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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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