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H00 ? H 3 F ? (3HH1 + 2H 2 Hu )F ? (H11 + 2HH1u )F = 0.
Having solved these systems we constructed explicit forms of operators Q for
special forms of the function F (u). Let us adduce some of obtained operators and
ans?tzes:
a
F (u) = exp u,
Q1 = x1 ?1 + ?u , u = ln x1 + ?(x0 ),
?(x0 )
Q2 = ?0 + 2 tg x0 ?u , exp u = ;
cos2 x0
426 W.I. Fushchych

F (u) = uk ,
u u u
?1 ? 4x?1 ?u , x0 exp + x1 + ? x2 exp
Q1 = ?0 + exp = 0,
0
0
2 2 2
Q2 = (k + 1)x1 ?1 + u?u , uk+1 = x1 ?k+1 (x0 );
F (u) = u?1/2 ,
Q1 = ?0 + x1 u1/2 ?u , 2u1/2 = x0 x1 + ?(x1 ),
a2
x2 x?2 + 1 x0 x3 + ?(x1 ),
x2 ?0 a1 x5 )u1/2 ?u , 1/2
Q2 = + (4x0 + u =
1 1 01 1
2
where a1 , a2 , a3 are constants.
The most simple solutions of the equation (78), constructed by means of the above
ans?tzes are or the form
a

exp u = (x2 + a1 ) cos?2 x0 , exp u = x1 exp x0 ,
1

if F (u) = exp u;

uk+1 = xk+1 x1 ,
0

if F (u) = uk ;
x4
0
u = W (x0 )x2 ,
u = x0 x1 + + a1 , 1
12
if F (u) = u;
x4
= x0 x1 + 1 + a1 ,
1/2
W (x1 )x2 , 1/2
u = 2u
0
24
a1 3
= x2 x?2 + 3a1 x0 x3 + x1 + a2 x?1 + a3 x2 ,
u1/2 01 1 1
1
6
if F (u) = u1/2 .
So we had classified and reduced the nonlinear wave equations (78) by means of
conditional symmetry.
8. Three-dimensional acoustics equation. Bounded sound beams are described
by a nonlinear equation of the form [26]

u00 ? (F (u)u1 )1 ? u22 ? u33 = 0. (79)

In the case when F (u) = u it coincides with the Khokhlov–Zabolotskaya equation

u00 ? (uu1 )1 ? u22 ? u33 = 0. (80)

Let us add to (79) an additional condition in the form of a first-order nonlinear
equation

u0 u1 ? F (u)u2 ? u2 ? u2 ? 0. (81)
1 2 3

Theorem 12 [26]. The equation (80) with the condition (81) is invariant under the
infinite-dimensional algebra with the operator

(82)
X = ai (u)Ri , i = 1, 12,
Conditional symmetry of equations of nonlinear mathematical physics 427

where ai (u) are arbitrary smooth functions of the dependent variable u,
Rµ+1 = ?µ , µ = 0, 3, R5 = x3 ?2 ? x2 ?3 ,
R6 = x2 ?1 + 2x0 ?2 , R7 = x3 ?1 + 2x0 ?3 , R8 = xµ ?µ ,
F (u)
R9 = 4x0 ?0 + 2x1 ?1 + 3x2 ?2 + 3x3 ?3 ? 2 ?u , R10 = F (u)x0 ?1 ? ?u ,
F (u)
R11 = x2 ?0 + 2(x1 + F (u)x0 )?2 , R12 = x3 ?0 + 2(x1 + 2F (u)x0 )?3 .
Operators R1 , . . . , R8 are Lie symmetry operators for the equation (80), R9 , . . .,
R12 are operators of the conditional symmetry for the equation (79). Using conditional
symmetry operators of the equation (79) R9 , . . . , R12 it is possible to construct wide
classes of exact solutions. For example, the operator X = ?0 + a(u)?1 generates the
following ans?tzes:
a
(83)
u = ?(?1 , ?2 , ?3 ), ?1 = a(u)x0 + x3 , ?2 = x2 , ?3 = x3 .
The ansatz (83) reduces the four-dimensional equation (79), (81) to three-dimansional
ones
da(?)
(a(?) ? ?)?11 ? ?22 ? ?33 + ? 1 ?2 = 0,
1
d?
(84)
??
(a(?) ? ? ?
?)?2 ?2 ?2 = 0, ?i = , i = 1, 3.
1 2 3
??i
Taking a(u) in some concrete form it is possible in some cases to construct the
general solution of (84). Let a(u) = u + 1, then we get a system
?11 ? ?22 ? ?33 = 0, (85)

?2 ? ?2 ? ?2 = 0. (86)
1 2 3

The system (85), (86) can be naturally called the Bateman (1914) — Sobolev–Smirnov
(1932–1933) equations, because Bateman, Sobolev and Smirnov investigated this
system in detail. The equations (85), (86) has the general solution which is given
by Sobolev–Smirnov formula
(87)
? = c1 (?)?1 + c2 (?)?2 + c3 (?)?3 ,
where c1 , c2 , c3 are arbitrary functions satisfying the following conditions:
c2 ? c2 ? c2 = 0, c2 + c2 = 0.
1 2 3 2 3

Thus the formula (87) gives the class of exact solutions for the three-dimensional
nonlinear equations (85), (86).
9. Conditional symmetry of the Dirac equation. Let us consider the nonlinear
Dirac equation
{?µ pµ ? ?(??)}?(x) = 0 (88)
?
and put on its solutions a condition ?? = 1. Then (88) becomes a linear equation
?
with a nonlinear additional condition:
?
(?µ pµ ? ?)? = 0, (89)
?? = 1.
428 W.I. Fushchych

The system (89) is conditionally invariant under the operators [9]
Q1 = P0 ? ??0 , Q2 = P3 ? ??3 . (90)
In the case under consideration the equation of the type (6) has the form
Q1 ? = 0 and Q2 ? = 0. (91)
The operator Q1 generates the ansatz
(92)
?(x) = exp(?i??0 x0 )?(x1 , x2 , x3 ),
where ?(x1 , x2 , x3 ) is a four-component vector-function depending on three variables
only.
10. Conditional symmetry of Maxwell’s equation. Let us consider a linear
system [5]

?E ?H
= ?rot E. (93)
= rot H,
?t ?t
It can be verified directly that the system (93) is not invariant under the Lorentz
transformations. However if we add to the system (93) the well-known additional
conditions
(94)
div E = 0, div H = 0,
the system (93), (94) becomes a Lorentz-invariant one. The point of view on Max-
well’s equations which was set forth [1–9, 21] stressesthe naturality of the notion of
conditional invariance and its importance for a wide class of equations of mathematical
physics [22].
Conclusion. Investigation of conditional symmetry of partial differential equation
has been started recently. The adduced results show that we can anticipate on this
way the qualitatively new understanding of symmetry of an equation, of symmetry
classification or partial differential equations, of reduction of multi-dimensional nonli-
near equations to equations with less number of independent variables, of process of
linearization of nonlinear equations.
The principle of relativity, or equivalence of all inertial reference frames, is one of
the most fundamental laws of physics, mechanics, hydromechanics, biophysics. Saying
in the language of mathematics this principle represents the invariance of an equation
of motion whether under the Galilei transformations or under the Lorentz ones. Partial
differential equations which do not, satisfy this principle usually are not considered
in physical theories. Such equations cannot be used for mathematical description of
motion of real physical systems.
The concept of conditional invariance enables to get essentially wider classes of
equations satisfying relativity principle. Equations which are non-compatible in usual
sense with the relativity principle can satisfy it conditionally, that is, non-trivial
conditions on solutions of these equations exist, which pick out subsets of solutions of
the initial equation, invariant or under Galilei transformations, or tinder Lorentz ones.
Description and detailed investigation of classes of equations conditionally invariant,
under Galilei and Poincar? groups and their subgroup seem to the author a rather
e
significant problem of mathematical physics.
Conditional symmetry of equations of nonlinear mathematical physics 429

Conditional symmetry, for example, of a scalar equation, enables to construct
ans?tzes which increase the number of dependent variables (antireduction). It allows
a
not only to carry out reduction by number of independent variables but to increase the
number of dependent variables. We should like to stress that such ans?tzes change
a
essentially the structure of nonlinearity of the initial equation. And, certainly, they
cannot be constructed by means of the classical Lie method. The process of linearizati-
on, for example, of the nonlinear Navier–Stokes system in our approach is considered
as change of a nonlinear equation for a linear system
?u
+ ?u + ?p = 0, (95)
div u = 0,
?t
with a nonlinear additional condition

(u ?)u = 0 or {(u ?)u}2 = 0. (96)

The linear Navier–Stokes equation with the nonlinear additional conditions has a
nontrivial conditional symmetry. Evidently it is also possible to choose as an additional
condition for the Stokes–Stokes equation the following equations:

(u ?)u + ?p = 0.

We are going to devote further papers to detailed investigation of conditional linerisa-
tion of nonlinear partial differential equations.
In conclusion I adduce the list (which is far from being complete) of nonlinear
equations having nontrivial conditional symmetry

(1988, 1990)
u0 + u11 = F (u), u0 + uu11 = 0,

?
? ?, (1990)
Su + F (|u|)u = 0, S=i
?x0

(1988)
u00 = u?u,

2u = F (u), (1989)

u01 ? (F (u)u1 )1 ? u22 ? u33 = 0, (1990)

u00 ? (F (u)u1 )1 = 0, (1991)

u00 = C(x, u, u)?u, (1987)
1

u0 ? ?[F (u)?u] = 0, (1988)

u0 + F (u)uk + u111 = 0, (1991)
1

? 2 ?(u) N ??(u)
(1992)
u0 + + = F (u),
2
?x1 x1 ?x1

3
u1 = ?u3 , (1992)
u0 + u11 +
2x1
430 W.I. Fushchych

N
(1992)
u0 + uu11 + uu1 = ?1 u + ?2 ,
x1
1
u0 + (u ?)u = ? ?p,
?
(1992)
?0 + div (?u) = 0,
1
p = f (?), p = ??2 ,
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