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?x ?u
with ? being a smooth function.
Thus, the problem of finding the conditional symmetry of a equation reduces to the
solution of equations (3.3), (3.4). As a rule, the determining equations for calculating
?µ and ? are nonlinear equations.
As is known, in the classical approach ?µ , ? satisfy a linear system of differential
equations which, because of being overdetermined, can be solved.
Ansatz ’95 343

3.1 Conditional symmetry of the Maxwell equations
The first equation where we had noticed conditional symmetry was the Maxwell
subsystem [13]

?E ?H
= ?rot E. (3.7)
= rot H,
?t ?t
It is possible to prove by means the standard Lie method that the maximal invari-
ance algebra of system (3.7) is an 8-dimensional extended Euclid algebra AE1 (4) with
basis elements:
?
Jab = xa pb ? xb pa + Sab , D = xµ P µ , (3.8)
Pµ = i ,
?xµ
where Sab are 6?6 matrices, which realize a reduced representation of the Lie algebra
of the group SU (2).
Thus, system (3.7) is not invariant with respect to the Lorentz transformations,
which are generated by operators

Joa = xo Pa ? xa P0 + S0a , (3.9)

Sab , S0a are matrices which realize a finite-dimensional representation of the Lie
algebra of the Lorentz group S(1, 3).
Theorem 1. (Fushchych W. and Nikitin A. 1983 [13]). System (3.7) is conditionally
invariant under the Lorentz boosts (3.9) if and only if the solutions of (3.7) satisfy
the conditions

(3.10)
div E = 0, div H = 0.

Thus, system (3.7) only together with equations (3.10) is invariant under the Lorentz
group.
Note 2. 90 years ago H. Lorentz (1904, April 23), H. Poincar? (1905, June 5, July 23),
e
A. Einstein (1905, June 30) discovered the theorem about invariance of the full
Maxwell system (3.7), (3.10) with respect to rotations in the four-dimensional pseudo-
Euclidean space-time. This theorem is a mathematical formulation of the fundamental
Lorentz–Poincar?–Einstein principle of relativity.
e

3.2 Conditional symmetry of linear Schr?dinger systems
o
Let us consider the multicomponent system of disconnected Schr?dinger equations:
o

p2
S? = p0 ? a ?r = 0, r = 1, 2, . . . , n,
2m
? ? (3.11)
, pa = ?i
p0 = i , a = 1, 2, 3,
?x0 ?xa
? = (?1 , ?2 , . . . , ?n ), ? = ?(x0 = t, x1 , x2 , x3 ).

It is evident that every separate Schr?dinger equation (3.11) is invariant with
o
respect to a scalar representation of the group G2 (1, 3), a full Galilei group.
344 W.I. Fushchych

Let us consider a problem of existence of nontrivial vector, spinor, tensor represen-
tations of the full Galilei group, which are realized on the set of solutions of system
(3.11).
We demand system (3.11) be invariant with respect to the following linear repre-
sentations of the algebra AG2 (1, 3)
? ?
Pa = ?i
P0 = i , , M = im,
?x0 ?xa
1
Ja = xa pb ? xb pa + Sa , Sa = ?abc Sbc ,
(3.12)
2
Ga = x0 pa ? xa p0 + ?a , D = 2x0 P0 ? xk Pk + ?0 ,
1
A = x0 D ? x2 P0 + mx2 ? ?a xa ,
0 k
2
where matrices Sa , ?0 , ?a satisfy the commutation relations [29]

[Sa , Sb ] = i?abc Sc , [?a , ?b ] = 0, [?0 , Sa ] = 0,
(3.13)
[?a Sb ] = i?abc Sc , [?0 , ?a ] = i?a .

Theorem 2 (Fushchych and Shtelen, 1983, [19]). System of equations (3.11) is condi-
tional invariant under representation AG2 (1, 3) (3.12) if

3 1
?0 ? i ? ?k Pk ? = 0, (3.14)
2 m

(?2 + ?2 + ?2 )? = 0. (3.15)
1 2 3

Q-conditional symmetry of Lorentz noninvariant nonlinear
3.3
wave equation
Let us consider the following wave equation (Fushchych and Tsyfra 1987, [15])

Lu ? 2u + F (x, u) = 0 (3.16)
1

2 2 2 2
?0 ?u ?1 ?u
F (x, u) = ? + +
x0 ?x0 x1 ?x1
1
(3.17)
2 2 2 2
?2 ?u ?3 ?u
+ + , xµ = 0,
x2 ?x2 x3 ?x3

?µ are arbitrary parameters.
Equation (3.16) is invariant only with respect to scale transformations and trans-
lations:

xµ > xµ = eb xµ , u > u = e2b u, u > u = u + c,

b is a real parameter.
Let us consider a Lorentz-invariant ansatz

? = xµ xµ = x2 ? x2 ? x2 ? x2 . (3.18)
u = ?(?), 0 1 2 3
Ansatz ’95 345

This ansatz, despite the fact that (3.16) is not invariant with respect to the Lorentz
group, reduces equation (3.16) to ODE
2
d2 ? d? d?
+ ?2 (3.19)
? 2 +2 =0
d? d? d?
whose solutions are given by the functions

? = ?2 ? ?2 ? ?2 ? ?2 ,
0 1 2 3
?(?) = 2(??2 )1/2 tan?1 ?(??2 )?1/2 for ? ?2 > 0,
(?2 )1/2 + ?
2 ?1/2
?(?) = ?(? ) for ? ?2 < 0.
ln
(?2 )1/2 ? ?
What is the reason of such reduction? From the classical point of view, ansatz (3.18)
must not reduce the Lorentz non–invariant equation (3.16) to ODE.
The reason of all this is the fact that equation (3.16) is conditionally invariant with
respect to the Lorentz group.
Theorem 3 (Fushchych and Tsyfra, 1987 [15]). Equation (3.16), (3.17) is conditio-
nally invariant with respect to the Lorentz group if the following six conditions are
? ?
? x? (3.20)
Jµ? u = 0, Jµ? = xµ , µ, ? = 0, 1, 2, 3.
?x? ?xµ
Thus, equation (3.16) together with the additional condition (3.20) is invariant
with respect to the Lorentz group. The condition (3.20) picks out the subset from the
whole set of solutions which is invariant with respect to the Lorentz group.

3.4 Conditionally conformal symmetry
of the Poincar?-invariant d’Alembert equation
e
Let us consider the nonlinear d’Alembert equation with an additional condition

2u + F (u) = 0, (3.21)

?u ?u
(3.22)
= F1 (u).
?xµ ?xµ

Theorem 4 (Fushchych, Zhdanov, Serov 1989 [18]). Equation (3.21) is conditionally
invariant under the conformal group if

F = 3?(u + c)?1 , (3.23)

?u ?u
(3.24)
= ?,
?xµ ?xµ

where ?, c are arbitrary constants. The operators of conformal symmetry are
?
Kµ = 2xµ D ? (x? x? ? u2 ) (3.25)
, µ = 0, 1, 2, 3
?xµ
346 W.I. Fushchych

? ?
D = xµ (3.26)
+u .
µ
?x ?u
Remark 3. Formulae (3.25), (3.26) give a nonlinear representation for the conformal
algebra AC(1, 3).
An ansatz for the system

2u = u?1 , ?µ u? µ u = 1 (3.27)

has the form (Fushchych and Zhdanov, 1989 [4])

u2 = (aµ xµ + g1 )2 ? (bµ xµ + g2 )2 , (3.28)

where g1 = g1 (?µ xµ ), g2 = g2 (?µ xµ ) are arbitrary smooth functions, aµ , bµ , ?µ are
arbitrary complex parameters satisfying the condition

aµ aµ = ?bµ bµ = 1, aµ bµ = aµ ?µ = bµ ?µ = ?µ ?µ = 0.

Remark 5. The problem of compatibility and construction of solutions of the d’Alem-
bert–Hamilton system are considered in detail in [5, 6].

3.5 Conditional symmetry of the nonlinear heat equation
Let us consider the equation

u0 + ?[f (u)?u] = 0, (3.29)
f (u) = const.

Ovsyannikov L. (1962, [20]) carried out the complete classification of the one-
dimensional equation (3.29). Dorodnitsyn A., Knyaseva Z., Svirshchevskii S. (1983,
[21]) carried out group classification of the three-dimensional equation (3.29) From
the analysis of these results it follows.
Conclusion 1. (Fushchych 1983 [2]). Among equations of the class (3.29), there are
no nonlinear equations invariant with respect to Galilei transformations which are
generated by the operators
?
(3.30)
Ga = x0 ?a + M (u)xa ,
?u
M (u) is constant.
Theorem 5 (Fushchych, Serov, Chopyk 1988 [16]). The equation (3.29) is conditional
invariant under the Galilean operators (3.30) if

(?u)2
(3.31)
u0 + = 0,
2M (u)
u
(3.32)
M (u) = .
2f (u)

Conclusion 2. The nonlinear equation (3.29) with the additional condition (3.31) is
compatible with the Galilei relativity principle.
Ansatz ’95 347

Conclusion 3. If
1k 2m 1?k
(3.33)
f (u) = u, M (u) = u ,
2m kn + 2
f (u) = eu , (3.34)
M (u) = 1,

where m, k are arbitrary constants, kn + 2 = 0, then equation (3.29) is conditionally
invariant with respect to Galilei transformations.
Q-conditional symmetry of the one-dimensional equation
u0 ? u11 = F (u)
was studied in detail (Fushchych and Serov, 1990, [22, 23]). Recently these results
were obtained by Clarkson P. and Mansfield E. (1994, [24]).
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