a possible reason for this was absence of an adequate mathematical

technique to handle the classification problem. In the paper [4] we suggested a new

approach to SV in partial differential equations which enabled us to solve the problem

of SV in two-dimensional wave equation with time independent potential [4]. In the

present paper we give the complete solution of the problem of SV in the Schr?dinger

o

equation (1) obtained within the framework of the above said approach.

Solution with separated variables is looked for in the form of the ansatz [4]

(2)

u = Q(t, x)?0 (t)?1 (?1 (t, x))?2 (?2 (t, x)),

where ?0 (t), ?1 (?1 (t, x)), ?2 (?2 (t, x)) are smooth functions satisfying ordinary diffe-

rential equations (ODE)

d?0

= U0 (t, ?0 , ?1 , ?2 ),

dt

(3)

d 2 ?a d?a

= Ua ?a , ?a , ; ? 1 , ?2 , a = 1, 2,

2

d?a d?a

and Q, ?1 , ?2 are functions to be determined from the requirement that ansatz (2)

reduces Eq. (1) to ODE, ?1 , ?2 ? R1 are arbitrary parameters (separation constants).

It is important to emphasize that functions Q, ?1 , ?2 do not depend on the parameters

?1 , ?2 .

Доповiдi НАН України, 1994, № 11, P. 38–44.

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

Because of the lack of space we have no possibility to adduce all necessary com-

putations. That is why, we shall restrict ourselves by pointing out main steps of

realization of the approach to SV suggested in [4].

First of all, we note that the substitution

?1 > ?1 = ?1 (?1 ), ?2 > ?2 = ?2 (?2 ), Q > Q = Q?1 (?1 )?2 (?2 ) (4)

does not alter the structure of relations (2), (3). That is why, we introduce the

following equivalence relation: (?1 , ?2 , Q) is equivalent to (?1 , ?2 , Q ) provided (4)

holds with some ?a , ?a .

Substituting (2) into (1) with account of equalities (3) and splitting obtained rela-

tion with respect to independent variables ?0 , ?a , ?aa , ?a , a = 1, 2 we conclude that

up to the equivalence relation (4) equations (3) take the form

d?0

= (?1 R1 (t) + ?2 R2 (t) + R0 (t))?0 ,

dt

d 2 ?a

= (?1 B1a (?a ) + ?2 B2a (?a ) + B0a (?a ))?a

2

d?a

and what is more, functions ?1 , ?2 , Q satisfy the over-determined system of nonlinear

partial differential equations

2

1) ?1xb ?2xb = 0,

b=1

2

2) [Ba1 (?1 )?1xb ?2xb + Ba2 (?2 )?2xb ?2xb ] + Ra (t) = 0, a = 1, 2,

b=1

2 2

(5)

3) 2 Qxb ?axb + Q i?at + ?axb xb = 0, a = 1, 2,

b=1 b=1

2

4) [B01 (?1 )?1xb ?2xb + B02 (?2 )?2xb ?2xb ]Q + iQi +

b=1

2

Qxb xb + R0 (t)Q ? V (x)Q = 0.

+

b=1

Thus, to solve the problem of SV for the linear Schr?dinger equation it is necessary

o

to construct the general solution of the system of nonlinear equations (5). Roughly

speaking, to solve a linear equation one has to solve a system of nonlinear equations!

This is the reason why so far there is no complete description of all coordinate systems

providing separability of the four-dimensional d’Alembert equation.

But in the case involved we have succeeded in integration of nonlinear system (5)

for ?1 , ?2 , Q. First, we have established that the general solution of equations 1–3

from (5) determined up to the equivalence relation (4) splits into four inequivalent

classes

1) ?1 = A(t)x1 + W1 (t), ?2 = B(t)x2 + W2 (t),

? ? ? ?

i A2 B2 i W1 W2

Q(t, x) = exp ? x1 + x2 ? x1 + x2 ,

4 A B 2 A B

Variable separation in the two-dimensional Schr?dinger equation

o 285

стр. 69 |