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starting equation (5). The system of equations (5), (8), with F = 0, is consistent.

Theorem 3 ([10, 15], 1988, 1989). The equation (5) is conditionally invariant

under the conformal group, if

3?

(9)

F= ,

u+c

?u ?u

(10)

= ?,

?xµ ?xµ

where ?, c are arbitrary constants. The operators of conformal symmetry are

?

Kµ = 2xµ D ? (x? x? ? u2 ) , µ = 0, 1, 2, 3,

?xµ (11)

? ?

µ

D=x +u .

?xµ ?u

Remark. It is important to note, that the operators (11) differ principally from the

conformal operators for equation (5), when F = 0 or F = ?u3 . In those cases, the

conformal operators are

? ?

?

Kµ = 2xµ D ? x? x? µ , D = xµ (12)

.

?xµ

?x

The operators (11) are non-linear, whereas those in (12) are linear.

Thus the wave equation (5), (9), with non-linear condition (10), has a symmetry

possessed by neither the solution set for the linear equation, nor that for the nonlinear

equation.

4. Criteria for conditional symmetry

Let us consider some PDE

L(x, u(1) , u(2) , . . . , u(n) ) = 0,

u(1) = (u0 , u1 , . . . , un ), u(2) = (u01 , u02 , . . . , unn ), ..., (13)

?2u

?u

uµ = , uµ? = , ....

?xµ ?xµ ?x?

Definition 1 (S. Lie, 1884). Equation (13) is invariant with respect to the opera-

tor (6) if

(14)

Xs L = ?L,

where Xs is the s-th prolongation of (6), and ? = ?(x, u) is an arbitrary function.

Let us denote by the symbol

(15)

Q = Q1 , Q2 , . . . , Qr

some set of operators which does not belong to the invariance algebra (IA) of equa-

tion (13).

12 W.I. Fushchych

Definition 2 ([2], 1987). Equation(13) is said to be conditionally invariant under

the operators Q from (15), if there exists a supplementary condition on the solutions

of (13) of the form

(16)

L1 (x, u, u(1) , . . . , u(n) ) = 0

such that (13) together with (16) is invariant under the Q.

Thus one has the following conditions

(17)

Qs L = ?0 L + ?1 L1 ,

(18)

Qs L1 = ?2 L + ?3 L1

or

(19)

Qs L = 0, Qs L1 = 0.

L=0 L=0

L1 = 0 L1 = 0

An important class of supplementary conditions (16) is that for which the equation

L1 = 0 is a quasi-linear equation of first order

L1 (x, u, u(1) ) ? Qu = 0, (20)

? ?

Q = y µ (x, u) (21)

+ z(x, u)

?xµ ?u

with y µ , z being smooth functions. In this case, we shall say that (13) is Q-conditio-

nally invariant.

In this way, the problem of finding the conditional symmetry of (13) reduces to

the solution of the equations (17), (18). The conditions (16), (20) can be considered

as equations for the construction of ansatzes for the starting equation (13). The

problem of calculating the conditional symmetry is far more complicated than the

usual method of Lie for finding the symmetry of the full solution set. In the case

of conditional symmetries, the defining equations are, as a rule, non-linear equations

which can be solved in only some cases. Fortunately, for most of the equations of

non-linear mathematical physics, one can construct partial solutions of the defining

equations.

5. A list of equations with non-trivial conditional symmetry

Conditional symmetries began to be exploited only quite recently, and the first

publications appeared only in 1983 [1, 2]. Now, the number of articles in this are

is increasing rapidly with each year, and therefore it is difficult to give a complete

list (for 1992) of important equations of mathematical physics possessing conditional

symmetry. So I shall only give those equations which we have studied and which are

interested from our Kievan point of view. We have put in brackets the year(s) when

the conditional symmetry of the given equation was found. More detailed information

about ansatzes and solutions of the above equations are to be found in the original

articles, a list of which are given in [2, 9, 11].

?

? ?u(u ? 1),

2

?

?

? ?(u3 ? 3u + 2),

(1988, 1990)

1. u0 + u11 = F (u) =

? ?u3 ,

?

?

?

?u(u3 + 1).

Conditional symmetries of the equations of mathematical physics 13

2. iu0 + ?u + F (|u|)u = 0,

F (|u|) = ?1 |u|4/r + ?2 |u|?4/r , F (|u|) = ?3 ln(u? u), (1990)

?1 , ?2 , r arbitrary, real; ?3 arbitrary, complex.

(1987, 1988)

3. u00 = u?u, u00 = c(x, u, u(1) )?u.

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

4.

u0 + ?(F (u)?u) = 0. (1988)

5.

u0 + F (u)uk + u111 = 0. (1991)

6. 1

N

7. u0 + (?(u))11 + (?(u))1 = F (u),

x1

3

u1 = ?u3 , (1992)

u0 + u11 +

2x1

N

u0 + uu11 + uu1 = ?u + ?2 .

x1

1

u0 + (u?)u = ? ?p,

8.

? (1992)

1

p = ?2 .

?0 + div (?u) = 0, p = f (?),

2

?

? µ ?µ ? + F (??)? = 0. (1989)

9.

(1 ? u? u? )2u + uµ u? uµ? = 0. (1989)

10.

6. Conditional symmetry and exact solutions of KdV type equations

To illustrate the constructive nature of conditional symmetries, we shall examine

the equation

u0 + F (u)uk + u111 = 0, (22)

1

where F (u) is a smooth function, k = 0 is an arbitrary, real parameter. When F (u) =

u, k = 1, equation (22) coincides with the standard KdV equation.

Theorem 4 ([11], 1991). Equation (22) is Q-conditionally invariant with respect to

the following operators

Q = xr ?1 + H(x, u)?u (23)

0

with r an arbitrary, real parameter, in the following cases

?1/k

?1 k

(2?k)/k (1?k)/2

u1/2 ; (24)

1. F (u) = ?1 u + ?2 u , H(x, u) =

2

H(x, u) = (k?1 )?1/k ;

F (u) = (?1 ln u)1?k , (25)

2.

F (u) = (?1 arcsin u + ?2 )(1 ? u2 )(1?k)/2 ,

3.

(26)

H(x, u) = (k?1 )?1/k (1 + u2 )1/2 ;

14 W.I. Fushchych

F (u) = (?1 sinh?1 u + ?2 )(1 + u2 )(1?k)/2 ,

4.

(27)

H(x, u) = (k?1 )?1/k (1 + u2 )1/2 ;

H(x, u) = (k?1 )?1/k , (28)

5. F (u) = ?1 u,

where r = 1/k, k = 0, ?1 , ?2 are arbitrary, real parameters.

Exploiting the operator of conditional symmetry (23), one can construct ansatzes

for the solutions of equation (22), some of which I now exhibit.

The ansatz

2

?1/k

x1 k?1 x0

u= + ?(x0 )

2 2

gives the solution

2

?1/k

x1 k?1 x0 ?1/k

? ?2 /?1

u= + ?x0

2 2

when F (u) is as in (24). The ansatz

u = exp ?(x0 ) + (k?1 x0 )?1/k x1

gives the solution

k(k?1 )?3/k 1?3/k ?1/k

+ (k?1 x0 )?1/k x1 ? ?2 /?1

u = exp ? x0 + ?x0

k?2

when F (u) is as in (25) with k = 2. The ansatz

u = sin ?(x0 ) + (k?1 x0 )?1/k x1

gives the solution

k(k?1 )?3/k 1?3/k ?1/k

+ (k?1 x0 )?1/k x1 ? ?2 /?1

u = sin x0 + ?x0

k?2

for k = 2.

Theorem 5 ([12], 1990). The equation

u01 ? (F (u)u1 )1 ? u22 ? u33 = 0 (29)

is invariant under under the infinite-dimensional algebra

(30)

X = ai (u)Ri , i = 1, . . . , 12,

where ai (u) are arbitrary, smooth functions, if one adds to (29) the condition

u0 u1 ? F (u)u2 ? u2 ? u2 = 0. (31)

1 2 3

The operators Ri are given as follows:

Rµ+1 = ?µ , µ = 0, . . . , 3, R5 = x3 ?2 ? x2 ?3 , R6 = x2 ?1 + 2x0 ?2 ,

R7 = x3 ?1 + 2x0 ?3 , R8 = xµ ?µ ,

Conditional symmetries of the equations of mathematical physics 15

F (u) ?

R9 = x0 ?0 + 2x1 ?1 + 3x2 ?2 + 3x3 ?3 ? 2 ? , R10 = F (u)x0 ?1 ? ?u ,

? (u) u

F

R11 = x2 ?0 + 2(x1 + F (u)x0 )?2 , R12 = x3 ?0 + 2(x1 + F (u)x0 )?3 .

7. Antireduction

In [10], we have begun work on antireduction. By the term antireduction of a

PDE we understand the finding of such ansatzes which transform the given PDE into

a system of equations for some (unknown, and to be found) functions. In this process,

the number of independent variables may remain the same, or be reduced (dimensional

reduction), but the number of dependent variables increases. As a rule, one usually

exploits the converse of this, that is, one reduces to a system with fewer dependent

variables (reduction of components). To illustrate the effectiveness of antireduction,

we consider the equation for short waves in gas dynamics

2u01 ? 2(x1 + u1 )u11 + u22 + 2?u1 = 0. (32)

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