# Симметрии дифференциальных уравнений. Формальные операторы

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• 7

. .

.

-

. , - , -, . , , - .

XIX - , , XIX , , . XX , , , , [7, 10, 11] ( ) -. , , -, . , , , , - , (., , [8, 9, 12]).

- , - , . , , . -, , . , , - , I0 I1.

[15], - . , a priori -

• 8

, -- [13]. , - , -. .

1. .

- n- ,

)( 1= n(n) ,y,yx,y,Fy , (1)

n- () [F]. X : C C ()

,),,,(y

yyxX= (2)

(, ) ( ) - . ,

,),( dxyx dxdxyx ),(

)(ky , ,

=

+

+= .

)!1()1(

0

)(1

k

kk

k ykxydx

, - . , . .

( ) ),,(),( yxdxyxDx =

xD

.y

yx

D0k

(k)1)(k

x

=

+

+

=

, - , - . , )( xy , , )( +

• 9

1. (2) (1),

[ ] ,0),,,,( ][)1()( = Fnnn yyyxFyX (3)

.][ )(

= kkxn y

DX (4)

(4) n- (2), ][F

, (3) [F] (: (1), (1)).

2. 0)(

(k)(k)

yH,,y,yx,y,H -

k- k- - (2), H

.0][ =HXk

(5)

. (5) - , (1).

2. .

1 . )( )1()( nn ,y,yx,y,Fy - n- , n () (2).

. 2, n-

.0][0

)( =

=

n

kk

kx y

HD (6)

(6) H

H = )( 1)()( nn ,y,yx,y,Fy

(6) - . n- ,

,2211 nnCCC +++= (7)

nkk ,1, = n - , , -, y , - (, , - - ).

nkk ,1, = (6) , . . , , , (3). .

• 10

. n- (1), , n - (2).

2 [1]. 1-

)(x,yFy = (8)

( ) ,

.exp ydxyFX

= (9)

. (3) (8)

[ ] ,0),(][1=

FyxFyX

. .

,0][ =

yFDx

.exp

= dxyF

. .

, 1- -

,x,yFx,yx,yX yx += )()()(

)(x,y .

xDx,yXXX )( =

(8) ()

.0]),()[,( yyyyxFyxX =

. 2-

maxxm

myyy +++

= 2)3()1(2 (10)

, ,

.3

2)1(23

1exp 12

11y

m ym

xymaxdxym

mX

++

+

= + (11)

• 11

[6], - [12]

,auu m=

. , - , .

(9) , - .

3.

[ ] yk dxyyyxX = ),,,,(exp )( (12)

() - . , - (12) - .

3 . ),,,( )(kyyxzz = - k- (2). ][zDz x= - (k + 1)- .

. 2, z

.0][][ )( =

++

+

kkxx y

zDyzD

yz

, :

.0][][ )( =

++

+

kkxxx y

zDyzD

yzD

, , Dx dx, Dx dy , i > 0 [ ] .,D 1)(iy(i)yx =

+

+

+

+

yzD

yzD

yzD

yzD

yz

xxxx ][][][][2

.0][][][ )(1

)1()( =

+

+ + k

kxk

kxk

kx y

zDy

zDyzD

z, , , . .

.0][][][ )(1

)( =

+

++

+

+ kkxk

kxx y

zDyzD

yzD

yz

)y,y,z(x,z (k)= ,

,][ )()1(

kk

x yzy

yzy

yzy

xzzDz

++

++

== +

• 12

,y

zyz

1)(k(k) +

=

)1( +ky ][zDx . z - , (k + 1)- -.

. 3 , - , - 1- , -.

3. .

4 [1, 2]. n- (1), , -

=

=

),,,(),,,,,( )2()1(

yyxHzzzzxGz nn

(13)

(1)

.exp yy

y dxHH

X

=

(14)

. 1. (1) (14). ,

n-

.0][][ )( =

++

+

nnn

xn

xn

yzD

yzD

yz

,exp

=

dxHH

y

y = )(nyt ,,( yxF

), )1( nyy .

++

+

)1(1 ][][ n

nnx

nx

n

yzD

yzD

yz

.0][][][ )1(1 =

++

+

n

xnnxx

n Dy

FDyFD

yF

tz

[F] - ,

.0][][ )1(1 =

++

+

nnn

xn

xn

yzD

yzD

yz

2, , . . ),,,,,( 121 = nn zzztxGz - [F] t = 0,

• 13

),,,,( 121 = nn zzzxGz

= ),,,( 121 nzzzxG ).,,,,0,( 121 nzzzxG 3 )1( = ii zz ,

,,2 ni = ),,(1 yyxHzz == -, .

2. (1) (13). -, ),,( yyxHz = - (14), , (1).

. 4 . [11]. , - .

. (2), (1), n - (1) , n k + 1 , ),,,,( )(kyyyxHz = - k- , z , ).,,,,( )1( = knkkk zzzx z

. z (2) - (1)

.0][][ )( =

++

+

nnxx y

zDyzD

yz

4 (n)y ),,,,( )1()( = nn yyyxFyt .

.0][][ )1(1 =

++

+

nnxx y

zDyzD

yz (15)

),,,,( )(kk yyyxHz = (2) (1). , 3 ,)(sksk zz =+

1,1 = kns , (2) (15). )1()1()( ,,, + nkk yyy ,zp ri =

1, = nkr , (15)

.0][][ )1(1 =

++

+

kkxx y

zDyzD

yz (16)

k- , (16) (-) x, (2), n- (1) ).,,,,( )1( = knkkk zzzx z

. (16) - , k1.

5 . n - - (1), , -

=

=

,0/),,,,,(),,,,,(

)()(

)1()(

kk

knkn

yHyyyxHzzzzxGz

(17)

• 14

(1) (2), z ),,,,( )(kyyyxH = (2). (1) (17), (2), ,,,,( yyxH

), )(ky k - (1).

. 1. 4. 2. . ,

(1) z = H(x,y,y', , y(k)).. k- o

.0][][ )( =

++

+

kkxx y

zDyzD

yz

. k . (1) . )1(,, knzz , z. , (1) (2).

. , ),,,,( )(kyyyxH (2) , (1) - .

4. .

, 4 5 - - , . - , , , . , - , (- ). , , -. ( ), .

1. .

yyxyyxX = )],(),([

(12),

+=

yyyy yxyx

2)()( .

• 15

(11).

xymmymAxx

mm m

3)1(4)1(2

)3()1(4 2122 +

++++

++

+

, (10) 2x(yy' y),

xymmymyxy

3)1(4)1(2 2

++

++

(11) ( )

,3

1exp2 1 Ddxymmx

+

, (10):

.3

)1(2)1(23

1exp~ 1

+

++

+

= y xmmymxdxy

mmX

I0 x,

)(,3

221

zwxwm

zwym

=+

+=+

(10)

.03

22

1 23

2 =+

++

+ mwm

zwdzdwAzm

, (10) - (11) , .

, ,

.)(( 2

y

Fyy yy

+=

, (12) , - dx . , - (11) . , (10), - (11), , - .

2. . , (2), ( ), (17)

• 16

==

).,,,,(,0

)1(nyyyxHzz

(18)

(- , . . ). .

1)(yAyy mn = (19)

121

)(2

1 ++

= ml ym

Ayl

P (20)

(12)

( ) .)(exp 1 ylm dxyyX =

, (19),

,exp yy ydxyyX =

=

. . , , (20), - x , - (19).

3. . , - dx , (12) , , . - , , . . , , , , 100- -. .

6 [2].

31 )]([)()( += yxgyxgyxfy (21)

f(x), g(x) ( )

),( yxFy = ,

[ ] .),(),(exp yyxdxyxX =

+==+

.)(),(),(

1

2

yxgCxzyxfzz

. X

[ ]{ ++++= yyxy ydxyxX )(),(exp2

[ ] }.)()22(2 22 yyyyyyxyxxxx yyy +++++++++

• 17

),( yxFy = y ),( yxF , -

.0)()22(2 22 =++++++++ yyyyyyxyxxxx FFyy

y , - , - :

=++++=++

=

.02,022

,0

2yyxxxx

yyxy

yy

FF

)()( xbyxa += ,

,)(

)(2 22

bayxcbyayaa

+++

=

)(),(),( xcxbxa . - 1- ),( yxF :

),2(1 2+++

=

xxxx

y

dydF

(

++= 32 )2(])(2[)()(),(

aabababaaaxfbayyxF

,)(4)(

)(2)2(2])(3[

33

22

3

22

bayaacba

bayaacacabbaabaaa

+

+

+

)(xf . X:

.)(2)( 22

2

2 bayaacba

bayababa

bayyu

+

++

+

=

, du/dx , y , - ),( yxF :

+

++=+

=

++

.2

)()()(

,

2

2

22

2

aacbabay

ababaubay

dxdybay

fuaaau

dxdu

ybay +