ñòð. 108 |

? = 2(x1 ? x0 ). Having substituted this value of parameter ? into (18), we obtain

the solution of equation (11):

1

u(x0 , x1 ) = [2(x1 ? x0 )]? 2 .

Linear equation (12) has a remarkable property: any operator of invariance algebra

of this equation maps it’s solution into another solution, i.e. the following generating

formula takes place

2 1

z(t, x) = Q z(t, x), (20)

12

where z, z are solutions of equation (12), Q is an operator that belongs to algebra (13).

Let us use formula (20) and the relation between the solutions of equations (11)

and (12) to construct generating solutions formula for equation (11). If we, for examp-

le, choose operator ?x instead of Q in (20), then we get one of the formulas which

describe the generating solutions of nonlinear equation (11)

?1

1

? u(x0 , ? )

2 1

u(x0 , x1 ) = ?[u(x0 , ? )]3 (21)

,

??

1 2

where u(x0 , x1 ) and u(x0 , x1 ) are solutions of equations (11) while function ? =

? (x0 , x1 ) is determined by the equation

1

u(x0 , ? ) = x?1 . (22)

1

460 W.I. Fushchych, N.I. Serov, V.A. Tychinin, T.K. Amerov

So equation (11) solutions of the form

?1

1 1 2

1

x0 x?1 ? ln x0 x1

2 2

u(x0 , x1 ) = 1

are multiplied into parametrical solutions:

?1

1

3

2

ln ? ? ln ? = x2 x2 ? 2

2

u(x0 , x1 ) = x0 ? , 01

2

by means of formulas (21), (22).

In the case Q = ?t , it follows from (18)–(20) that

1

[u(x0 , ? )]5

2

u(x0 , x1 ) = (23)

,

1

1 1

? [u(x0 , ? )]2 u0 (x0 , ? )

2[u? (x0 , ? )]2

where ? = ? (x0 , x1 ) is defined by the condition

1

1

u? (x0 , ? ) + x1 [u(x0 , ? )]3 = 0. (24)

Note. If we choose anyone of operators (13) in the capacity of Q in formula (20)

then the generating solutions formula for equation (11) is constructed analogously.

The synthesis of Galilei local transformations:

z = z exp{?ax ? a2 t} (25)

t = t, x = x + 2at,

and non-local relation (18), (19) leads to the new generating solutions formula of

equation (11)

1

u(x0 , ? )

2

u(x0 , x1 ) = (26)

,

1

? ?1

?ax1 u(x0 , ? ) + x1

where a is arbitrary real parameter and ? = ? (x0 , x1 ) is functional parameter which

is a solution of the following equations

1 1 1

?2

[u(x0 , ? )]2 ?0 = ?1 ?11 + 2a u2 (x0 , ? ). (27)

?1 = ,

1

? ?1

?ax1 u(x0 , ? ) + x1

It should be noted that x0 is a parameter of first equation (27) and that is why this

equation can be considered as first-order ODE with separable variables. Because of

this the second of the equation (27) is only the correlating condition of obtained ?

with respect for x0 . The following example show the effectivity of formulas (26)–

(27). The constant solution u1 (x0 , x1 ) = 1 being generated by these formulas takes

the form of following implicit solution:

1 a

= ln x1 + a2 x0 .

ln +

?1 ? a ?1 ? a

(x1 u) (x1 u)

4. Nonlinear superposition principle. Solutions of equation (12) have linear

superposition principle. Using formulas (18), (19) we get nonlinear superposition

1 2

principle for the solutions of equation (11). Let u(x0 , x1 ), u(x0 , x1 ) are the pair of

On non-local symmetries of nonlinear heat equation 461

solutions of equation (11) then third solution of this equation can be obtained by the

formula

1 1 1

(28)

= + ,

3 1 1 2 2

u(x0 , x1 ) u(x0 , ? ) u(x0 , ? )

k k

where ? = ? (x0 , x1 ) are functional parameters which can be obtained from the condi-

tions

22

1 1 1 2

u(x0 , ? )d ? = u(x0 , ? ) d ?,

k (29)

?11 k

1 2 k k

= k u?2 (x0 , ? ),

? + ? = x1 , ?0 k = 1, 2.

?2

1

1

The substitution u(x0 , x1 ) = leads equation (11) and formulas (28), (29)

U (x0 ,x1 )

to the following form

U0 = U 2 U11 , (30)

3 1 2

1 2

U (x0 , x1 ) = U (x0 , ? ) + U (x0 , ? ), (31)

1 2

d? d?

= ,

1 2

1 2

U (x0 , ? ) U (x0 , ? )

(32)

k

?11 k

1 2 k k

2

? + ? = x1 , ?0 = U (x0 , ? ), k = 1, 2.

k

?21

Example. Having two simplest stationary solutions

1 2

U (x0 , x1 ) = x1 , U (x0 , x1 ) = 2x1

of equation (30) and using formulas (31)–(32) we can obtain nonstationary solution

of that equation

3

?2x0

U (x0 , x1 ) = ±e 1 ? 2x1 e2x0 ± 1 ? 2x1 e2x0 .

2

5. Non-Lie ans?tzes for equation u0 = ?1 (u? 3 u1 ). It follows from (10), that

a

transformations (6)–(8) map equation

2

u0 = ?1 (u? 3 u1 ) (33)

into the equation

4

zt = ?x (z ? 3 zx ). (34)

Symmetry of equation (34) in class of Lie transformations is wider then that of

equation (33) (see Theorem 1). By analogy with section 1 let us use Lie symmetry of

equation (34) for a construction of non-local ans?tzes which reduce equation (33).

a

462 W.I. Fushchych, N.I. Serov, V.A. Tychinin, T.K. Amerov

Let us concisely adduce the results of our analysis. Ans?tzes for function z:

a

z = x?3 ?(?),

1) ? = t,

1

z = x?3 ?(?),

2) ? = at + ,

x

1

3

z = t 4 x?3 ?(?),

3) ? = a ln t + ,

x

(35)

3

z = (x2 + 1)? 2 ?(?),

4) ? = t + ? arctg x,

3

z = (x2 ? 1)? 2 ?(?),

5) ? = t + ? arctg x,

3

?3

z = t (x2 + 1)

6) ?(?), ? = ln t + ? arctg x,

4 2

3

?3

z = t (x2 ? 1)

7) ?(?), ? = ln t + ? arctg x.

4 2

Ans?tzes for function u:

a

2

u = [?1 (x0 )x2 + ?2 (x0 )]? 3 ;

1) 1

3

[x1 + ?1 (x0 )][?2 (x0 )] 4 = ?? ?3 (?) + ?3 (?),

2) ? ?

?1

? = ?2 (x0 ) + ?, ? = u; (36)

?

3 3

[x1 + ?1 (x0 )][?1 (x0 )] 4 = [?3 (? )] 2 ?4 (?)d?,

3) ? ?

? = ?2 (x0 ) + ?3 (? ), ?1 = u.

Reduced equations which where obtained by the substitution of ans?tzes (36) into the

a

equation (33):

?1 + 4(?1 )2 = 0,

1) ?

(37)

?2 ? 2?1 ?2 = 0;

?

1

?1 = ?1 (?2 ) 4 , ?2 = ?2 (?2 )2 ,

2) ? ? ? ?

(38)

3

1

3(?3 )? 3 + ?3 ?3 + ?2 ?3 ? ?1 = 0;

? ?

4

?1 = 0, ?2 = ?2 (?2 )2 ,

3) ? ? ?

... 3 3

2 ? ? ? 3(?3 )2 = 2?1 (?3 )4 ,

? ? ? (39)

4 3

4 7 1

(?4 )? 3 ?4 ? (?4 )? 3 (?4 )2 + 3?1 (?4 )? 3 + ?2 ?4 ? ?4 = 0,

? ? ?

3 4

where ?1 , ?2 , ?3 are arbitrary constants. In particular, having integrated the system

of equations (38) with ?2 = 0, we obtain the parametrical solutions of equation (33)

of the form:

?c1 5 c3 x1 + c2 x0

1

41

u= ,

3

? (? ? 4c3 x0 )

(40)

4

ñòð. 108 |