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relativistic correction to kinetic energy.
?
3) Let us give the example of potential V , which, being substituted into (6.8),
leads to the equation, which is Galilei invariant and is equivalent in approximation
1/m2 to the Breit one,

S (1) · xS (2) · x
e(1) e(2) (1) (2)
? 1 ? 2?3 ?3 ·S
(1) (2)
V= aS +b +
x2
x
? ? ?
?(1) ?(2) ?(2)
? ?
1 ? ?(1) ? ?(2) S ?
(1)
+i?e(1) e(2) + +
2m(1) m(2) 2m(2)
? x
?(1) ?(x)
??
S (2) + 2?i?(1) ?(2) 1 + 2S (1) · S (2)
+ ,
x3
m(1) m(1) m(2)

where
1 1
(?) (?)
?
?? = ?1 ? i?2 , ? = 1 + c, b = 1 + c,
4 2
?1
2? = a + b ? 2, c = m 2 + m2 m(1) m(2) m(1) + m(2) .
(1) (2)

?
Applying the BGC reduction for the Hamiltonian H = E + V , one obtains the
(1) (2)
operator, which coincides for ? = 1 (1 + ?1 )(1 + ?1 )? with the approximate Breit
4
Hamiltonian in the c.m. frame.
One may conclude from the above that two-particle GIWE can be successfully
applied for the description of interacting particles. The application of such equations
to concrete physical problems will be considered in future publications.

Appendix
Explicit expression for the operator (S · x)/x in spherical spinor basis
Spherical spinors ?s j??m are (2s + 1)-component functions with the components
j

jm
(?s j??m )µ = Cj?? m?µsµ Yj?? m?µ , (A.1)
j

jm
where Cj?? m?µsµ are Clebah–Gordan coefficients, Yj?? m?µ are spherical harmonics.
Substituting (A.1) into (4.11), choosing x = x0 = (0, 0, 1) and taking into account that
??
[17]

2(j ? ?) + 1
(S · x0 )µµ = (S2 )µµ = µ?µµ , (A.2)
Yj?? 0 (?0 ) =
x , ?
4?
378 W.I. Fushchych, A.G. Nikitin

one comes to the following system of linear algebraic equations for a?? [18]:

(a?? ? µ??? ) 2(j ? ?) + 1Cj?? (A.3)
= 0,
0sµ
?

where
?, ? = ?s, ?s + 1, . . . , ?s + 2nsj , nsj = min(s, j),
µ = ?nsj , ?nsj + 1, . . . , nsj .
The solution of system (A.3) is given by formulae (4.12). For s ? 3
one has
2
specifically
1
= ?1;
s = 0, a?? = 0, s= , a 1 ? 1 = a? 1 1
2 2 2 2 2


j j+1
a10 = a01 = ? a0 ?1 = a?1 0 = ?
s = 1, , , j = 0;
2j + 1 2j + 1

3 j+1 j
=? a? 3 ? 1 = a? 1 ? 3 = ?
s= , a3 = a1 , ;
1 3
2 3j 3(j + 1)
2 2 2 2 2 2 2 2



(2j + 3)(2j ? 1)
1 1
=?
a 1 ? 1 = a? 1 , j= ;
1
3 j(j + 1) 2
2 2 2 2


1 1
a? 1 ? 3 = a? 3 ? 1 = ? , j= .
3 2
2 2 2 2


The remaining coefficients a?? for s ? 3
are equal to zero.
2

1. Tamm I.E., Dokl. Akad. Nauk SSSR, 1942, 29, 551 (in Russian).
2. Corben H., Schwinger J., Phys. Rv., 1940, 58, 953.
3. Wightman A.S., Invariant wave equations, LNP, 1980, 73, 1.
4. Levi-Leblond J.-M., Commun. Math. Phys., 1967, 6, 286.
5. Hagen C.R., Hurley W.J., Phys. Rev. Lett., 1970, 26, 1381.
6. Hurley W.J., Phys. Rev. D, 1971, 4, 2339.
7. Fushchych W.I., Nikitin A.G., Lett. Nuovo Cimento, 1976, 16, 81.
8. Fushchych W.I. and Nikitin A.G., Salogub V.A., Rep. Math. Phys., 1978, 13, 175.
9. Fushchych W.I., Nikitin A.G., Teor. Mat. Fiz., 1980, 44, 34 (in Russian); Theor. Math. Phys., 1981,
44, 584 (in English).
10. Fushchych W.I., Nikitin A.G., Fiz. Elem. Chastits At. Jadra, 1982, 12, 1157 (in Russian).
11. Nikitin A.G., Acta Phys. Pol. B, 1982, 13, 369.
12. Breit G., Phys. Rev., 1929, 34, 553.
13. Foldy L.L., Wouthyusen S.A., Phys. Rev., 1950, 78, 29.
14. Achijezer A.I., Beretecky V.B., Quantum electrodynamics, Moscow, Nauka, 1981.
15. Fock V.A., Foundations of quantum mechanics, Moscow, Nauka, 1976.
16. Barker W.A., Glover F.N., Phys. Rev., 1955, 99, 317.
17. Edmonds A.R., Angular momentum in quantum mechanics, Princeton, 1957.
18. Nikitin A.G., Teor. Mat. Fiz., 1983, 57, 257 (in Russian).
W.I. Fushchych, Scientific Works 2000, Vol. 2, 379–381.

Some exact solutions of the
many-dimensional sine-Gordon equation
W.I. FUSHCHYCH, Yu.N. SEHEDA

In the present paper we construct the multiparametrical families of exact solutions
of the many-dimensional nonlinear d’Alembert equation

2U = sinh U, (1)

where 2 = ? 2 /?x2 ? · · · ? ? 2 /?x2 . This equation is concerned with some problems
n
0
of field theory [1]. In the case of n = 1 the analysis and the physical interpretation of
solutions of this equations are given in [2].
Up to date the inverse-scattering method is applied for solving two-dimensional
nonlinear equations (KdV, sine-Gordon, nonlinear Schr?dinger and some others) main-
o
ly and the attempts to extend this method for solving many-dimensional equations
are not so successful.
To construct some classes of exact solutions of the many-dimensional equation (1),
we use group-theoretical ideas of Lie which were applied fruitfully by Birkhoff [3],
Sedov [4] and Ovsyannikov [5] to nonlinear equations of hydrodynamics.
The maximal local invariance group of eq.(1) is the Poincar? group P (1, n) of
e
1,n
rotations and translations of the (1 + n)-dimensional space R .
We look for the solutions to eq.(1) of the form

(2)
U (x) = ?(?),

where ? is a function of the invariant variable ? only (for more details see [6]). We
use the following set of invariants which were presented in [7]. (Below the summation
convention is employed. The parameters ?? , ?? , . . . are arbitrary real constants.)

? = (x? x? )1/2 , (3a)
1/2
?? ? ? = ?1,
? = (?? y ? )2 + y? y ? (3b)
,
1/2
? = (?? y ? )2 ? y? y ? ?? ? ? = 1, (3c)
,

?? ?? = l = ±1, ,
? = ?? x? , (3d)
1
?? ? ? = l = ±1,
(?? y ? )2 + a?? y ? , ?? ?? = ?? ? ? = 0, (3e)
?=
2
? = ?? y ? + a ln ?? y ? , a = 0,
(3f)
?? ? ? = l = ±1,
?? ?? = ?? ? ? = 0, y ? = x? + a? ,

a, a? are arbitrary constants.
Lettere al Nuovo Cimento, 1984, 41, 14, P. 462–464.
380 W.I. Fushchych, Yu.N. Seheda

Substituting (2) into the many-dimensional partial differential eq.(1) we reduce it
to the ordinary differential equations
N1
(4a)
U+ U = sinh U
?
(cases (3a), (3b))
N2
?U ? (4b)
U = sinh U,
?
(case (3c))
l = ±1 (4c)
U = l sinh U,
(cases (3d), (3e)).
Here N1 and N2 are natural numbers depending on the value of the space dimen-
sion n.
When N1 = 0 and N2 = 0 eqs.(4a) and (4b) cannot be solved in explicit form.
Taking l = 1 we have from (4c)
du
v
w= ,
2 cosh U + C
C is an arbitrary constant. The solutions of eq.(1) are found by in inversion of elliptic
integrals [8]:
v
C ?2
C +2
?1
U = 2 tgh {sn (z, k)}, ?, k 2 = (5a)
z= , C > 2,
2 C +2
v
U = 2 tgh?1 {sin z}, (5b)
z = 2?, C = 2.
Writing the integration constant in the form
du
v
w=
2 cosh U ? C
we have analogously
2 ? C sn2 (?, k) C +2
?1
k2 = (5c)
U = cosh , , 0 < C < 2,
2 cn2 (?, k) 4
v
C/2 ? sn2 (z, k) C +2 4
U = cosh?1 ?, k 2 = (5d)
, z= , C > 2,
cn2 (z, k) 2 C +2
U = 4 tgh?1 exp[?], (5e)
C = 2,
1
U = cosh?1 {cn (?, k)}?1 , k2 = (5f)
C = 0, .
2
When l = ?1 we have the solution
C2
U = cosh?1 cn (z, k) + sn2 (z, k) ,
2
v (5g)
C ?2
C +2
?, k 2 =
z= , C > 2.

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