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Thus, system (23) is invariant under the Galilei transformations generated by
operator (20) for an arbitrary function F (|?|).
Let us now apply these results to obtain invariant solutions of system (23) with
? = 1 in two-dimensional space-time in the case where F (|?|) = 0:

?? ? 2 ? ?? ?V ?V
?V (24)
i + =V , i = 0.
2
?t ?x ?x ?t ?x
The invariance algebra of system (24) includes the translation operators, Galilei,
dilation, and projective operators:

P0 = ?t , P1 = ?x , G = t?x ? i?V + i?V ? ,
D = 2t?t + x?x ? V ?V ? V ? ?V ? ,
A = t2 ?t + tx?x ? (ix + tV )?V + (ix ? tV ? )?V ? .

1) The one-dimensional subalgebra G+?P0 is associated with the symmetry ansatz
i
? = ?(2?x ? t2 ), V = ? t + U (2?x ? t2 ). (25)
?
178 W.I. Fushchych, Z.I. Symenoh

Ansatz (25) reduces system (24) to the following system of ordinary di?erential equa-
tions:
1
? 2?U U = 0, (26)
2?? = U ? ,
?
where ? ? ? = 2?x ? t2 . The general solution of system (26) has the form
??
?? ,

1 ? 1
(C1 + 2 ?)3/2 d? + C3 , (27)
U= C1 + ?, ? = C2 exp
?2 3 ?
where C1 , C2 , C3 are arbitrary constants. Thus, we obtain the partial solution of
system (24), where ? has the form (27), and

i 1
V =? t+ C1 + ?.
?2
?
2) The subalgebra

G + ?(Z3 + Z4 ) = t?x ? i?V + i?V ? + ?(?? + ??? )

is associated with the symmetry ansatz
x x
V = ?i (28)
?=? + ?(t), + U (t).
t t
Ansatz (28) reduces system (24) to the following system of ordinary di?erential equa-
tions:
? U
?
i? =
? U, U+ =0
t t
with the general solution of the form
C1 C1 ?
U= , ?=i + C2 ,
t t
where C1 , C2 are arbitrary constants. Thus, we get the partial solution of system (24):
x C1 x C1 ?
V = ?i + , ?=? +i + C2 .
t t t t
3) The subalgebra

G + ?(Z1 + Z2 ) = t?x ? i?V + i?V ? + ?(??? + ? ? ??? )

is associated with the symmetry ansatz
x x
V = ?i (29)
? = exp ? ?(t), + U (t).
t t
Ansatz (29) reduces system (24) to the following system of ordinary di?erential equa-
tions:
?2 ? U
?
i? +
? ? = U ?, U+ =0
t2 t t
Symmetry of equations with convection terms 179

with the general solution
i?2
C1 i
C1 ? ?
U= , ? = C2 exp ,
t t t
where C1 , C2 are arbitrary constants. Thus, we get the partial solution of system (24):
i?2
x C1 ?x i
V = ?i + + C1 ? ?
, ? = C2 exp .
t t t t t
4) The subalgebra
A + ?i(Z1 ? Z2 ) = t2 ?t + tx?x ? (ix + tV )?V + (ix + tV ? )?V ? +
+ i?(??? ? ? ? ??? )
is associated with the symmetry ansatz
? x x1 x
? = exp ?i , V = ?i + U (30)
? .
t t t t t
Ansatz (30 reduces system (24) to the following system of ordinary di?erential equa-
tions:
? ? ?? = 0.
U = 0,
2
where ? ? ??? , ? = x . Consider the following two cases:
?
2 t
4a) ? > 0. In this case, system (24) has the following solution:
vx vx
x ?
V = ?i , ? = exp ?i + C2 exp ? ?
C1 exp ? ,
t t t t
where C1 , C2 are arbitrary constants.
4b) ? < 0. In this case, system (24) has the following solution:
v v
x ? x x
V = ?i , ? = exp ?i ?? ??
C1 cos + C2 sin ,
t t t t
where C1 , C2 are arbitrary constants.
5) The one-dimensional algebra
A + ?(Z3 + Z4 ) = t2 ?t + tx?x ? (ix + tV )?V + (ix + tV ? )?V ? + ?(?? + ??? )
is associated with the symmetry ansatz
? x x1 x
? =? +? , V = ?i + U (31)
,
t t t t t
which reduces system (24) to the following one:
U = 0, ? + i? = 0.
?2?
where ? ? x
?? 2 , t. Solving this system, we obtain the exact solution of
?=
system (24):
? x2
x ? x
V = ?i , ? = ? ? i 2 + C1 + C2 ,
t t 2t t
where C1 , C2 are arbitrary constants.
The paper is partly supported by the International Soros Science Education Prog-
ram (grant No. PSU061097).
180 W.I. Fushchych, Z.I. Symenoh

1. Fushchych W.I., Cherniha R.M., The Galilean relativistic principle and nonlinear partial di-
?erential equations, J. Phys. A: Math. Gen., 1985, 18, 3491–3503.
2. Fushchych W.I., Cherniha R.M., The Galilei-invariant nonlinear systems of evolution equations,
J. Phys. A: Math. Gen., 1995, 28, 5569–5579.
3. Fushchych W., Chopyk V., Cherkasenko V., Symmetry and exact solutions of multidimensional
nonlinear Fokker–Planck equation, Dopovidi Akademii Nauk Ukrainy, 1993, ¹ 2, 32–42.
4. Fushchych W., How to extend symmetry of di?erential equations?, in Symmetry and Solutions
of Nonlinear Equations of Mathematical Physics, Kyiv, Inst. of Math., 1987, 4–16.
5. Fushchych W., New nonlinear equations for electromagnetic ?eld having velocity di?erent from c,
Dopovidi Akademii Nauk Ukrainy, 1992, ¹ 4, 24–27.
6. Fushchych W., Ansatz’95, J. Nonlinear Math. Phys., 1995, 2, ¹ 3–4, 216–235.
7. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of
nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993.
8. Ovsyannikov L.V., Group analysis of di?erential equations, Academic Press, New York, 1982.
9. Olver P., Application of Lie groups to di?erential equations, New York, Springer, 1986.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 181–185.

On new Galilei- and Poincar?-invariant
e
nonlinear equations for electromagnetic ?eld
W.I. FUSHCHYCH, I.M. TSYFRA

Nonlinear systems of di?erential equations for E and H which are compatible wi-
th the Galilei relativity principle are proposed. It is proved that the Schr?dinger
o
equation together with the nonlinear equation of hydrodynamic type for E and H
are invariant with respect to the Galilei algebra. New Poincar?-invariant equations
e
for electromagnetic ?eld are constructed.

1. It is usually accepted to think that the classical Galilei relativity principle does
not take place in electrodynamics. This postulate was accepted more then 100 years
ago and it is even di?cult to state the following problems:
1. Do systems of di?erential equations for vector-functions (E, H) or (D, B) which
are invariant under the Galilei algebra exist?
2. Is it possible to construct a successive Galilei-invariant electrodynamics?
3. Do the new relativity principles di?erent from Galilei or Poincar?–Lorentz–
e
Einstein ones exist?
The positive answers to this questions are given in [1–6]. But from the physical
and mathematical points of view this fundamental problems still require detailed
investigations. In the paper we continue these investigations. Further we give theorems
on local symmetries of the following systems of di?erential equations
?D ?B
= ?rot E,
= rot H,
(1)
?t ?t
div D = 0, div B = 0;

a1 D + a2 2D = F1 E 2 , B 2 , B E E + F2 E 2 , B 2 , B E B,
(2)
b1 H + b2 2H = R1 E 2 , B 2 , B E E + R2 E 2 , B 2 , B E B;

?E ?H
= rot H + N1 ?P1 , = ?rot E + N2 ?P2 , (3)
?t ?t
?P1 ?P2
(4)
div E = N1 , div H = N2 ,
?t ?t
where N1 , N2 , P1 , P2 are functions of w1 = E 2 ? H 2 , w2 = E H;
?F1 (?† ?)
?Ek ?Ek
+ Hl = ,
?t ?xl ?xk
(5)
?F2 (?† ?)
?Hk ?Hk
+ El = , k = 1, 2, 3;
?t ?xl ?xk
J. Nonlinear Math. Phys., 1997, 4, ¹ 1–2, P. 44–48.
182 W.I. Fushchych, I.M. Tsyfra
?
?? ? 1
2
?E ?H
=? ?l ? ie?(E ? H) ?
i +
? 2m
?t ?xl ?xl
(6)
?E ?H e
+ e?(E ? H) ? ?? ?(E ? H)?,
?t ?t 2m

? are the Pauli matrices, ? is a wave function;
?
?? ? 1
2
El Hl
=? ?l ? ie ?1
i + ?2 +
? 2m
?t E2 H2
(7)
?1 ?2 e E H
??
+ e?1 + ? ? ?3 + ?4 ?,
2m
E2 H2 E2 H2

where ?, ?1 , ?2 , ?3 , ?4 , ? are functions of E 2 , H 2 , E H.

? ?
m(v 2 )v = a1 (E + v ? H) + a2 (H ? v ? E), (8)
+ vl
?t ?xl

where v = (v1 , v2 , v3 ), a1 , a2 are smooth functions of v 2 , E 2 , H 2 , v E, v H, E(v ? H),
H(v ? E).
Equation (8) can be considered as a hydrodynamics generalization of the classical
Newton–Lorentz equation of motion.
2. To study symmetries of the above equations (1)–(4), we use in principle the
standard Lie scheme and therefore all statements are given without proofs. But it
should be noted that the proofs of theorems require nonstandard steps and long
cumbersome calculations which are omitted here.
As proved in [9], system (1) of undetermined equations for D, B, E, H is invariant
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