CONTACT INTEGRABLE EXTENSIONS OF SYMMETRY PSEUDO-GROUP AND COVERINGS FOR THE R-th DOUBLE MODIFIED DISPERSIONLESS KADOMTSEV-PETVIASHVILI EQUATION Текст научной статьи по специальности «Математика»

CONTACT INTEGRABLE EXTENSIONS OF SYMMETRY PSEUDO-GROUP AND COVERINGS FOR THE R-th DOUBLE MODIFIED DISPERSIONLESS KADOMTSEV-PETVIASHVILI EQUATION1

O.I. MOROZOV, M.V. PAVLOV

We find contact integrable extensions and coverings for the r-th double modified dispersionless Kadomtsev-Petviashvili law equation. One of the coverings provides a Backlund auto-transformation and a recursion operator for the equation under the study.

Key words: Lie pseudo-group, differential covering, contact integrable extension, Backlund transformation.

We consider the r-th double modified dispersionless Kadomtsev-Petviashvili law equation

_ . /(к + 1)Цу Ut | K | (K + l)2 2(К+1Д _ /Uy _ л uyy ^-tx ' 1 ^ ¡кiixUy -г 2K + 2 ^x / ^xx ^ \w ^x ) (1)

\ X X / X

with k £ {—2, —3/2, —1}. This equation arises from the differential covering [5, 6, 7],

/(K + 2)2 2(k+1) . s K+1 ,K + 1 2 ^

t = 1 2k + 3 UX J+ (K + 2)wxu%+1-Wyjux <Uy = ~(u%+1 + wx)ux

over the r-th modified dispersionless Kadomtsev-Petviashvili law equation [1], written in the form

™yy = Wtx + Q (K + 1) + wy) ™xx + KWX Wxy, (3)

with r = k(k + 1)_1, see [2, 4, 14, 12]. Namely, excluding w from (2) yields Eq. (1).

We apply the method of contact integrable extensions [11], to find differential coverings of Eq. (1). The method starts from computing Maurer-Cartan law forms and structure equations for the symmetry pseudo-group by the methods described in [13, 3, 9, 10]. The structure equations are given in Appendix. These equations are not involutive. Their involutive completion includes equations for differentials of forms ... , The completion may be obtained by the normal prolongation [16, 13, 15], of system (23). Those equations are too big to be written in full here. The Maurer-Cartan law forms 90, ... , 623, f1, are

6Q — WjjUj^ Çdu Uç dt u% dx Wy dy^,

61 = ux2K~2 (dut — utt dt — utx dx — uty dy) — (k + 2) (uy uxK~2 — 1) 93 + ((k + 1) (k + 2) (uyuxK~2 - (2k + 3)-1) - utux2K~3)) 62 + (utux2K~3 + (k + 1)2(k + 2) (uyuxK~2 - (2k + 5)(2k + 3)_1) e0

02 — Wj^ ^dWj ^tx ^^ Wjj dx w^y dy^f

93 = uxK~2 {dUy — utx dt — uxy dx — E dy) — (uy uxK~2 — k — l) 02 —(uy uxK~3 + (k + l)2) 90,

1 The work of the first author was partially supported by the joint grant 09-01-92438-KE_a of RFBR (Russia) and

Consortium E.I.N.S.T.E.IN (Italy).

The work of the second author was partially supported by the grant of Presidium of RAS "Fundamental Problems of Nonlinear Dynamics" and by RFBR grant 14-01-00389.

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— иххих4к 4 (dutt — uttt dt — uttx dx — utty dy) — 2 (к + 2)(uyuxK 2 — 1) 013 -(2 utu~2K_3 - (к + 2)((к + 2) u2u~2K_4 - (2k + 3) iiyu~K_2

+(2к2 + 9к + 8)(2к + 3)"1)) 012 + Л110 0O + ¿m öi + ¿112 62 + ¿из 63 -(u?u*4k_6 + (к + 1)2(k + 2)2 (uyuxK~2 - (2к + 3)-1)2 +2 (к + 1)(к + 2)(2к + 3)-1 u"2K"3) 022 - 2 (к + 2) ((UyUxK~2 - l)utux2K~3 -(к + 1)(к + 2)(u2u-2K_4 + 2 (к + 2)(2к + З)_3 u~K_2 - 2к - 3)) 023, Ö12 = иххих2к 2 Çdutx — uttx dt — utxx dx — utxy dy) — (uyuxK 2 — 1) 02з +((к + 1)(k + 2)(uyu"K_2 - (2к + 3)_1) - utux2K~3) (022 + 03) - 0i

_2 -2K-1 -1 —2k—2 1 -1 -2K-3 -K-l

C^txx^xx^x 2 UtxUxxUx — (к + 2) (UXyUxx ((к + 2)UyUx + К ti-^ )

-(к + 2) (u2u"2k_4 - (k + 1) uyu~K_2) + к (к + 1))) 0O,

013 = иххихък~ъ (duty — utty dt — utxy dx — ID)t(^) dy) — (2к + Ъ)(иуихк~2 — 1) 012

-(UyUxK-2 + (K + l)2) 0! - ((UyUx3K~5 -(K + l) Ux2k~3) Ut-(K+ 1) (UyUx2K~4

-(к + 2)(2к + 3)_1((2к2 + 5k + 4)uyuxK~2 -к- 1)) 022 + ^130 0O + ^132 в2 +Л133 03 - (utu?K~3 - (к + 2) (u2u-2K"4 - (2k + 3) UyUxK~2 +2(k + 1)(K + 2)(2K + 3)"1)) 023,

022 — Wj^j^ '^^txx ^^ ^xxx ^Îx ^xxy rfy^ 2 62 ^xxx^XJC

^23 = мл:К iAuxy ~ utxy dt — Uxxy dx — ШХ(Е) dy) — (UyUxK 2 — К — 1) 022 _ 03

1 _ _ _ _ _

+ 2 (K "xy^x^x* 1 - (к + 4) UyUxK 2 - К (к + 1)) 02

(Mxxy^xx ^x UXyUxx Ux + UyUx + (к + 1) ) 0Q, ^ — ^XX^X tÎt^

= ^xx^x1 dx + (utuxK~3 + (k + 2) (UyUx2K~4 — UyUx2 + 2 (к + 1)2(2k + 3)_1)) f1 +(uyu^-2 - к - 1)

where E is the right-hand side of Eq. (1), ID)t, ID)X are restrictions of the total derivatives on Eq. (1), and ^110, ^112, ^ii3, ^130, ^132, ^133 are functions of derivatives of u of the first and the second orders. These functions are too long to write them in full. The coefficients of the structure equations depend on the invariants

U1 = (K + 2) (uyU"K_2 — UxyUxxUx*'1 + K + l),

= uxxu% dy + (к + 2) (UyU:

—K—2

- 1)

(4)

U2 = Щххиххих2к 1 _ (K + 2)

- (2uyuxK~2 -(k + 1)(k + 2)) U1 + 2 (к + 1)(к + 2) uyu~k_2

98

—2-.—2K—2 in*. 1(—2k—3

X^ X I fai ^^ X

(к + 2)(2к + 3)_1ux) + 2(K + 1)(к + 2)(2к2 + к - 2)(2к + 3)_1,

i/^ — ^XJC Wj^ ^XXX^XJC ^^ ^WyH ^/C H ) 2 (^/C H i/^

-(k + 1)(K2+K + 2),

UA = (k + 1) (к - (к + 2) (i/3 - (к + 1) {uxxxux2ux + к2 + Sk + 2))),

1 ___ ___ __

= 2 + 3{uxutxy - uty) + utxux£ux2K 2 ((к + 3)^ - (к + 2) (uyuxK 2

+(к + 1)(к + 3))) + ((2к + 3) иуихк~2 - 1) i/i2 - (к + 2)((к + 3) иуихк~2 +2к + 1)U2- ((к + l)-1utu-2K-3(K(K + l)-1uyu~K-2 + 2к2 + 5к + 4) + (к + 1) (к и2их2к~4 - (2к + 3)_1((2к4 + 9к3 + 7к2 - 13к - 18) иуихк~2

¿у

-2 (2к4 + 45к3 + 42к2 + 53к + 25)))) + (щих2к~3((к + l)-1^-*"2 - 1)

-(к + 2)(u2u-2k~4 - (2к + 3)_1((2к2 + 5к + 4)иуихк~2 -к- 1))) U3 +((к + 2)-2 щих2к~3((к + l)_1uyu-K-2 + 1) + иуих2к~4 -2(2к2 + 7к + 1)(2к + 3) иуихк~2 + (2к2 + 8к + 7)(2к + З)"1) U4 +utu"2K_3((K + 6) uyu"K_2 + (к + 1)(4к2 + 9к + 6))

+ (к + 1)2(к + 2)((3к + 2) иуих2к~4 - (2к + 3)_1(4(к2 + Зк + 3)иуихк~2 + 8к3 +36к2 + 57К + 30))).

We find contact integrable extensions [11], of the form

/3 7 3

S=1 j=l

3/3

+

j=i \k=o

^ Fjk вк + G¡ a\ Л (5)

where X* denotes summation over all i,j G N such that 1 <i <j<3 and (i,j) ^ (3,3). We consider two types of such extensions. The first type consists of extensions whose coefficients in right-hard side of (5) depend on the invariants U..., i/5. The coefficients of extensions of the second type depend on U1, ..., Us and on one additional function W with the differential of the form

3 7 3

dw=YJ^i 9i+YJ*iij9ij+YJjs (6)

¿=0 5—1 7=1

We require Eqs. (23) and (5) or Eqs. (23), (5), and (6) to be compatible. This condition gives two contact integrable extensions of the first type

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dMi = ((к + 2)2 («! - (к + 1) 0O) -в1-(к + 2) 03) A^1 + Л + «K + 2) («! - (к+l) 0O) - 03) Л^3 + («! + 02 + 022 + ((К + 1)(к Ui -(к + 2) У3) -У4 - (к + 1)(к + 2)(к2 + 6к + 4))(к + 1)_2(к + 2)"2 0О +((к + 1)((к + l)U2 + (k + 2) í/3 - к(к2 + 3k + 3)U1- к(к + 1)(к + 2)) +У4) (к + I)"2 ^ + ((к + 1)(к2 U1 + (к + 2) í/3) - к U4

-к(к + 2)(3к + 2)(к + 1)2)(к + 1)"2(к + 2)-1 ^3) Л (7)

= ((W + K + 2)2 a3 - (W + K + 2) (023 + (K + 1)(K + 2)0O) - 0i) A^1 + a3 A <f2 +((W + K + 2)a3 - (K + 1)(K + 2)0O - 03) A + (a3 + 02 + 022 + ((* + 1)(k U1 -(K + 2) y3) -U4-(K + 1)(K + 2)(K2 + 6K + 4))(K + 1)"2(K + 2)"2 0O -(((K + 1)(K u1 - (K + 2) t/3) -y4 - (K + 1)(K + 2)(K2 + 6k + 4)) W2 +(k + 2)((K + 1) ((K - 1)^ - 2 (K + 2) + (K + 1)(K2 + 5K + 3)) - 2 U4)) W +(K + 2)2((K + 1)(K(K2 + 3K + 3) U1 - (K + 1) U2-(k + 2) (i/ - K2(K + 1))) -y4)(K + 1)_2(k + 2)-2 Ç1 - ((K + 1)(K ^ - (K + 2) (t/3 + (K + 1)(K2 + 6K + 4))) -y4)(K + 1)"2(K + 2)"2 <f3) A Û>3, (9)

dW = -(k + 1)W (a3 + 0O + 02) + + (W + K + 2)(Z + (K + 1) W) +(W + k + 2)((W + k + 2)Z + (k + 1)W(W -(k + 2)_1 U1 + 3K + 4)) ^ +(Z - (K U1 - (K + 2) (t/3 + (K + 1)2(K + 6))

-(K +l)-1y4)(K + 2)-1^)w3 (10)

with a parameter Z.

The inverse third fundamental Lie theorem in Cartan's form [16, §26; 15, p. 394], ensures existence of forms m2, satisfying Eqs. (7), (8), and (9). Since the forms (4) are known explicitly, it is not hard to find the forms . We have the following solutions to Eqs. (7), (8), and (9), respectively:

and

dM2 = ((к + 1)2(к + 2)2 61 - 62 + (k + 1)(k + 2) 03) A^1 + a2A^2

+((к + 1)(к + 2) 02 - 03) A + (a2 + 02 + 022 + ((* + 1) (к U1-(k + 2) t/3)

-U4 -Çk + 1)(k + 2)(к2 + 6к + 4))(к + 1)~2(к + 2)"2 0O

+{U2 - (к + 1)(к + 2) UJ Ï1 + (k(k + 1)U1- U4

-(к + 1)2(к + 2)(3к + 2))(к + 1)_1(к + 2)-2 ^3) А ш2,

(8)

or one contact integrable extension of the second type

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шл =

U-хЧх

Щ (

dq — I--I- (к + 2) I uyu£

\ux \

~ Gr+u*+1)4x dy)'

+

K+l

2K + 3

u2K+2fj

(11)

=

^(dr " ■- '+ '1)(K + 2) M " 2^3 U'"+2))

б^Х

-(^-(к + ЦиГ1)^),

(12)

and

=

ds —

,2k+3

'(k + 2)2 v 2k + 3

(к + 1)(к + 2)

_(lc + 2)(| + ui+1)

к+1 1 ~к+2

+ |s.

б^Х

(13)

with W = s^1«-*-1.

The forms (11), (12), (13) are equal to zero if and only if the following overdetermined systems of pdes are satisfied:

qt = ^- + (K + 2) (uyu% + Чу ~ ( ^ ) 4x>

K + l

2k + 3

(14)

= -(* + !)(* +2) (uyu«-^u2r2) = №-(K + 1) rx,

\ Ltv '

(15)

(к + 2У

4 2к+ 3

,2k+3

(k + 2)(^ + uKx+A s£+2 /

(щ (к + 1)(k + 2) _ _

+ ( — + (K + 2)+ -—-^Ц-—-ulK+2 I s, \ur ^ 2k + 3

0

(16)

Sy s^ + i + Ux I

These systems are compatible whenever u is a solution to Eq. (1), so these systems define differential coverings over (1).

Научный Вестник МГТУ ГА_Том 19, № 02, 2016

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Expressing ut and uy from (14) and cross-differentiating yields

Чуу = 4tx + I (K + 1) ~2 - —) Чхх - к — qXy. (17)

\ Чх Чх/ Чх

Previously Eq. (17) and the Backlund transformation (14) were found in [12] by means of another method.

From Eqs. (15) we have

ut = |-+ (K + + + и*К+2 11 ^

= (^ + (|с + 1)иГ1)

Uy

v "x

(18)

The compatibility condition for this system is (ut)y - (uy)t = ~(k + 1)(k + 2) u£+2rx_2 (Gtx-k(k + 2) u%+1 (ry rxx - rx rxy)) = 0, (19) where

( ry rt\ ry

G = ryy-rtx-[(K + l)-I--\rxx + K- rxy.

\ 'x'x/ 'X

When k = 0, system (18) is compatible whenever G = 0, that is, whenever r is a solution to Eq. (17). When k + 0, Eq. (19) entails = H with

H = -k~1(k + 2)_2 G rx (Ty rxx - rx rxy)~1.

Substituting this into (18) gives a system of pdes with the compatibility condition

k (2k + 3) r2 Ht-k(k + 2) rx (2(k + 2)(2k + 1) rxH + (2k + 3) ry) Hy

+k ((k + 1)(k + 2)2(2k + 1) r2 H2 + 2(k + 2)(2k + 1) rxryH

-(2k + 3)(rtrx -(k + 2) ry2)) Hx-(k + 1) ((2k2 + 5k + 1) rxG

+k(2k + 3)(rxrtx - rtrxx)) H — (2k + 3)ryG = 0. (20)

Thus Eqs. (15) define a Backlund transformation from Eq. (1) to the third order equation (20) for r.

Finally, excluding u from (16) shows that s is a solution to the same equation (1). So, (16) defines a Backlund auto-transformation for Eq. (1).

The Backlund auto-transformation (16) allows one to find a recursion operator for symmetries of Eq. (1), see [8]. Charactersitics of (local contact) symmetries are solutions to the linearization

- - /(K + 1)Uv ut (K +1)2 9iwn\ - n _

ID)2 (<p) = DtDx(<p) + I-+ K u*uy + 2k + 3 Ux I °*(<P) ~ IT

V X X / X

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-к + Ш)жШ>у(^) + (2(k + 1) + к u^j ID)y(^)

(fut и2 , 2(k + 1)3 , Л

+ (Ur2 (K +1} ¡f + KU* u> + )

+к ((к + - ^f) О Ш>х(<р) (21)

of Eq. (1). Then the linearization

Dt(tfO = (k + 2) + Dy(<p) + ((* + 2)2S2(k+1) - (K + 2)2 № + u«+1) s%+1

\ j \ /

ut l K + 1 2fK-+n\\ - sx - ut

+ ^-+(k + 2) uy + j) D^WO + ^ HM<P) - Sx

X J X X

+(* + 2) ^ - (k + 1) M*) s*+1 + (ku«-1^ + sx Dx(<p),

= (—-(* + 2) s£+1 + + — Dy(<p) - Pj - (* + 1) "i) Dx0p) (22)

of system (16) maps solutions ^ of (21) to its solutions and vice versa. Thus (22) is a recursion operator for symmetries of (1). It is easy to verify by direct computation, that each local contact symmetry of Eq. (1) has a lift to a symmetry of system (16). So the covering defined by system (16) does not contain non-removable parameters [8]. Therefore Eq. (1) provides an example of pde that possesses a recursion operator which corresponds to a covering without non-removable parameters.

APPENDIX

The structure equations of the symmetry pseudo-group of Eq. (1) read

dd0 = (022 + (u2 ~(k + l)2 (u1 -2(k + 1)) f1 - (u4 + (K + 1)(K + 2)u3-K(K + 1) u1 +(k + 1)2(k + 2) (2K + 3)) (K + 1)_1 (K + 2)_1 ^3) A0O+^1 A01+^2 A02 + (K + 1)2(K + 2) (k2 + 6K + 4)) (K + 1)_2(k + 2)2 -(U4-K(K + 1) U1 + A 03, d01 = (K + 1) (2 01 + (K + l)(/c + 2)2 62 -(K + 2) 03)) A0o+^1A 0n + A 013

+(K + 1)(K + 2)02 A 03 - (2 (K + 1) (02-2(u1-2(k + 1xk + 2)) ^ + ^2)

+((2 k + 3)U1 - (K + 1)(k + 2)(3K + 4))(K + 2)"1 ^3) A0X + A (t/3 0O + U1 03 + 012),

d02 = 0O A 022 + + 1) (u1 -2(k + 1)(k + 2)) ^ + i ux ^ A 02

A 012 + A 022 + A 023,

d03 = ((K + 1) (03 - (K + 1)(K + 2)02 + (K + 1)(K + 2)((K + 3)^ - (JC + 2) (U2 + 2)) f1) + (U2 + U4) + + 2) 022) A0O + ((K + 1)03+^! A 02

+(k + 1)(2 (^ - 2 (K + 1)(K + 2)) (^ + (K + 2)_1 ^3) - <f2) A03 + (1A 013 +^2A023 +^3A012,

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d0n =^ia(2 + ^2<f3 + ((4 U4-(k + 1)(k -2 )U±- к(к + 1)2(к2 - 4)

+(2к + 1) U2) в1-(к + 1)2(к + 2) (Ü! - 2 (к + 1)(к + 2)) (02 + (к + 1)(к + 2) 03)

+(к2 - 1)(к + 2) 013 + (к + 1)(к (2У5 + 3 (k + 2) U2) - U1U2

-(к + 1)2(к + 2)(3 к - 2)((к + 3)^ - 2 (k + 1)(k + 2))) + ((к + 1)(к U1

-(к + 2)U3 + (k + 1)(k + 2)(3к2 + 6к + 4)) - У4)(к + 1)_2(к + 2)~2 0n) Л 0O

+((к + 1)(4 U1 + (к + 1)(к + 2) (11k + 14)) 02 - (к + 1)(2 U1 - (к + 1)(к + 2)) 03

—(2к + 1) 012 + 4(к + 1)(k + 2) 023 - (4 U4 - (к + 1)(2к2 + 3 к - 4) U1

+(2к + 1) У2 + 4к(к + 1)2(к + 2)- (2к + 3)(к + 2)"1 Ufr

-(2(2к + 3)(к + 2)-1 + (Зк + 2)(к + 1) (U2 - (к + 3)(к + 1) U1

+2(к + 2)(к + I)2) ^3) Л0Х + ((к + 2)(к + I)2 - 2 (к + 1)(к + 2)) 03

+(4к + 5) 0П - (к + 1)(к + 2) 013) Л02 + ((к + 2) 013 - (2 U5

+(к + - 2(к + 2)(U2 + (к + 1)(к + 4)^! + 2(к + 1)2(к + 2)))) ^2) Л 03

"(022 - ((к + 1)(к + 9) и1-и2- 14(к + 2)(к + I)2) ^ - ((к + 1)(к и1

-(к + 2) (У3 - (Зк2 + 6к + 4)(к + 1))) - У4)(к + 1)-2(к + 2)-2

+(3 (U1 ~(к + 1)(к + 2)) + (к + 1)-х(к + 2)-1 У4) ^3) Л 0П

+((к + 1)^ - 2 (к + 1)(к + 2)) 012 - 2 иг 013) Л

¿012 (0О + (2)+^уЛ(3 + ( (i/i - (к + 4)(к + 2)(к + 1)) 022

1

+(к + 1)((к + 2) 023) + (i/4 + - (к + 1)(к2 + 2к + 2) и± + к (к + 1)2(к + 2)) 02

+((к + 1) (к U1 - (к + 2) (i/3 - к2(к + 1))) - У4)(к + 1)-2(к + 2)-2 012) Л 0Х

+((2 к + 3) 012 - (к + 1)(к + 2) 023 - i U, (к + 2)03 - + У2 - (к + 1) U,

-U4 - к(к + 2)(к + I)2) ^2) Л02 + (к + 2) (023 - (к + 1) 022) Л 03 -(022 - ((к + 1)(к + 5) U1-U2- 6(к + 2)(к + I)2) ^ + ((к + 1)(к U1

1

-(к + 2)(У3 -к2(к + 1)) - и4)(к + 2)-2(к + I)-2 - - ((Зк + 8)(к + 1) U±

+2 u4 - 4(к + 2)(к + 1)2)(к + 1)_1(к + 2)-1 ^3) Л 012

+2(к + 1)(U1 -2(к + 2)(к + 1)) (022 Л$2 + 023 Л ^3) - U1 023 Л

¿013 + ((к + 1)2(к + 2)2 023 - У3 0Х - (к + 1)(к + 2) 012

(к + 1)(к + 2)((к + 4)(к + 1) U± -4(U2 + (k + 2)(к + 1))) 02

-((к2 -1)U1 + U2-U4- 2(к + 2)(2к + 1)(к + I)2) 03 + ((к + 1) (к U1

+(023 - (к + 1)(к + 2) 022 + (i/2 + U4) + i (У! - 2(к + 2)(к + I)2) 02

+У3 ^2) Л 01 + ((Зк + 4) 013 - (к + 1)(к + 2) 012 - - i (к + 1)((3к + 4) U,

-4 (к + 1)(к + 2)) 03) Л02 + (012 + (к + 1)(к + 2) 023 + ((к + 1)(2 +(к + 2)((к2 - k- 4)U1-U2- 2к(к + 1)(к + 2))) - 2(к + 2)У4)(к + 2)"1 +((к + 1)(к + 2)(5к + 2)((к + 1)(к + 3)U1-U2- 2(к + 2)(к + I)2)

-4(к + 1)У5)(к + 2)"1 f ) Л 03 + | U, $2 Л 012 - (022 - ((к + 1)((к + 7) U! — 10(к + 2)(к + 1)) - U2) $1 + ((к + 1)(кУх - (к + 2)(У3 - (2к2 + Зк + 2)(к + 1))

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+U4)(K + 1)-V + 2)"1 A 013 + (K + 1 )(U± - 2(K + 1)(K + 2)) 023 A <f2 -(K + 2) (U3 - (2K2 + 3K + 2)(K + 1)) - U4)(K + 1 )~2(K + 2)"2 013) A 0O —U4)(K + 1)~2(K + 2)~2 $2 - ((3K + S)(K + 1)U1 - 2(K + 2)(2K + 3)(K + l)2 dB 22 (00 + (U4 -(K + 1)(K U± + (K + 2) U3

+(K + 1)(K + 2)(K2 + 6K + 4)))(K + 1)_2(K + 2)"2 022 A (0O + +(((K + 1)2(U1 + 2K + 4)) - U2) f1 - ((K + 1)(K U1 + (K + 1)(K + 2)(3K + 2)) -tf4)(K + l)-V + 2)-H3)A022

dB23 =V4A^3+V6A (0O +(2)+^7A(1+j (t/i 022 + (jc U4 - (K + 1)(K2 U±

—2(K + 2)U3+ K(K + 2)(3K + 2)(K + 1)2)(k + 1)_1(K + 2)"2 02 - 2 (K(K + 1 )i/12 +((K + 2)U3 + U4 + (K + 2)(K2 -3K- 2)(K + l)2) U1 + K(K + 1)(K + 2)(ü4 +(K + 2) y3))(K + 2)-2 - 2 (U4 -(K + 1)(K U1 + (K + 2) t/3

+(k + 1)(K + 2)(3K + 2)))(K + 1)"2(K + 2)"2 023) A 0O + i (6(K + 2) 023

-8 (k + 1)(K + 2) 022 - 2 (zc t/4 - (K + 1)(K2 Ux - 2(K + 2) U3

+K(K + 1)(K + 2)(3K + 2))((K + 1)~1(K + 2)_1 + (K U¡ - (K + 2) (2(k + 1)(K2

+4k + 2)^ + 2(K + 4)U2 + 4 U4))(K + 2)'1 A 02 + (K(K + 1)U1 + (K + 2)U3

-U4 -(K + 2)(3K + 2)(K + 1)2)(k + 2)"1 03 A <f3 + (023 - i - 2(K +

-2(k + 1)(K + 2)) A 022 + (((K + 3)(K + 1)U± -U2- 2(K + 2)(K + l)2) ^ +(U4 -(K + T)(K U± + (K + 2) U3 + (K + 2)(3K + 2)(K + 1))(K + 1 )~2(K + 2)"2 1

+ - (3(k + 1)(K + 2)U1 + 2U4 - 2(K + 1)2(K + 2)2)(K + 1 )~1(K + 2)"1 A 023

df1 = (022 + (2K + 3)02- ((K + 1)((K + 3)U1 - 2 (K + 2)) + U4) ((K + 1)(k + 2))_1f3 +((k + 1) (K U1-(K + 2) (U3 - K2(K + 1))) - U4) ((K + 1)(K + 2))"2 (0O + ^2)) A f1, d^2 = (0X -(K + 1)(K + 2) 03-(K + 1)2(K + 2)2 0O) + (02 + 022 + (U4 - K(K + 1) -(K + 1 )2(K + 2)(3K + 2))(K + 1)_1(k + 2)_1 - (U4 + (K + 1)(K + 2) U3 -K(K + 1) U1(K + 1 )2(K + 2)(K2 + 6K + 4))(K + 1)~2(K + 2)~2 0O +(U2 -(K + 1)(K + 2) f!) A + (03 -(K + 1)(K + 2) 03) A

= + 2) + 1)(K 0O - 2 02) + 03) + (K(K + 4)U1-U2-2(K + 1)2(K + 2)) -U, A^1 + (022 + (K + 2)02- (U4 + (K + 1)(K + 2) U3 - K(K + 1) U, +(K + 1)2(K + 2)(3K + 2))(K + 1)_2(K + 2)-2) (0O + ^2)) A

(23)

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КОНТАКТНЫЕ ИНТЕГРИРУЕМЫЕ РАСШИРЕНИЯ ГРУППЫ ПСЕВДОСИММЕТРИЙ И НАКРЫТИЯ R-ro ДВАЖДЫ МОДИФИЦИРОВАННОГО БЕЗДИСПЕРСИОННОГОУРАВНЕНИЯ КАДОМЦЕВА - ПЕТВИАШВИЛИ

Морозов О.И., Павлов М.В.

Найдены контактные интегрируемые расширения группы псевдосимметрий и накрытия r-того дважды модифицированного бездисперсионного уравнения Кадомцева - Петвиашвили. Одно из накрытий приводит к преобразованию Бэклунда и оператору рекурсии для исследуемых уравнений.

Ключевые слова: псевдогруппа Ли, дифференциальное накрытие, контактное интегрируемое расширение, преобразование Бэклунда.

СВЕДЕНИЯ ОБ АВТОРАХ

Морозов Олег Игоревич, 1963 г.р., окончил МГУ им. М.В. Ломоносова (1986), член Московского математического общества, доктор физико-математических наук, профессор кафедры высшей математики МГТУ ГА, автор 49 научных работ, область научных интересов - дифференциальные уравнения, симметрии, псевдогруппы Ли, электронный адрес: [email protected].

Павлов Максим Валентинович, 1962 г.р., окончил Московский физико-технический институт (1985), кандидат физико-математических наук, ведущий научный сотрудник Новосибирского Государственного Университета, автор семидесяти научных работ, область научных интересов - интегрируемые нелинейные многомерные дифференциальные уравнения, электронный адрес: [email protected].