ñòð. 124 |

µ = 0, n ? 1, ?, ?, ? = 0, m ? 1.

a = 1, N ,

Under conditions (8) the operators in (3) may be rewritten in the following non-Lie

form [8]:

(9)

Qa = ?aµ (x)?xµ + ?a (x), a = 1, N ,

? ??a /?u? are (m ? m) matrices and the system (2) takes the

m?1

?

where ?a = ?,?=0

form

(10)

?aµ (x)uxµ + ?a (x)u = 0, a = 1, N .

Here u = (u0 , u1 , . . . , um?1 )T is a column function.

526 W.I. Fushchych, R.Z Zhdanov

In this case, the set of functionally independent first integrals of the system (2)

with R1 = R2 may be chosen as follows [7]:

?j = bj? (x)u? , j = 1, m,

(11)

i = m + 1, m + n ? R1

?i = ?i (x),

and, moreover, det bj? (x) m m?1 = 0.

i=1 ?=0

Substituting (11) in (7) and solving for the variables u? , ? = 0, . . . , m ? 1, we have

u? = A?? (x)?? (?m+1 , ?m+2 , . . . , ?m+n?R1 )

or (in matrix notation)

(12)

u = A(x)?(?m+1 , ?m+2 , . . . , ?m+n?R1 ).

It is easily verified that the matrix

m m?1 ?1

(x) = bj? (x) j=1 ?=0

satisfies the following system of PDE:

Qa A ? ?aµ (x)Axµ + ?a (x)A = 0, (13)

a = 1, N ,

and that the functions ?m+1 (x), ?m+2 (x), . . . , ?m+n?R1 (x) form a complete set of

functionally independent first integrals of the system of PDE

(14)

?aµ (x)?xµ = 0, a = 1, N .

The ansatz (7) is said to reduce the system of PDE (1) if substitution of (7) in (1)

yields a system of PDE for the functions ?0 , ?1 , . . . , ?m?1 that contains only the new

independent variables ?m+1 , ?m+2 , . . . , ?m+1?R1 .

Definition 3. The system of PDE (1) is conditionally invariant with respect to the

involutive set of differential operators (3) if the over-determined system of PDE

(1), (2) is Lie invariant with respect to a one-parameter transformation group with

generators Qa , a = 1, . . . , N .

Before stating the reduction theorem, we prove several auxiliary assertions.

Lemma 1. Suppose that the operators (3) form an involutive set. Then the set of

differential operators

(15)

Qa = ?ab (x)Qb , a = 1, N

with det ?ab (x, u) N

a,b=1 = 0 is also involutive.

We prove the assertion by direct computation. In fact,

[Qa , Qb ] = [?ac Qc , ?bd Qd ] = ?ac (Qc ?bd )Qd ? ?bd (Qd ?ac )Qc + ?ac ?bd fcd Qd1 =

d1

= f c Qc = f c ??1 Q .

? ?

ab ab cd d

Here ??1 are the elements of the inverse of the matrix ?ab (x, u) N .

a,b=1

cd

Lemma 2. Suppose that the differential operators (3) satisfy the condition R1 = R2

and that the conditions

(16)

[Qa , Qb ] = 0, a, b = 1, N

Conditional symmetry and reduction of partial differential equations 527

are satisfied. Then there exists a change of variables

µ = 0, n ? 1, ? = 0, m ? 1

u ? = g ? (x, u), (17)

xµ = fµ (x, u),

that reduces the operators Qa to the form Qa = ?xa?1 .

Proof. It is known that for any first-order differential operator

Q = ?µ (x, u)?xµ + ? ? (x, u)?u? ,

where ?µ and ? ? are sufficiently smooth functions, there exists a change of variab-

les (17) that reduces the operator Q to the form Q = ?x0 (cf. [1]). Consequently, the

operator Q1 from the set (3) is reduced to the form Q1 = ?x0 by means of the change

of variables (17). From the condition [Q1 , Qa ] = 0, a = 2, . . . , N , it follows that the

coefficients of the operators Q2 , Q3 , . . . , QN do not depend on the variable x0 , whence

the operator Q2 reduces to the operator Q2 = ?x1 under the change of variables

µ = 1, n ? 1,

x0 = x0 , xµ = fµ (x1 , . . . , xn?1 , u ),

? = 0, m ? 1,

u? = g ? (x1 , . . . , xn?1 , u ),

without the form of the operator Q1 changing.

Repeating the above procedure N ? 2 times completes the proof.

Lemma 3. A system of PDE of the form (1) that is conditionally invariant with

respect to a set of differential operators ?xµ , µ = 0, N ? 1, possesses the structure

UA = FAB WB (xN , xN +1 , . . . , xn?1 , u, u, . . . , u) + FAµ u?µ ,

?

x

r

1 (18)

? = 0, m ? 1, µ = 0, N ? 1,

A = 1, M ,

?

where FAB and FAµ are arbitrary smooth functions of x and u, u, . . . , u, WB are

r

1

FAB M

arbitrary smooth functions, and, moreover, = 0.

A,B=1

We shall prove the lemma with N = 1. By Definition 3, the system (1) is condi-

tionally invariant under the operator Q = ?x0 if the system

UA (x, u, u, . . . , u) = 0, A = 1, M ,

r

1 (19)

? = 1, m ? 1

u?0 = 0,

x

is Lie invariant with respect to a one-parameter translation group with respect to the

?

variable x0 . Denoting by Q the r-th extension of Q, the Lie invariant criteria for the

system of PDE (19) under this group assume the form (cf. [1, 2])

? ? = 0, m ? 1, (19a)

QUA = 0, A, B = 1, N ,

UB = 0

u?0 = 0

x

? ?, ? = 0, m ? 1.

Qu?0 (19b)

= 0, B = 1, N ,

UB = 0

x

u?0 = 0

x

Direct computation shows that the relations

? ?

Q ? ?x0 , Qu?0 ? ?x0 (u?0 ) = 0

x x

528 W.I. Fushchych, R.Z Zhdanov

hold (recall that in the extended space of the variables x, u, u, . . . , u variables x0 and

r

1

u?0 are independent), whence, using the method of undetermined coefficients, we may

x

rewrite (19a) and (19b) in the form

?UA /?x0 = RAB UB + PA u?0 ,

?

(19c)

A = 1, M ,

x

?

where RAB and PA are arbitrary smooth functions of x, u, u, . . . , u.

r

1

The system (19c) may be considered a system of inhomogeneous ordinary diffe-

rential equations for the functions UA , A = 1, . . . , M . Integrating (19c) with respect

?

to PA = 0, we have

(0)

UA = FAB WB , A = 1, M ,

where WB , B = 1, . . . , M , are arbitrary smooth functions of the variables x1 , x2 , . . .,

xn?1 , u, u, . . . , u; F = FAB MA,B=1 is the fundamental matrix of the system (19c)

r

1

(which is known to satisfy the condition det F = 0).

Further, by applying the method of variation of an arbitrary parameter, we deduce

(18) with N = 1, where

(F )?1 Pc dx0 , ? = 0, m ? 1.

? ?

FA0 = FAB A = 1, M ,

BC

The lemma is proved.

Theorem 1. Suppose that the system of PDE (1) is conditionally invariant with

respect to the involutive set of operators (3). Then the ansatz invariant with respect

to the set of operators (3) reduces this system.

Proof. By the definition of the quantity R1 , R1 ? N . We denote by ? the difference

N ? R1 . Then R1 equations of the system (2) are linearly independent (without loss

of generality, we may assume that it is the first R1 equations which are linearly

independent), and the other ? equations are linear combinations of these first R1

equations.

By the condition that R1 = R2 , there exists a nonsingular (R1 ? R1 ) matrix

||?ab (x, u)||R1 such that

a,b=1

n?1

?

? ?aµ u?µ ? ?a , ? = 0, m ? 1.

?ab (?bµ u?µ ?

u?a?1 ??

?b ) = + a = 1, R1

x x x

µ=R1

By the definition of conditional invariance, the system of PDE (1), (2) is invariant

with respect to one-parameter transformation groups with generators (3), whence the

equivalent system of PDE

UA (x, u, u, . . . , u) = 0, A = 1, M ,

r

1

n?1

(20)

?

?aµ u?µ ? ?a = 0, ? = 0, m ? 1

u?a?1 + ?? a = 1, R1 ,

x x

µ=R1

is invariant with respect to a one-parameter group with generators

n?1

? ?? (21)

Qa = ?ab Qb = ?xa?1 + ?aµ ?xµ + ?a ?u? .

µ=R1

Conditional symmetry and reduction of partial differential equations 529

In fact, the action of a one-parameter transformation group with infinitesimal

operator Qa on the solution manifold of the system (20) is equivalent to an identity

transformation.

Since the set of operators (21) is involutive (Lemma 1), there exist functions

c

fab (x, u) such that

c

(22)

[Qa , Qb ] = fab Qc , a, b, c = 1, R1 .

Computing the commutators on the left side of (22) and equating the coefficients

c

of the linearly independent operators ?x0 , ?x1 , ?xR1 ?1 gives us fab = 0, with a, b, c =

1, . . . , R1 . Consequently, the operators Qa commute. Hence, by Lemma 2, there exists

a change of variables (17) that reduces these operators to the form Qa = ?/?xa?1 .

Expressed in terms of the new variables x and u (x ), the system (20) takes the

form

UA (x , u , u , . . . , u ) = 0, A = 1, M ,

r

1 (23)

? = 0, m ? 1,

?

uxa?1 = 0, a = 1, R1 .

Moreover, the system of PDE (23) is conditionally invariant with respect to the set

of operators Qa = ?xa?1 , a = 1, . . . , R1 , whence, by Lemma 3, the system (23) may

be rewritten in the form

? ?

UA = FAB WB (xR1 , . . . , xn?1 , u , u , . . . , u ) + FAµ uxµ ,

r

1

? = 0, m ? 1, µ = 0, R1 ? 1,

A = 1, M ,

? = 0, m ? 1,

?

ux = 0, a = 1, R1 ,

ñòð. 124 |