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[Q6+a , Q6+b ]+ = 2?ab Q12+b , [Q6+a , Qb ]+ = 2?ab Q6+b ,
[Q9+a , QB ] = [Q12+a , QB ] = 0, B = 1, 2, . . . , 15.
In conclusion let us give the explicit form of the motion constants of the electro-
magnetic field which correspond to the symmetry operators (5.6). Due to the Maxwell
234 W.I. Fushchych, A.G. Nikitin

equations the following bilinear combinations do not depend on x0 and so are conser-
ved in time

d3 x [(rot H)a (rot H)b ? (rot E)a (rot E)b +
d3 x ? T Qab ? =
Iab =

+Ea p2 Eb ? Ha p2 Hb ,
(5.11)
? ? d x Ea p H b + H a p E b ?
3 T 3 2 2
Iab = d x ? Qab ? =

?(rot E)a (rot H)b ? (rot H)a (rot E)b ] .

In contrast with the classical motion constants (energy, momentum, etc) the integral
combinations (5.11) depend not only on E and H but also on the derivatives of these
vectors.
So starting from the symmetry operators (5.6) found above we obtain ten new
constants of motion for the electromagnetic field in vacuum given by relations (5.11).
These motion constants, in contrast to the Lipkin ones [25, 4, 23, 26], have nothing
to do with the Lorentz or conformal invariance of the Maxwell equations inasmuch as
the corresponding symmetry operators (5.6) do not belong to the enveloping algebra
of the algebra C(1, 3) ? H.
Acknowledgment. We would like to thank the referee for his useful comments.
Note added in proof. In the formulation of theorem 3 we have omitted six symmetry
operators of the massless Dirac equation, which have the form Qµ? = ?Q?µ =
[K? , Aµ ]. So this equation has 52 linearly independent symmetry operators Q ? M1
All the symmetry operators Q ? M1 for the Dirac equation with m = 0 belong to the
enveloping algebra of algebra P (1, 3) inasmuch as operator B (3.3) can be represented
as D? = 1 ?µ??? J µ? J ?? ? on the set of the Dirac equation solutions.
4


1. Ames W.F., Nonlinear partial differential equations in engineering, Vol. 1, New York, Academic,
1965.
2. Danilov Yu.A., Preprint IAE-1976, Kurchalov Inslilute of Atomic Energy, 1968.
3. Da Silveira, Nuovo Cimento A, 1980, 56, 385–395.
4. Fradkin D.M., J. Math. Phys., 1965, 6, 879–890.
5. Fushchych W.I., Preprint ITP-E-70-32, Institute for Theoretical Physics, Kiev, 1970.
6. Fushchych W.I., Teor. Mat. Fiz., 1971, 7, 3–12 (Theor. Math. Phys., 7, 3–11).
7. Fushchych W.I., Nuovo Cimento Lett., 1973, 6, 133–138.
8. Fushchych W.I., Nuovo Cimento Lett., 1974, 11, 508–512.
9. Fushchych W.I., On the new method of investigation of the group properties of systems of partial di-
fferential equations, in Group-Theoretical Methods in Mathematical Physics, Kiev, Naukova Dumka,
1978, 5–44.
10. Fushchych W.I., Dokl. Akad. Nauk SSSR, 1979, 246, 846–850.
11. Fushchych W.I., Ukr. Mat. Zurn., 1981, 33, 834–837 (Ukr. Math. J., 33, 632–635).
12. Fushchych W.I., Nikitin A.G., Nuovo Cimento Lett., 1977, 19, 347–352.
13. Fushchych W.I., Nikitin A.G., Nuovo Cimento Lett., 1978, 21, 541–545.
14. Fushchych W.I., Nikitin A.G., Nuovo Cimento Lett., 1979, 24, 220–224.
15. Fushchych W.I., Nikitin A.G., J. Phys. A: Math. Gen., 1979, 12, 747–757.
On the new invariance algebras and superalgebras 235

16. Fushchych W.I., Nikitin A.G., Czech. J. Phys. B, 1982, 32, 476–480.
17. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Kiev, Naukova Dumka, 1983
(Engl. transl., Dordrecht, Reidel, 1986).
18. Fushchych W.I., Onufrijchuck S.P., Dokl. Akad. Nauk SSSR, 1977, 235, 1056–1059.
19. Fushchych W.I., Vladimirov V.A., Dokl. Akad. Nauk SSSR, 1981, 287, 1105–1108.
20. Fushchych W.I., Vladimirov V.A., J. Phys. A: Math. Gen., 1983, 16, 1921–1925.
21. Gelfand I.M., Zetlin M.L., Dokl. Akad. Nauk SSSR, 1950, 71, 1017–1020.
22. Ibragirnov N.H., Dokl. Akad. Nauk SSSR, 1969, 185, 1226–1228.
23. Kibble T.W.B., J. Math. Phys., 1965, 6, 1022–1025.
24. Kotelnikov G.A., Nuovo Cimento B, 1982, 72, 68–78.
25. Lipkin D.M., J. Math. Phys., 1964, 5, 696–700 .
26. Michelsson J., Niederle J., Lett. Math. Phys., 1984, 8, 195–205.
27. Nikitin A.G., On the invariance group of Kemmer–Duffin–Petiau equations for particle with ano-
malous , in Group-Theoretical Methods in Mathematical Physics, Kiev, Naukova Dumka, 1978,
81–95.
28. Nikitin A.G., Symmetries of the Weyl equation, in Group-Theoretical Studies of Mathematical
Phsics equations, Kiev, Naukova Dumka, 1985, 31–35.
29. Nikitin A.G., Segeda Yu.N., Fushchych W.I., Teor. Mat. Fiz., 1976, 29, 82–94 (Theor. Math. Phys.,
29, 943–954).
30. Ovsjannikov L.V., The Group Analysis of Differential Equations, Moscow, Nauka, 1978.
31. Stra?ev V.I., Vest. Akad. Nauk BSSR, 1981, 5, 75–80.
z
32. Stra?ev V.I., Shkol’nikov P.L., Izv. Vuzov Fiz., 1984, 4, 81–88.
z
W.I. Fushchych, Scientific Works 2001, Vol. 3, 236–240.

On some exact solutions
of the three-dimensional non-linear
?
Schrodinger equation
W.I. FUSHCHYCH, N.I. SEROV
Some exact solutions of the three-dimensional non-linear Schr?dinger equation are
o
found. The formulae for generating solutions of the Schr?dinger-invariant equations
o
are adduced.

The linear heat equation and its complex generalisation, i.e. the Schr?dinger equati-
o
on
(P0 ? Pa /2m)u = 0, Pa = ?i?/?xa ,
2
(1)
P0 = i?/?x0 , a = 1, 3,
where
x0 ? t, x = (x1 , x2 , x3 ) ? R3
u = u(x0 , x),
and m is the particle mass, is invariant under the generalised Galilei group G2 (1, 3).
The basis elements of the Lie algebra AG2 (1, 3) have the following form:
Pa = ?i?/?xa , Jab = xa Pb ? xb Pa , (2)
P0 = i?/?x0 ,

(3)
Ga = x0 Pa + mxa , I = u?/?u, a, b = 1, 3,
3
D = 2x0 P0 ? xP + i, (4)
2
3 1
A = x0 x0 P0 ? xP + i + mx2 . (5)
2 2

The same algebra for the one-dimensional equation had been found over a hundred
years ago by S. Lie [8]. For the three-dimensional equation (1) this algebra had been
found by Hagen [7] and Niederer [9] (see also Fushchych and Nikitin [4, 5]). The
elements D and A generate the scale and projective transformations respectively.
We denote the group generated by operators (2)–(4) and its Lie algebra by symbols
G1 (1, 3) and AG1 (1, 3). The group and the algebra generated by (2)–(5) are denoted
as G2 (1, 3) and AG2 (1, 3).
We now consider the following non-linear generalisation of (1):
(P0 ? Pa /2m)u + F (x, u, u? ) = 0,
2
(6)
where F is an arbitrary differentiable function. To construct the families of exact
solutions of (6) we have to know the symmetry of (6) which obviously depends on
the structure of the non-linearity.
J. Phys. A: Math. Gen., 1987, 20, L929–L933.
Exact solutions of the three-dimensional non-linear Schr?dinger equation
o 237

Theorem. Equation (6) is invariant under the following algebras:

(7)
iff
AG(1, 3) F = ?(|u|)u,

where ? is an arbitrary smooth function, and

F = ?|u|k u, (8)
iff
AG1 (1, 3)

where ?, k are arbitrary parameters, the operator of scale transformations D having
the form D = x0 P0 ? xP + 2i/k, k = 0, and

F = ?|u|4/n u, (9)
iff
AG2 (1, 3)

where n = 3 is the number of spatial variables in the Schr?dinger equation [2, 3].
o
To give the proof of the theorem, which we omit because of its clumsiness, it
is necessary to apply the Lie method to (6). The detailed account of this method is
given by Ovsyannikov [10] and Bluman and Cole [1]. We can make sure that (6) with
non-linearities (7)–(9) admits the groups G, G1 and G2 by direct verification.
Later on we shall construct the exact solutions of the Schr?dinger equation with
o
non-linearity (9), i.e.

(P0 ? Pa /2m)u + ?|u|4/3 u = 0.
2
(10)

It follows from the theorem that only the equation with fractional non-linearity is
invariant under the group G2 (1, 3).
Following Fushchych [2] we seek solutions of (10) with the help of the ansatz

(11)
u(x) = f (x)?(?1 , ?2 , ?3 ),

where ? is the function to calculate. This function depends only on three invariant
variables ?1 , ?2 and ?3 being the first integrals of the Euler–Lagrange system of
equations:

dx0 dx1 dx2 dx3 du
(12)
=1 =2 =3 = ,
? 0 (x, u) ? (x, u) ? (x, u) ? (x, u) ?(x, u)

where ? 0 , ? 1 , ? 2 , ? 3 and ? are coordinates of the infinitesimal operator of the group
G2 (1, 3), i.e. the following functions:

? = (ax0 + b)x + gx0 + ? ? x + d,
? 0 = ax2 + 2bx0 + d0 ,
0
12 3
? = ? Im ax + gx + (ax0 + b) u,
2 2

where a, b, g, ?, d0 and d are parameters of the group G2 (1, 3).
Functions f (x) also are found from the system (12). The method of seeking f (x)
and variables ? is given in more detail in [2, 6].
Ansatz (11) reduces (10) to the equations for function ? which depends only on
three variables ?1 , ?2 and ?3 . Thus to construct solutions of (10) using ansatz (11)
it is necessary to have the explicit form of the function f (x) and the new invariant
238 W.I. Fushchych, N.I. Serov

variables ?1 , ?2 and ?3 . Not going into details we write them. Depending on relations
between parameters of the group G2 (1, 3) there are nine sets f (x) and ?(x):
1
?3/4 ?1/2
f (x) = 1 ? x2 imx0 x2/ 1 ? x2 ?1 = (?x) 1 ? x2
1) exp , ,
0 0 0
2
?1
?3 = tanh?1 x0 + tan?1 (?x/?x);
?2 = x2 1 ? x2 ,
0

1
?3/2
exp ? ix2 x?1 , ?1 = (?x)x?1 ,
2) f (x) = x0 0 0
2
?2 = x2 x?2 , ?3 = x?1 + tan?1 (?x/?x);
0 0

1
?3/4 ?1
exp ? imx0 x2 1 + x2
f (x) = 1 + x2
3) ,
0 0
2
?1/2 ?1
?1 = (?x) 1 + x2 ?2 = x2 1 + x2
, ,
0 0
?1 ?1
?3 = ? tan x0 + tan (?x/?x);
?3/4 ?1/2
?2 = x2 x?1 ,
4) f (x) = x0 , ?1 = (?x)x0 , 0
?3 = ? ln x0 + tan?1 (?x/?x);
?3/4 ?1/2 ?1/2 ?1/2
5) f (x) = x0 , ?1 = (?x)x0 , ?2 = (?x)x0 , ?3 = (?x)x0 ;
?3 = ?x0 + tan?1 (?x/?x);
?2 = x 2 ,
6) f (x) = 1, ?1 = ?x,
?2 = x 2 ,
7) f (x) = 1, ?1 = ?x, ?3 = x0 ;
1
f (x) = exp ? im?x/x0 ,
8) ?1 = ?x + x0 ?x,
2
?2 = ?x + x0 ?x, ?3 = x0 ;
9) f (x) = 1, ?1 = ?x, ?2 = ?x, ?3 = x0 ,
where ?, ?, ? are constant vectors satisfying the conditions
?2 = ? 2 = ? 2 = 1, ?? = ?? = ?? = 0.
We adduce the explicit form of the reduced equations for the function ?, obtained
from ansatz (11) in all nine cases:
L? + 6?2 ? 2im?3 + m2 ?2 ? ? 2?m|?|4/3 ? = 0,
1)
L? ? ?11 + 4?2 ?22 + (?2 ? ?1 )?1 ?33 + 4?1 ?12 ,
2

L? + 6?2 + 2im?3 ? 2?m|?|4/3 ? = 0,
2)
L? + 6?2 + 2im?3 ? m2 ?2 ? ? 2?m|?|4/3 ? = 0,
3)

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. 56
( 145 .)



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