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Q L = ?0 L + ?1 (Qu).

s

Definition 2. We say that the equation (3) is Q-conditionally invariant if the system

(3), (9) is invariant under the operator (10).

Let us turn now to the simplest one-dimensional acoustics equation.

2. Conditional symmetry of the equation (1).

Theorem 1 [18]. The equation (1) is Q-conditionally invariant under the operator

(10) if its coefficient functions

j 0 ? A(x), j 1 ? B(x), z ? h(x)u + q(x), x = (x0 , x1 )

satisfy the following differential equations:

Case 1. A = 0, B = 0:

B B

h = 2 B1 ? A0 + A1 , q=2 B0 ,

A A

2 h 2 h q q

hh0 ? B0 = q11 ?

h00 + A00 + hA00 + 2 A11 + 2 A1 ,

A A A A A A 1

1

h h

h11 = A11 + 2 A1 ,

A A1

(12)

q q q

q00 + 2 q0 ? A00 + 2 B0 = 0,

A A A1

B B h

B11 ? 2h1 ? A11 + 2 A1 + 2 A1 = 0,

A A1 A

B B B

h0 ?

B00 + 2 A00 + 2 B0 = 0.

A A A 1

The subscripts denote the corresponding derivatives.

Case 2. A = 0, B = 0 (without loosing generality one may choose B = 1):

h11 + 3hh1 + h3 = 0,

h0 = 0,

q11 + hq1 + 3h1 + 2h2 q = 0, (13)

q00 ? qq1 ? hq 2 = 0.

Case 3. A = 1, B = 0:

h1 = 0, h00 + hh0 ? h3 = q11 , q(q0 + hq) = 0,

(14)

q00 + h0 q ? h2 q ? 0.

Thus a problem of study of Q- conditional symmetry of the equation (1) is reduced

to search of the general solution for the equations (12)–(14). Let us emphasize that

418 W.I. Fushchych

coefficient functions j, z of the operator Q unlike coefficient functions ?, ? (4) satisfy

a system of nonlinear equations. This fact makes difficult to describe conditional

symmetry of given equations. Nevertheless it is possible to construct their partial

solutions.

We had found 12 inequivalent operators of conditional symmetry for the equa-

tion (1) [8]. Two of them have the form

Q1 = x2 x1 ?1 + ux2 + 3x2 + b5 x5 + b6 ?u , (15)

0 0 1 0

W = W 2, (16)

Q2 = ?1 + [W (x0 )x1 + f (x0 )]?u , f = W f,

W is the Weierstrass function.

The operator (15) generates the ansatz

U = x1 ?(x0 ) + 3x?2 x1 ? b5 x3 + b6 x?2 . (17)

0

0 0

The ansatz (17) reduces the nonlinear equation (2) to linear differential equation

(ODE)

x2 ? (x0 ) = 6?(x0 ) (18)

0

operator (16) gives rise to the ansatz

1

W (x0 )x2 + f (x0 )x1 + ?(x0 ) (19)

u= 1

2

reducing the equation (1) to linear ODE with the Weierstrass potential

(20)

? (x0 ) = W ?(x0 ).

Note 2. In an analogous way we had constructed families of exact solutions for the

multi-dimensional equation [8]

(21)

u00 = u?u.

Conclusion 3. Ans?tzes generated by operators of conditional invariance often

a

reduce the initial nonlinear equation to a linear one. The reduction by Lie operators,

as a rule, does not change the nonlinear structure of the equation under study.

3. Conditional invariance of the d’Alembert equation. Let us consider the

nonlinear equation

2u = F1 (u), (22)

u = u(x0 , x1 , x2 , x3 ),

where F1 (u) is an arbitrary smooth function. The equation (22) is invariant under the

conformal group (that is the maximal invariance group admitted by (22)) iff F1 = 0

or F1 = ?u3 . Let us impose on the solutions of (22) the Poincar?-invariant eikonal

e

constraint

?u ?u

(23)

= F2 (u),

?xµ ?xµ

where F2 is a smooth function.

Conditional symmetry of equations of nonlinear mathematical physics 419

Theorem 2 [15]. Provided F1 = F2 = 0 the equation (22) with the condition (23) is

invariant under the infinite-dimensional Lie algebra with coefficients of the opera-

tor (4) having the form

? µ (x, u) = c00 (u)xµ + cµ? (u)x? + dµ (u), ?(x, u) = ?(u),

where c00 (u), cµ? (u), dµ (u), ?(u) are arbitrary smooth functions.

Consequently, the additional condition (23) (F2 = 0) picks out from the whole

set of solutions of the linear d’Alembert equation (F1 = 0) the subset having the

unique symmetry properties. Besides, an arbitrary smooth function of a solution of

the system (22), (23) (F1 = F2 = 0) is its solution too.

Theorem 3. The system (22), (23) is invariant under the conformal group C(1, 3)

iff

F1 = 3?(u + C)?1 , (24)

F2 = ?,

where ?, C are constants.

Thus, the additional eikonal constraint (23) extends the class of nonlinear wave

equations admitting conformal group. It means that we can construct wide classes of

exact solutions of the equation (22) using the subgroup structure of the group C(1, 3).

Note 3. The system (22), (23) had been completely integrated in [16].

Let us consider the Lorentz non-invariant wave equation

Lu ? 2u + F (x, u, u), (25)

1

3

2 2 2 2

?0 ?u ?a ?u

F =? (26)

+ .

x0 ?x0 xa ?xa

a=1

The maximal invariance group admitted by the equations (25), (26) is the following

two-parameter group

xµ > xµ = ea xµ , u > u = u + b,

where a, b are group parameters.

An additional condition of the type (6) is chosen in the form

Jµ? = xµ ?? ? x? ?µ , (27)

Jµ? u = 0, µ, ? = 0, 1, 2, 3.

By direct check one can assure that the equations (25), (27) are invariant under the

Lorentz group O(1, 3). It means that the Lorentz-invariant ansatz

? = xµ xµ = x2 ? x2 ? x2 ? x2 (28)

u = ?(?), 0 1 2 3

reduces the nonlinear wave equation (25) to the following ODE

2

d2 ? d? d?

+ ?2 ?2 = ? µ ? µ .

? 2 +2 = 0,

d? d? d?

420 W.I. Fushchych

The solution of the above equation is given by the formulae

?(?) = 2(??2 )?1/2 tan?1 ?(??2 )?1/2 , ?2 < 0,

(?2 )?1/2 + ?

?(?) = ?(?2 )?1/2 ln ?2 > 0,

,

(?2 )?1/2 ? ?

?(?) = C1 ? ?1 + C2 , ?2 = 0,

where C1 , C2 are constants.

Thus, the condition (27) selects from the set of solutions of the Lorentz non-

invariant equation (25) a subset invariant under the six-parameter Lorentz group.

This essential extension of the symmetry makes it possible to construct wide classes

of exact solutions of the nonlinear wave equation (25).

4. Conditional invariance of the nonlinear Schr?dinger equation. Let us consi-

o

der the nonlinear equation of the form

?

S?i (29)

Su + F (|u|)u = 0, + ?1 ?.

?x0

The equation (29) is invariant under the Galilei algebra AG(1, 3) having the basis

elements

Pa = ?a , Jab = xa Pb ? xb Pa ,

P 0 = ?0 , a, b = 1, n,

(30)

1

Ga = x0 Pa + xa R1 ,

2?1

where

? ?

? u? ?

R1 = i u .

?u ?u

In the class of nonlinear equations (29) there are two well-known ones having

wider symmetry algebra than the equation (29) has [17, 18]:

Su + ?2 |u|r u = 0, (31)

Su + ?3 |u|4/n u = 0, (32)

where ?2 , ?3 , r are arbitrary parameters, n is the number or space variables in the

equation (29).

The equation (31) is invariant, under the extended Galilei algebra AG1 (1, n) =

AG(1, n), D having the basis elements (30) and

2

(33)

D = 2x0 P0 + xa Pa + R2 ,

r

where

? ?

+ u? ? .

R2 = u

?u ?u

The equation (32) is invariant under the generalized Galilei algebra AG2 (1, n) =

AG1 (1, n), A having the basis elements (30), (33) and

x2 nx0

R1 ? x2 = x2 + · · · + x2 .

x2 P0

A= + x0 xa Pa + R2 ,

0 1 n

4?1 2

Conditional symmetry of equations of nonlinear mathematical physics 421

Theorem 4 [18]. The Schr?dinger equation (29) is conditionally-invariant under

o

the operator

Q1 = ln(uu??1 )R1 + xa Pa ? cR2 , (34)

c = const,

provided

F (|u|) = ?4 |u|?4/r + ?5 |u|4/r ,

where ?4 , ?5 , r are arbitrary real parameters, and the condition

?1 ?|u|r+4 + ?6 |u|r = 0 (35)

holds.

Thus imposing on solutions of the nonlinear equation (29) an additional constraint

(35) we extend its symmetry.

Theorem 5 [18]. The equation (32) being taken together with the the equation (35)

is invariant under the algebra AG2 (1, n) and the operator Q1 (34).

5. Conditional symmetry of nonlinear heat equations. To describe nonlinear

processes of heat and mass transfer the one-dimensional equations of the form are

used

(36)

u0 + u11 = F (u),

(37)

u0 + uu11 = 0,

where F is a smooth function.

We look for operators of conditional symmetry in the form

(38)

Q = A(x, u)?0 + B(x, u)?1 + C(x, u)?u

with some smooth functions A, B, C.

Theorem 6 [19]. The equation (36) is Q-conditionally-invariant under the operator

(38) if functions A, B, C satisfy differential equations:

Case 1: A = 1

Buu = 0, Cuu = 2(B1u + BBu ),

3Bu F = 2(C1u + CBu ) ? (B0 + B11 + 2BB1 ), (39)

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