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with respect to the in?nite-dimensional algebra which contains the Poincar?, Galilei e
and conformal algebras as subalgebras. This fact allows us to impose some conditions
on functional dependence of D, B, E, H and to select equations invariant under the
Galilei algebra AG(1, 3).
Theorem 1. System (1) is invariant with respect to the Galilei algebra AG(1, 3) with
basis operators

? ?
P0 = ? t = , Pa = ?xa = ,
?t ?xa
Jab = xa ?xb ? xb ?xa + Ea ?Eb ? Eb ?Ea + Ha ?Hb ? Hb ?Ha +
+ Da ?Db ? Db ?Da + Ba ?Bb ? Bb ?Ba ,
Ga = t?xa + ?abc (Bb ?Ec ? Db ?Hc )

if

H = ?N (B 2 , B E)E + M (B 2 , B E)B,
D = N (B 2 , B E)B, (9)

where M , N are arbitrary functions of their variables.
On new Galilei- and Poincar?-invariant nonlinear equations
e 183

Choosing concrete form of M and N , we obtain families of Galilei-invariant equa-
tions (1) with conditions (9). So, when N = B E, M = 1, then (9) takes the form

(E H)2 EH EH
D= E+ H, B= E + H.
(1 ? E 2 )2 1 ? E2 1 ? E2

Corollary 1. The transformation rule for E and H has the form

E > E = E + u ? B, H > H = H ? u ? D,
D > D = D, B>B =B
under Galilei transformations, where u is a velocity of an inertial system with respect
to another inertial system.
Theorem 2. System (1), (2) is invariant with respect to the Poincar? algebra AP (1, 3)
e
with basis elements
P0 = ?x0 , Pa = ?xa ,
Jab = xa ?xb ? xb ?xa + Ea ?Eb ? Eb ?Ea + Ha ?Hb ? Hb ?Ha +
+ Da ?Db ? Db ?Da + Ba ?Bb ? Bb ?Ba ,
J0a = x0 ?xa + xa ?x0 + ?abc (Db ?Hc + Eb ?Bc ? Hb ?Dc ? Bb ?Ec )
if and only if

F1 = R2 = M (B 2 ? E 2 , B E), F2 = ?R1 = N (B 2 ? E 2 , B E),
a1 = b1 = a(B 2 ? E 2 , B E), a2 = b2 = b(B 2 ? E 2 , B E).
Theorem 3. System (3) is invariant with respect to the Poincar? algebra AP (1, 3)
e
with basis elements
P0 = ?t , Pa = ?xa ,
Jab = xa ?xb ? xb ?xa + Ea ?Eb ? Eb ?Ea + Ha ?Hb ? Hb ?Ha ,
J0a = t?xa + ?abc (Eb ?Hc ? Hb ?Ec )

if and only if E and H satisfy system (4).
System (5) was proposed in [4] and its symmetry has been studied in [10], when
F1 = 0, F2 = 0.
Corollary 2. System (5), (6) can be considered as a system of equations describing
the interaction of electromagnetic ?eld with a Schr?dinger ?eld of spin s = 1/2.
o
Theorem 4. System (5), (6) is invariant with respect to the Galilei algebra AG(1, 3)
whose basis elements are given by formulas
P0 = ?t , Pa = ?xa ,
Jab = xa ?xb ? xb ?xa + Ea ?Eb ? Eb ?Ea +
(10)
1
+ Ha ?Hb ? Hb ?Ha + ([?a , ?b ]?)n ??n ,
4
Ga = t?xa + ?Ea + ?Ha + imxa ?k ??k .
2
if ? is a function of W = E ? H .
184 W.I. Fushchych, I.M. Tsyfra

Theorem 5. Equation (7) is invariant with respect to the Galilei algebra AG(1, 3)
with the basis elements Pµ , Jab (10) and
Ga = t?xa ? Ea Ek ?Ek ? Ha Hk ?Hk + imxa ?k ??k . (11)
E2H 2
if ?1 , ?2 , ?3 , ?4 , ? are functions of W = .
(E H)2
Corollary 3. Operators Ga (11) give the nonlinear representation of the Galilei
algebra. Thus, one can consider system (5), (7) as a basis of the classical Galilei-
invariant electrodynamics. The ?elds E, H, ? are transformed in the following way
E
E>E = ,
1 + ? a Ea
H
H>H = no sum over a,
1 + ? a Ha
2
?a
? > ? = exp imxa ?a + im t
2
under transition from one inertial system to another, ?a is group parameter.
Theorem 6. System (8) is invariant with respect to the Poincar? algebra AP (1, 3)
e
with basis elements
P0 = ?t , Pa = ?xa ,
Jab = xa ?xb ? xb ?xa + Ea ?Eb ? Eb ?Ea + Ha ?Hb ? Hb ?Ha + va ?vb ? vb ?va , (12)
J0a = t?xa + ?abc (Eb ?Hc ? Hb ?Ec ) + ?va ? va (vk ?vk )
if
m0
m(v 2 ) = v .
1 ? v2
and a1 , a2 are functions of W1 , W2 , W3 , where W1 = E H, W2 = E 2 ? H 2 , W3 =
1?v 2 [(v E) + (v H) ? v H ? E ? 2E(v ? H)].
1 2 2 22 2

Corollary 4. From this theorem we obtain the dependence of a particle mass from
v 2 , as a consequence of Poincar?-invariance of system (8).
e
Theorem 7. System (8) is invariant with respect to the Galilei algebra AG(1, 3) with
Pµ , Jab from (12) and
Ga = t?xa + ?va
only if m = m0 = const, a1 = a2 = 0.
Corollary 5. Operators (12) give a linear representation for E and H [8] and a
nonlinear representation for velocity v. The explicit form of transformations for v
generated by G1 is
v1 + ? 1 v2 v3
v1 > v 1 = v2 > v2 = v3 > v3 =
, , .
1 + ? 1 v1 1 + ? 1 v1 1 + ? 1 v1
Remark 1. In conclusion we note that there exists the nonlinear representation of
the Galilei algebra AG(1,3), generated by the operators Pµ , Jab from (12) and

G(1) = t?xa ? Ea Ek ?Ek ? Ha Hk ?Hk ? va vk ?vk .
a
On new Galilei- and Poincar?-invariant nonlinear equations
e 185

Acknowledgements. Our work was carried out under ?nancial support from
INTAS.

1. Fushchych W., On additional invariance of relativistic equations of motion, Theor. Math. Phys,
1971, 7, ¹ 1, 3–12.
2. Fushchych W., On additional invariance of the Dirac and Maxwell equations, Lett. Nuovo Ci-
mento, 1974, 11, ¹ 10, 508–512.
3. Fushchych W.I., On additional invariance of the Klein–Gordon–Fock equation, Proc. Acad. Sci.
of USSR, 1976, 230, ¹ 3, 570–573.
4. Fushchych W.I., New nonlinear equation for electromagnetic ?eld having the velocity di?erent
from c, Proc. Acad. Sci. of Ukraine, 1992, ¹ 4, 24–27.
5. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of
nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993, 436 p.
6. Fushchych W., Ansatz’95, J. Nonlinear Math. Phys., 1995, 2, ¹ 3–4, 216–235.
7. Fushchych W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, New York,
Allerton Press Inc., 1994.
8. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, Reidel Publ.
Comp., 1987.
9. Fushchych W., Tsyfra I., On symmetry of nonlinear equations of electrodynamics, Theor. Math.
Phys., 1985, 64, ¹ 1, 41–50.
10. Fushchych W., Tsyfra I., Boyko V., Nonlinear representations for Poincar? and Galilei algebras
e
and nonlinear equations for electromagnetic ?elds, J. Nonlinear Math. Phys., 1994, 1, ¹ 2,
210–221.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 186–200.

Reduction of self-dual Yang–Mills equations
with respect to subgroups of the extended
Poincar? group
e
V.I. LAHNO, W.I. FUSHCHYCH

For the vector potential of the Yang–Mills ?eld in the Minkowski space R(l, 3), we
construct the ansatze that are invariant under three-parameter subgroups of the
?
extended Poincar? group P (1, 3). We perform the symmetry reduction of self-dual
e
Yang–Mills equations to systems of ordinary di?erential equations.

1 Introduction
Classical SU (2)-invariant Yang–Mills equations (YME) comprise a system of twelve
nonlinear partial di?erential equations (PDE) of the second order in the Minkowski
space R(1, 3). On the other hand, once the Yang–Mills potentials satisfy the self-
duality conditions, the YME are automatically satis?ed. This allows one to construct
a broad subclass of solutions to the YME using the condition of self-duality, which
amounts to a system of nine ?rst-order PDE,
i
?µ??? F ?? , (1)
Fµ? =
2
where Fµ? = ? µ A? ? ? ? Aµ + eAµ ? A? is the Yang–Mills strength-tensor, ?µ???
is the rank-four antisymmetric tensor, and e is the gauge coupling constant, with
µ, ?, ?, ?, = 0, 3. Equations (1) are called the self-dual Yang–Mills equations (SDYME).
Self-duality properties have allowed exact solutions to YME to be explicitly con-
structed, starting with the ansatze for the Yang–Mills ?elds proposed by Wu and
Yang, Rosen, ’t Hooft, Corrigan and Fairlie, Wilczek, and Witten. One should also
note the Atiyah–Drinfeld–Hitchin–Manin construction that has been applied in the
construction of instanton solutions to YME (see reviews [1, 2] and the bibliographies
cited therein).
Recently, increasing interest has been given to SDYME and the corresponding Lax
pairs in the Euclidean space R(4) in view of the possibility of reducing them to classical
integrable equations (Euler–Arnold, Burgers, Kadomtsev–Petviashvili, Liouville, and
others). This problem was considered, in particular, in [3–5], where reduction with
respect to translations was performed. In [6], SDYME were reduced with respect to all
subgroups of the Euclidean group E(4), while in [7, 8], SDYME and the corresponding
Lax pairs in four- dimensional Minkowski space with the signature (+ + ??) were
reduced with respect to Abelian subgroups of the Poincar? group P (2, 2).
e
In this paper, we continue our investigation of the problem of the symmetry
reduction of YME and SDYME in the Minkowski space R(1, 3). It is known [9] that the
maximal symmetry group (according to Lie) of the YME is the group C(1, 3)?SU (2);
Theor. and Math. Phys., 1997, 110, ¹ 3, P. 329–342.
Reduction of self-dual Yang–Mills equations 187

this group also preserves SDYME (1). The presence of high symmetry allows one to
apply the method of symmetry reduction [10, 11] to the equations and, further, to
obtain exact solutions. Several conformally invariant solutions of YME were found
in [12] (see, also, [13]). A systematic investigation of conformally invariant reductions
of YME and SDYME was initiated in [14, 15], where YME and SDYME (1) were
reduced, with respect to three-parameter subgroups of the Poincar? group P (1, 3), to
e
systems of ordinary di?erential equations (ODE) and new solutions to the YME were
constructed. The uni?ed form of the P (1, 3)-invariant ansatze made it possible [16] to
perform a direct reduction of the YME to systems of ODE and to obtain conditionally
invariant solutions of the YME. In this paper, we consider the symmetry reduction
of SDYME (1) to systems of ODE that correspond to three-parameter subgroups of
?
the extended Poincar? group P (1, 3).
e
The paper is organized as follows. In Section 2, we consider the general procedure
for constructing linear ansatze. Section 3 is devoted to the derivation of the uni?ed
?
form of P (1, 3)-invariant ansatze and to the reduction of SDYME (1) to systems of
ODE. In the last section, we consider some of the reduced systems and obtain exact
real solutions of (1).

Linear form of P (1, 3)-invariant ansatze
?
2
As noted above, SDYME (1) are invariant under the conformal group C(1, 3), in which
the generators
? ?
Jµ? = xµ ?? ? x? ?µ + Amµ ? Am?
P µ = ?µ , ,
?Am ?Am
? µ
(2)
?
D = xµ ?µ ? Am ,
µ
?Amµ

?
span a subgroup isomorphic to the extended Poincar? group P (1, 3). Here, ?µ =
e
?
?xµ , with µ, ? = 0, 3 and m, n = 0, 3. Here and henceforth, we sum over repeated
indices (from 0 to 3 for the indices µ, ?, ?, ?, ? = 0, 3, and from 1 to 3 for m, n =
1, 3). The indices µ, ?, ?, ?, and ? are raised and lowered by the metric tensor gµ? =
diag (1, ?1, ?1, ?1).
?
Let AP (1, 3) be the extended Poincar? algebra whose basis is given by genera-
e
?
tors (2) and let AP (1, 3) be the extended Poincar? algebra generated by the vector
e
?elds
Jµ? = xµ ?? ? x? ?µ ,
(1) (1)
Pµ = ?µ , D = xµ ?µ .
In the classical approach, due to Lie [10, 11], symmetry reduction of SDYME (1)
?
to systems of ODE is associated with those subalgebras L of AP (1, 3) that satisfy
the condition r = r(1) = 3, where r is the rank of L and r(1) is the rank of the
?
projection of L onto AP (1) (1, 3). As can be easily seen, we have dim L = r = 3, which
means that in order to perform the reduction, we need to know the three-dimensional
?
subalgebras of AP (1, 3) satisfying the above condition. Taking into account that
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