ñòð. 125 |

R1

where det FAB = 0, whence

A,B=1

WA (xR1 , . . . , xn?1 , u , u , . . . , u ) = 0,

r

1

(24)

? = 0, m ? 1,

ux? = 0, A = 1, R1 , a = 1, R1 .

a?1

The ansatz of the field u ? = u ? (x ) invariant under the involutive set of operators

Qc = ?xa?1 , a = 1, . . . , R1 , is given by the formulas

? = 0, m ? 1.

u ? = ?? (xR1 , xR1 +1 , . . . , xn?1 ), (25)

Here ?? are arbitrary sufficiently smooth functions.

Substituting (25) in (24), we obtain

WA (xR1 , . . . , xn?1 , u , u , . . . , u ) ? WA (xR1 , . . . , xn?1 , ?, ?, . . . , ?) = 0, (26)

r

1 r

1

where ? is the set of partial derivatives of the functions ?? = ?? (xR1 , . . . , xn?1 ) of

s

order s.

Rewriting ansatz (25) in terms of the initial variables x and u(x)

? = 0, m ? 1,

g ? (x, u) = ?? (fR , (x, u), . . . , fn?1 (x, u)), (27)

yields the ansatz for the field u? = u? (x), ? = 0, . . . , m ? 1, invariant with respect

to the involutive set of operators (3) that reduces the system (1) to a system of PDE

with n ? R1 independent variables. The theorem is proved.

530 W.I. Fushchych, R.Z Zhdanov

Corollary. Suppose that the operators

N ?n?1

?

Qa = ?aµ (x, u)?xµ + ?a (x, u)?u? , a = 1, N ,

are the basis elements of a subalgebra of the invariance algebra of the system of

equations (1) and, moreover, that R1 = R2 . Then the ansatz invariant in the Lie

algebra Q1 , Q2 , . . . , QN reduces the system (1) to a system of PDE having n ? N

independent variables.

Proof. From the definition of a Lie algebra it follows that the operators Qa satisfy (4)

c

with fab = const. Consequently, they form an involutive set of first-order differential

operators, which renders the above assertion a direct consequence of Theorem 1.

By the above assertion, the classical reduction theorem for differential equations

by means of group-invariant solutions [1, 2, 9] is a special case of Theorem 1. If

any one of the operators Qa does not belong to the invariance algebra of the given

equation and if the conditions of Theorem 1 hold, a reduction via Qa -conditionally

invariant ans?tzes is obtained (numerous examples of conditionally invariant solutions

a

are constructed in [4–6, 10–14]).

We shall now consider several examples.

Example 1. The Lie-maximal invariance algebra of the Schrodinger equation

?3 u + U (x 2 )u = 0 (28)

with arbitrary function U is the Lie algebra of the rotation group having basis

elements

Jab = xa ?xb ? xb ?xa , (29)

a, b = 1, 3.

To obtain the ansatz invariant relative to the set of operators (29), the complete

set of first integrals of the following system of PDE must be constructed:

xa uxb ? xb uxa = 0, (30)

a, b = 1, 3.

This set contains 3 ? R1 functionally invariant first integrals, where

?x3

0 x2

?x1

3

= rank x3 0

R1 = rank ?ab (x) = 2.

a,b=1

?x2 x1 0

Consequently, the ansatz for the field u = u(x) invariant with respect to a Lie

algebra having basis elements (29) has the form

(31)

u(x) = ?(?),

where ? ? C 2 (R1 , C1 ) is an arbitrary smooth function and ? = ?(x) is the first

integral of the system of PDE (30). It is not hard to see that ? = x 2 satisfies (30)

and, consequently, is the first integral. Substitution of (31) in (28) yields an ordinary

differential equation for the function ?(?):

4? ? + 6? + U (?)? = 0.

? ?

Thus, the ansatz for the field u = u(x) invariant with respect to a three-dimensio-

nal Lie algebra with basis elements (29) reduces (28) to a (3 ? R1 )-dimensional PDE

(in this case, to an ordinary differential equation).

Conditional symmetry and reduction of partial differential equations 531

Example 2. Consider the nonlinear eikonal equation

u2 0 ? u2 1 ? u2 2 ? u2 3 + 1 = 0. (32)

x x x x

As shown in [15], the maximal invariance algebra of (32) is the 21-parameter

conformal algebra AC(2, 3). This algebra contains, in particular, a one-dimensional

subalgebra generated by the operator Q = x0 ?u ? u?x0 .

To obtain the ansatz invariant under the operator Q, the complete set of first

integrals of the following PDE must be constructed:

(33)

uux0 + x0 = 0.

The solution of (33) is sought for in the implicit form f (x, u) = 0, whence ufx0 ?

x0 fu = 0.

The complete set of first integrals of the latter PDE is ?0 = u2 + x2 , ?1 = x1 ,

0

?2 = x2 , ?3 = x3 . Solving f (?0 , ?1 , ?2 , ?3 ) = 0 with respect to ?0 , we have

u2 + x2 = ?(?1 , ?2 , ?3 ) (34)

0

Consequently, (34) gives the ansatz of the field u? = u? (x) invariant under the

operator Q. Solving (34) for u yields

u = {?x2 + ?(?1 , ?2 , ?3 )}1/2 . (35)

0

Let us emphasize that ansatz (34) cannot be represented in the form (12), since

the coefficients of Q do not satisfy condition (8).

Substituting (35) in (32) gives us a three-dimensional PDE for the function ? =

?(?):

?2 1 + ?2 2 + ?2 3 ? ?2 = 0.

? ? ?

Example 3. A detailed group-theoretic analysis of the nonlinear wave equation

utt = (a2 (u)ux )x , (36)

where a(u) is some smooth function, was performed in [16]. It was established that

the maximal invariance algebra of (36) has the basis operators

(37)

Q1 = ?t , Q2 = ?x , Q3 = t?t + x?x ,

whence the most general group-invariant ansatz for the PDE (36) is given by the

formula u = ?(?), where ? = ?(t, x) is the first integral of the PDE

{??t + ??x + ?(t?t + x?x )}?(t, x) = 0. (38)

Here ?, ?, and ? are arbitrary real constants. Using transformations from the group G

with generators of the form (37), Eq. (38) may be reduced to either one of the

following equations:

1) ??t + ??x = 0 (under ? = 0);

2) t?t + x?x = 0 (under ? = 0),

The first integrals of these equations are given by the formulas ? = ?x ? ?t and

? = xt?1 , respectively.

532 W.I. Fushchych, R.Z Zhdanov

Thus, there are two distinct group-invariant ans?tzes of the PDE (36) with arbit-

a

rary function a(u):

u(t, x) = ?(?x ? ?t),

1)

(39)

u(t, x) = ?(xt?1 ).

2)

Substitution of the above ans?tzes in (36) yields the ordinary differential equations

a

(? 2 ? ?2 a2 (?))? ? 2?2 a(?)a(?)?2 = 0,

1) ? ? ?

(? ? a (?))? ? 2? ? ? 2a(?)a(?)?2 = 0.

2 2

2) ? ? ? ?

It was established recently [17] that ans?tzes (39) do not exhaust the complete

a

set of ans?tzes reducing the PDE (36) to ordinary differential equations. This result

a

is a consequence of conditional symmetry, a property that is not found within the

framework of the infinitesimal Lie method.

Let us show, following [17], that (36) is conditionally invariant under the operator

Q = ?t ? ?a(u)?x , (40)

where ? = ±1.

Proceeding on the basis of the second extension of Q in (36), we have

?

Q{utt ? (a2 (u)ux )x } = ?aux {utt ? (a2 ux )x } + ?(aux + a?x )(u2 ? a2 u2 ), (41)

? ?? ? t x

whence it follows that the PDE (36) is Lie-noninvariant with respect to a group with

infinitesimal operator (40). But if the additional constraint

Qu ? ut ? ?a(u)ux = 0 (42)

is imposed on u(t, x), the right side of (41) vanishes. Consequently, the system (36),

(42) is Lie-invariant with respect to a group with generator (40), whence we conclude

that the initial PDE (36) is conditionally invariant under the operator Q.

The complete set of functionally independent first integrals of (42) may be chosen

in the form ?1 = u, ?2 = x + ?a(u)t.

Consequently, the ansatz invariant under the operator Q is given by the formula

?2 = ?(? 1 ), or

(43)

x + ?a(u)t = ?(u),

where ?(u) is an arbitrary sufficiently smooth function.

Substituting (43) in (36) leads us to conclude that the PDE (36) is satisfied

identically. Put differently, (43) gives a solution of the nonlinear equation (36) for an

arbitrary function ?(u). Recall that solutions that are obtained by means of the group-

invariant ans?tzes (39) contain two arbitrary constants of integration, and cannot, in

a

theory, contain arbitrary functions.

Thus, the conditional symmetry of PDE enlarges the range of possibilities for

reduction of PDE in an essential way.

Example 4. Consider the system of nonlinear Dirac equations

?

{i?µ ?µ ? ?(??)1/2k }? = 0, (44)

Conditional symmetry and reduction of partial differential equations 533

where ?µ , µ = 0, . . . , 3, are (4 ? 4) Dirac matrices, ? = ?(x0 , x1 , x2 , x3 ) a four-

dimensional complex column function, ? = (? ? )T ?0 , ?, k real constants, and ?µ =

?

?/?xµ , µ = 0, . . . , 3.

It is well known (cf. [5]) that the Lie-maximal invariance group of the system

of PDE (44) is the 11-parameter extended Poincar? group complemented with the

e

3-parameter group of linear transformations in the space ? ? , ? ?? . In [5, 10] it is

established that the conditional symmetry of the nonlinear Dirac equation is essen-

tially broader. From [10], it follows that the system: (44) is conditionally invariant

with respect to the involutive set of operators

1

(?0 ? ?3 ), Q2 = ?1 ?2 ? {B1 ?}? ??? ,

Q1 =

2 (45)

1

Q3 = (?0 + ?3 ) ? ?1 (x1 ?1 + x2 ?2 ) ? ?2 ?1 ? {B2 ?}? ??? ,

? ?

2

where B1 and B2 are (4 ? 4) matrices of the form

1

(1 ? 2k)?1 ?2 (?0 + ?3 ),

B1 = ?

2

B2 = ?k ?1 + (2?1 )(2?1 ? ?1 ?1 )(?1 x1 + 2(k ? 1)?2 x2 )(?0 + ?3 ) + (2?1 )?1 ?

?2

? ?

? ((2?1 ?2 ? ?1 ?2 )?1 + 2(?3 ?1 ? ?1 ?3 )?2 )(?0 + ?3 ),

?? ? ? ?

?1 , ?2 , and ?3 are arbitrary smooth functions of x0 + x3 , and {?}? denotes the ?-

th component of the function ?. Since the coefficients of the operators (45) satisfy

conditions (8), they may be rewritten in non-Lie form:

1

(?0 ? ?3 ), Q2 = ?1 ?2 + B1 ,

Q1 =

2

Q3 = 1 (?0 + ?3 ) ? ?1 (x1 ?1 + x2 ?2 ) ? ?2 ?1 + B2 .

? ?

2

Consequently, the ansatz of the field ?(x) invariant with respect to the set of

operators Q1 , Q2 , Q3 must be found in the form (12), where A(x) is a (4 ? 4) matrix

and ? = ?(x) a real function satisfying the following system of PDE

1

(Ax0 ? Ax2 ) = 0, ?1 Ax2 + B1 A = 0,

2

1

(Ax0 + Ax3 ) ? (?1 x1 + ?2 )Ax1 ? ?1 x2 Ax2 ? B2 A = 0,

? ? ?

2

?x0 ? ?x3 = 0, ?x2 = 0,

?x0 + ?x3 ? 2(?1 x1 + ?2 )?x1 ? 2?1 x2 ?x2 = 0.

? ? ?

Omitting the steps in integration of the above system, let us write down the final

result, the ansatz for the field ? = ?(x) invariant with respect to the involutive set

of operators (45):

?(x) = ?1 exp{(2?1 )?1 (?1 x1 + ?2 )?1 (?0 + ?3 ) +

k

? ?

(46)

+ (2?1 )?1 ((2k ? 1)?1 x2 + ?3 )?2 (?0 + ?3 )}?(?1 x1 + ?2 ).

?

This ansatz reduces me system of PDE (44) to a system of ordinary differential

equations for the 4-component function ? = ?(?),

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