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 +

    +=

    =+

    ,)(

    ),(

    1

    2

    yxguydxdy

    xfudxdu

    (21). .

  • 18

    , -, , -,

    ),( Cxz . . (21) -

    ,)( 3+= Ayyxfy

    const=g ( - 3- f g (21) , ). -

    ,)(22/1

    21

    += w

    dxxgCwy

    w , wxfw )(= . , , , . (21) (!) (!)

    ,42

    143

    21

    22

    +

    +

    =ggk

    gg

    gg

    ggf

    k , - (. . const=g ).

    , 100 , -. - 3- ,

    ,11

    )( nn

    n Ayy +

    =

    , , 3- , ),2( RSL n. , , . , 3-

    ++++=++ 221 )()1)(1()()2()()(_ yhynnyyxhnyxgyxfy nn

    .3)(]3)12[( 23312112 +++++ nnnnn yhyhhyfhhyyynhhn

    4. - . - - , - -, - . - , -- , [3, 4, 5, 13].

  • 19

    5. .

    1.

    ++++

    +++ 232 )2(23 yfgfyfy

    yhgfyy 02)2(

    22 =+++++

    yhghhyqfhgg

    ydxfyyhyX

    = exp

    =+++=+

    .0)(])([)(),(

    2

    2

    xhyzxgyxfyxqzz

    2.

    +++++++ yhgygffyyy ]2)(22[)( 22

    0)2()22(2 222342 =+++++++++ qhhyghgyfhgffgyyf

    [ ] [ ] == dxxgxGydxxfxGX y )(exp)(,)(2exp)(

    =+++=+

    .0)()()(),(

    2

    2

    zxhyxgyxfyxqzz

    3.

    0)()( 22 =+ yxFyyyyy (22)

    ( ) ., 221 yy dxyyXyX == (23)

    .0][2 = yyDx

    (23) X1 ( (22) ), (X2 ) , .

    (22)

    ==+

    .0),(2

    yzyxFzz

    , z - (23). X2 -

  • 20

    , . 2,

    ( ) ( )[ ] .012112 =

    ++

    yIdxyyy

    yIdxyy (24)

    dxy 2

    , - . (24)

    =

    =

    +

    .0

    ,0

    11

    11

    yIy

    yIy

    yIy

    , 0/1 = yI . . , -

    , ( , - ). , - .

    - . -, - , ( ) - . - , - . -, , [ , -

    yx F+ , 1- ),( yxFy = ),( yx= ]. - , - , ( ). - . , - , , , ,

    = .),,( dxyxI ,

    I .

    = (xy)F(I),

    , , - ,F,FF, -, , F . -

  • 21

    , ,F,FF, . :

    ( )[ ] ydxX += 21 exp

    ( ) ,21 ydxX +=

    - - , - .

    , [7] , . -, 2- - , . - , , ),,( yyxF . ),,(, 10 yyxzIxI == . - , 2- , , .0),,( = zzx - 1-

    0][ =

    +

    yzD

    yz

    x

    ,exp

    =

    dxzz

    y

    y

    .. . , ( -

    ) , . [14].

    1. . .

    // . ., 1994. . 190199. 2. . .

    // II . ., 1998. .137151.

    3. . ., . . // . II . ., 1999. . 177178.

  • 22

    4. . ., . . - // . II . , 1999. . 51.

    5. . ., . . - // IX - (2000). , 2000. . 222226.

    6. . ., . . . ., 2001.

    7. . . . ., 1983. 8. . . . . . .,

    1989. 8. 9. . . . . . .,

    1991. 7. 10. . . . ., 1978. 11. . . ., 1989. 12. Ibragimov N. H. Introduction to modern group analysis. Ufa, 2000. 13. Linchuk L. V. Symmetry analysis of functional-differential equations // Math. Research.

    Vol. 6. Theory and practice of differential equations. St. Petersburg, 2000. . 111117. 14. Polyanin A. D., Zaitsev V. F. Handbook of nonlinear partial differential equations.

    Chapman & Hall / CRC, 2004. 15. Zaitsev V. F. Universal description of symmetries on a basis of the formal operators //

    Math. Research. Vol.7. Theory and practice of differential equations. St.Petersburg, 2000. . 3945.

    V. Zaitsev

    SYMMETRIES OF DIFFERENTIAL EQUATIONS. FORMAL OPERATORS

    The paper is devoted to some aspects of the theory of formal operators. We in-

    troduce definitions, properties of formal operators and prove factorizations theo-rems. These theorems generalize well known methods of reduction and integration of ordinary differential equations admitting local operators. Therefore, we can construct universal description of all symmetric equations.

    517 . .

    ,

    . - , , - . - . - .

    1.

    , 1970 IX - , , , , , - . (~30 ), - 2/3 (~20 ).

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