N = 2 super Boussinesq hierarchy: Lax pairs and conservation laws

Physics Letters B 312 (1993) 463-470 North-Holland

PHYSICS LETTERS B

N -- 2 super Boussinesq hierarchy: Lax pairs and conservation laws S Belluccla, E Ivanovb'c, S Krlvonosa,c and A Plchugm c a I N F N - Laboratort Naztonah dt Frascatt, P 0 Box 13 1-00044 Frascatt, Italy b Physlkahsches lnstttut, Umversttat Bonn, Nussallee 12, D-5300 Bonn 1, Germany c Bogohubov Theorettcal Laboratory, JINR, Dubna, Head Post Office, P 0 Box 79, 101000 Moscow, Russmn Federatton

Received 24 May 1993 Editor R Gatto

We study the lntegrabfllty properties of the one-parameter family of N = 2 super Boussmesq equations obtained earher by two of us (E I & S K , Phys Lett B 291 (1992) 63) as a Hamfltonlan flow on the N = 2 super-W3 algebra We show that It admits nontrlwal h~gher order conserved quantities and hence gives rise to mtegrable hierarchies only for three values of the involved parameter, ~r = - 2 , - 1 / 2 , 5/2 We find that for the case c~ = - 1 / 2 there exists a Lax pair formulation in terms of local N = 2 pseudo-differential operators, while for ~ = - 2 the associated equation turns out to be b~-Hamlltonlan

I. Introduction D u r i n g the last few years there has been a considerable interest in the lntegrable e v o l u t i o n e q u a t i o n s assocmted with the W - t y p e algebras and superalgebras (see, e g , [ 1 - 8 ] ) Since the discovery [9,10] that the classical Vlrasoro algebra p r o v i d e s the second H a m l l t o m a n structure for the K d V hierarchy, there appeared a lot o f papers where various s u p e r s y m m e t r l c and W extensions o f this algebra were treated along a similar line and the relevant hierarchies o f the e v o l u t i o n e q u a t i o n s were d e d u c e d and analyzed In particular, integrable N = 1 [ 11 ], N = 2 [12], N = 3 [13] and N = 4 [14] s u p e r s y m m e t r l c K d V e q u a t i o n s have been constructed, with the N = 1, 2, 3, 4 super Vlrasoro algebras as the second H a m l l t o m a n structures In [ 1 ] it has been shown that the classical W3 algebra (with a n o n - z e r o central charge) defines a second H a m l l t o n l a n structure for the B o u s s m e s q hierarchy T h e Lax pair f o r m u l a t i o n o f the latter in terms o f the Gel'fand-D~ki1 pseudo-differential operators closely related to the H a m l l t o m a n f o r m u l a t i o n also has been given (see, e g , [2]) Obviously, s u p e r s y m m e t n c extensions o f the B o u s s m e s q e q u a t i o n should be associated, m the a b o v e sense, with super-W3 algebras In ref [8] two o f us (E I & S K ) have constructed, in a manifestly s u p e r s y m m e t r l c N = 2 superfield form, the m o s t general N = 2 super B o u s s m e s q e q u a t i o n for which the second H a m l l t o n i a n structure is given by the classical N = 2 super-W3 algebra [15,16] This e q u a t i o n t u r n e d out to c o n t a i n an arbitrary real p a r a m e t e r c~, m u c h hke the N = 2 super K d V e q u a t i o n [ 12] In this letter we address the q u e s t i o n o f eMstence o f the whole N = 2 Bousslnesq hierarchy, ~ e we e x a m i n e whether the e q u a t i o n constructed in [8] a d m i t s an infinite sequence o f c o n s e r v e d quantities in r e v o l u t i o n and a Lax pair f o r m u l a t i o n W e find that the n o n t r l v i a l higher o r d e r c o n s e r v e d quantities exist only for three values o f c~, n a m e l y for c~ = - 2 , - 1/2, 5 / 2 This again highly resembles the case o f N = 2 super K d V e q u a t i o n which IS k n o w n to give rise to the lntegrable hierarchies only for three special values o f the i n v o l v e d p a r a m e t e r [ 12] W e p r o v e the integrabihty o f the o p t i o n c~ = - 1/2 by finding the Lax pair for it (In terms o f the N = 2 pseudodifferentml o p e r a t o r s ) W e also show that the e q u a t i o n c o r r e s p o n d i n g to the choice c~ = - 2 possesses the first H a m l l t o n l a n structure T h i s property, together with the existence o f higher o r d e r c o n s e r v a t i o n laws, suggest that the c~ = - 2 e q u a t i o n is lntegrable as well Elsevier Science Pubhshers B V

463

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2. N = 2 super Bousslnesq equation Let us first briefly recall the basic points o f ref [8] in what concerns the N = 2 super Boussmesq equation and its relation to the N = 2 super-W3 algebra All the basic currents of N = 2 super- W3 algebra [ 15-17 ] are accomodated by the spin 1 supercurrent J ( Z ) and the spin 2 supercurrent T ( Z ) , where Z = (x, 0, 0) are the coordinates of the N = 2, ID superspace The supercurrent J ( Z ) generates the N = 2 super Virasoro algebra, while T ( Z ) can be chosen to be primary with respect to the latter The closed set o f SOPE's for these supeicurrents, such that it defines the classical N = 2 super-W3 algebra, has been written down in [8] Here we prefer an equivalent notation via the super Polsson brackets

(21)

{VA(ZI), VB(Z2)}(2 ) = "DAB(Z2)A(ZI2), where VA=I, 2 ~ (J, T) , and A (Zj2) denotes the N = 2 super delta-function

A(Zl2) =

012012(~(Z1

--

Z2)

(2 2 )

The subscript " ( 2 ) " of the super Polsson brackets indicates that they provide the second Hamlltonlan structure for the N = 2 super Boussinesq equation to be defined below The 2 × 2 super-differential operator DAB in (2 1) encodes the full information about the structure o f the classical N = 2 super-W3 algebra The explicit form o f its entries is as follows Dl, = - ½ c [D,D] 0 + D J D + D J D + JO + OJ , "~12 =

D T D + D T D + 2TO + OT,

D2I = D T D + D T D + 2TO + 2 0 T , '/)22

=

,

[D,D---]O 3

gC

- (5T-

--

2JO 3 - 679J~02 - 6~J7902 - 60JO 2 - ~ ( 8 0 J - 5T - B(2))'D0

2 [-D,D] J + B (2)) [D,D~] O - D ( S O J + 5 T + B(2))DO

+ (3 [-D,D] T - 6 0 2 j

+ V(3')O - ½0 ( S T - 2 [ ~ , V ] J + B (z,) [ D , ~ ]

- ' 3 0 D T + 3 0 2 D j + ~(7/2)) ~ + ( 3 0 D T - 3 0 2 D j - ~ 7 / 2 ) ) 79 I "-+~`-+(7/2) + - 2 0 3 J + 0 ~ , V ] T + IOUO' + 1D7 J + Ivy(v/2'

_

_

I [ ~ , ~ ) ] B(2>) +0

(23)

Here c is the central charge taking an arbitrary value at the classical level, and B(2)(Z)

,

~-/(7/2) ( Z ) ,

~(7/2) (Z) ,

g(3) (Z)

are the composite supercurrents with spin 2, 7/2, 7/2, 3, respectively, B(2) = 8 j 2 , c

~(7/2) = 80c ( J D J ) ~tU(7/2 ) = 8

+ c36 [D,D] J D J + c S j D T -

0 ( j - ~ j ) _ 7 2 T - ~ j + 3_..66[D, 79] J ~ J + 8 j D T C

U(3) = ~56- J T 464

72TDJc

C

C

-3-2 j ~-D,D] J + 1 2 8 j 3 + 1 2 0 D j D J

C

7-128j2Dj + c 4 0 j D J ' ~_j2~j

_ 40j~ J C

'

(24)

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19 August 1993

where 012 = 01 -- 0 2 ,

012 :

0 l -- 0 2 ,

Zl 2 :

Zl _ z 2 .+ 1 ( 0 1 0 2 _ 0 2 0 1 )

(2 5)

and the covarmnt splnor derivatives are defined by

Do- ooO ½00x, 77o- ooO lOOx,

{D,7?} = - 0 x ,

D 2 =792 = 0

(2 6)

The N = 2 super Bousslnesq equation can be defined as the system of two N = 2 superfield equations for the supercurrents T, J with the N = 2 super-W3 algebra (2 1), (2 3), (2 4) as the second Hamiltonlan structure In other words, it amounts to the following set of evolutmn equations

T = {T,H}(2),

J

=

{J,H}(2}

(2 7)

,

or, m the condensed notation,

dH VA= DAa~

(2 8)

The Hamlltonlan H in (2 7) and (2 8) is given by

. = f dZ (r +

(2 9)

We emphasize that (2 9) is the most general H a m l l t o m a n which can be constructed out of J and T under the natural assumptions that it respects N = 2 supersymmetry and has the same dimension 2 as the Hamlltonlan of the ordinary bosonic Boussinesq equation Note the presence of the free parameter c[ in (2 9) Now, using the Polsson brackets (2 1 ) and the definitions (2 3), (2 4), it is straightforward to find the explicit form o f the N = 2 super Bousslnesq system

T=-ZJ'" + ~,D]T' + 800(DJDJ) T

32j,[V,g)]j_

c

16j~,V]j,

T

+ 256j2j,

TT-

+ (?-2ct)DJDT+ (?-2oe) DJDT+ ( ? +4c~)J'T+ (2~4c+2{t) JT', J

=

2T' + c ~ ( 4 [ D , D ] J '

+4JJ')

(210)

The bosonlc sector o f e q s (2 10) is a coupled system of equations for two spin 2 currents and the spin 1 U(1 ) K a c - M o o d y current As was shown in ref [8], the standard Bousslnesq equation decouples from this system only for c~ = Presumably, th~s particular case corresponds to a N = 2 superextenslon of the Boussmesq equation constructed m terms of N = 1 superfields in [16] Note that the dependence on c m (2 10) is unessential and it can be removed by rescahng the superfields T and J For definiteness, in what follows we will put c = 8 On the contrary, the dependence on c[ is crucial for achieving mtegrabdaty m the next section we will see that only for three specml values of this parameter the above system results in integrable hierarchies

-4/c

3 Conservation laws Now we turn to the basic theme of the present paper, the analysis of mtegrablllty of the set (2 10) The standard signal o f mtegrablhty is the presence of an infinite sequence of mutually commuting nontrlvlal conserved 465

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q u a n t m e s In this section we report on the results of our study of the ~ssue of existence of the higher order conserved quantities for (2 10) with c = 8 In searching for such objects we made use of the standard method of undetermined coefficients One considers an integral of degree n constructed from all the possible independent densities of degree n, each multlphed by an undetermined coefficient (Two densmes are dependent if their difference is a total (super)derivative) The coefficients are then fixed by reqmrxng the integral to be a conservation law, that is time-independent In this way, with the heavy use of the symbohc m a m p u l a t l o n program Mathematlca [ 18 ], we have found the following first six conserved q u a n t m e s

Hi = f

dZ J,

//2 = f

dZ (T + &j2),

H3 = c~/

dZ ( J T + alJ 3 + aEDJ~J)

H4 = f

d Z ( T 2 + b i T J 2 + beTJ' + b3T'I)~J + b4J 4 + bsj2"D~J + b 6 J J " ) ,

115 = f

dZ (T79~T + ClT2j + c2TJ 3 + c3TJJ' + c4TJI)DJ + c s T D J ~ J + c6TJ t'

+ CvJ 5 + c8J3"l)~J + c9JZJ '' + cloJ27)~J ' + CllJ79~JT)~J + clZJ'DDJ"),

H6 = J f

dZ (dlT 3 + d2TT" + d3T2j 2 + daT2j ' + d s D T D T J + d6TZ79~J + d7TJ 4 + d8TJZ J '

+ dgTJZD~J + d I o T J D J D J + d l l T J J " + dl2TJ'J' + d I 3 T J ' D ~ J + dlaT'DJ'~J + d l s T D J ~ J ' + dI6TDDJDDJ + d l v T J " + d t 8 T D ~ J '' + dl9J 6 + d20J47)~J + dz1J3J '' + d22J379~J ' + d23JZ"D~JT)DJ + d24J2j ''' + d25JZ"D~J '' + d26Jl~l)JT)7)J' + d 2 7 j j l V ) ,

(3 1)

Here

d ~ d~-2 q- ~,5/2

(3 2)

The most striking result of this exercise as that the nontrivial higher order H. (n >~ 3) exist if and only if the parameter c~ takes one of the following three values = - 2 . - 1 / 2 , 5/2

(3 3)

We have then verified that H3 and H6 exist only for the two values o f ~ ~ = - 2 , 5/2 Notice that the special value of ~ at which the Bousslnesq equation in the bosonic sector decouples from two other equations is present a m o n g t h o s e in (3 3) for t h e c h o l c e c = 8 it i s j u s t ~ = 1/2 The corresponding values of the coefficients m H3-H6 are given m tables 1-4 The existence of these first hagher order nontrlvlal conservation laws xs a very strong indlcataon of the complete antegrabllity of the N = 2 super Boussmesq equation for the three values of ~ indicated in eq (3 3) and, hence, the existence of N = 2 super Bousslnesq hIerarchaes in these cases In the next sections we wall present the Lax pair for the ~ = - 1 / 2 case and the first H a m i l t o m a n structure for the ~ = - 2 case 466

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PHYSICS LETTERS B

Table 1 Coefficients of H 3

19 August 1993

Table 2 Coeffioents of Ha

-2 5/2

al

a2

~

bl

b2

b3

b4

b5

b6

-5/4 l

-5/2 2

-2 -1/2 5/2

-4 2 14

4 4 -8

8 8 -16

4 -1/2 17/2

-16 -1 -31

4 1 7

C9

CI0

CII

¢12

21/4 -7/2 21/4

-21/2 7 -21/2

21/4 1 4

Table 3 Coefficients of H5

-2 -1/2 5/2

CI

C2

C3

£4

C5

¢6

¢7

C8

3/2 1 -3/2

15/4 0 -15/2

-10 0 25/2

-20 0 25

-25/2 -5 10

5/2 0 -5/2

483/160 1/5 -33/10

497/24 4/3 113/6

-77/16 -2 -13/4

Table 4 Coefficients of H 6

dl

d2

-2 5/2

-1/9 1/117

1/6 1/26

d14

d15

d16

d3

d4

d5

d6

2/3 - 2 / 3 1 -4/3 23/78 - 7 / 3 9 5/13 -14/39

d17

d18

d19

d20

-2 2 -16/3 2/3 4/3 8/9 - 2 8 / 3 5/13 - 5 / 1 3 61/39 - 2 / 1 3 - 4 / 1 3 5/18 -89/39

d7

d9

dlo

-4/3 20/3 40/3 31/39 -101/39 -202/39

16 -55/13

d21

d8

d22

8/3 -32/9 115/234 -94/117

d23

d24

dll

dl2

d13

-10/3 -8/3 -16/3 1 28/39 61/39

d25

d26

d27

32/3 -16/9 -14/3 94/39 -73/234 -11/13

20/3 4/3

2/3 3/26

4. Lax pairs

In this section we construct a Lax pair for the N = 2 super Bousslnesq equation (2 10) (with c = 8) We start from the general multi-parameter form of the third order Lax operator

L = O3 + AI 0 2 + A 2

[7),~]0+A3

790+A4

~O + A 5 0 + A6 [ 7 9 , ~ ] + A 7

D + A8 ~ + A 9 ,

(41)

where Al, , A9 are arbitrary polynomials m J, T and their (super)derivatives with suitable dimensions and U ( 1 ) properties This means, in particular, that L must be a U ( 1 ) smglet For example, the first several coefficients in L can be p a r a m e t n z e d as follows

A~ = k ~ ) J ,

A2 -- k(Z)J,

A3 = k ( 3 ) ~ J ,

A4 = k(4)DJ,

A5 = k~5)J 2 + k~5)T + k~5)OJ + k~45)[D,D]J ere ,

(4 2)

where k are numerical coeffictents We search for those values of these parameters for which the Lax equation

Lt = fl L>.I,L

(43) 467

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r e p r o d u c e s the N = 2 super Bousslnesq set (2 10) Here, the subscript >/ 1 denotes a strictly differential part o f the o p e r a t o r #1 For fl = - 3 we h a v e f o u n d the following two solutions, b o t h c o r r e s p o n d i n g to the same value o f ~ = 1/2 L (1) = 0 3 + 3JO 2 - 3DJ790 + ( 2 J 2 - T + 3 0 j - ' [ D , ~ ] J ) L t2) = 0 3 + 3 j 0 2 +

3J[79,D]O

+ ( j 2 _ ½T + 3 0 j - ~ '

3DJ790+ (j2_7

[ 7 9 , D ] J ) [79,D] + ( D T -

0 + ( D T - 4 J D J - 2 0 D J ) 79,

'T + ~oJ

_

(4 4)

~[79,~1J) O

4 J D J - 2 0 D J ) 79

(4 5)

The solutions for the choice fl = 3 in (4 3) can be o b t a i n e d f r o m (4 4 ) - ( 4 5) through the substitutions

J-'-J,

T-,T,

D-'D,

79-'79

T h e y yield a Lax pair which is conjugate o f (4 3) Let us define the N = 2 super residue o f a generic N = 2 super pseudo-differential o p e r a t o r M

A =

~ l=--

(13, + 7,79 +

p,[D,~])O'

(4 6)

oc

as the coefficient o f [ D , D ] 0 - l Res .A = P - I

(4 7)

Then, following the reasoning o f [12], we can show that (4 3) implies the e q u a t i o n d /'Res dt J

Lk/3dZ = 0

(4 8)

This gives an infinite n u m b e r o f c o n s e r v a t i o n laws W e have checked, that for L(I) all residues o f the operators Res L k/3 are equal to zero, so L (~) is a d e g e n e r a t e d Lax operator, while for L(2) the expressions Res L k/3 r e p r o d u c e the c o n s e r v e d quantities for ~ = - 1/2 i n d e p e n d e n t l y found an the p r e v i o u s section N o t e that, despite the nonself-conjugacy o f the operators L and L k/3, the integrals in eq (4 8) are real the i m a g i n a r y parts o f the lntegrands m all cases p r o v e to be full d e r i v a t i v e s T h u s we h a v e p r o v e d the lntegrablhty o f the N = 2 super B o u s s m e s q e q u a t i o n for c~ = - 1 / 2 Its correct Lax f o r m is given by eq (4 3) with fl = - 3 and the Lax o p e r a t o r L (2) (4 5) (or its conjugate, with fl = 3 In (4 3)) T h e Lax o p e r a t o r s for the o t h e r cases listed in eq (3 3) (if existing) cannot be represented by local super-differential o p e r a t o r s (nonlocal Lax f o r m u l a t i o n s for super K d V e q u a t i o n s were considered, e g , in [20] )

5 First Hamiltonlan structure In the p r e v i o u s section we h a v e f o u n d a Lax pair for the N = 2 super Bousslnesq e q u a t i o n with c, = - 1 / 2 H e r e we study for which values o f a the set (2 10) can be given a first H a m i l t o n l a n structure T h e first H a m l l t o n l a n structure for the o r d i n a r y Bousslnesq e q u a t i o n can be o b t a i n e d f r o m the second one by shifting the stress tensor by a c o n s t a n t H e r e we wilt use the same idea We have checked that the Lax formulation of (2 10) m a more customary form Lt = fl[L2+/3,L], where + means restriction to the posture and zero parts of the N = 2 super pseudo-differential operator L 2/3, does not exist at all The Lax representation of the type (4 3) was proposed earher, e g, for the N = 2 super KdV equation m ref [ 19] 468

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In the case at hand there ~s a substantml freedom compared to the bosomc case since one can shift the supercurrents J and T by independent constants However, a close inspection of the second Hamlltonian structure (2 1 ), (2 3) shows that only shifting the supercurrent T yields a self-consistent structure

{ V A ( Z I ) , VB(Z2)}(I) : ~)AB(Z2)z~(ZI2) ,

(5 1)

where now DII = 0,

79,2 = 2 0 ,

7921 = 2 0 ,

D22 = - 5 [D,D----]0 + 7JO + 9 D J D + 9 D J D + 8 0 J

(5 2)

It is easy to check that this super Po~sson structure together with the proper degree H a m f l t o n l a n / / 4 from the set (3 1 ) reproduce the N = 2 super Bousslnesq equation only for ~ = - 2 ~r = _ l ( H 4 ) (. . . .

2) = -½ f

d Z ( T2 - 4 T J2 + 4 T J ' + 8 T D D J + 4 J 4 - 16JZZ)~J + 4 J J " )

(5 3)

In this case eq (2 10) can be represented m the form which follows from (2 8) via the substitutions DAB --~ 7"gAB, H + H This proves that our N = 2 super Bousslnesq equation is bl-Hamfltonian for a = - 2 In a number o f cases the existence of two Hamiltonlan structures for the same equation already Implies an infinite tower of higher order conservation laws (see, e g , [9] ) It would be of interest to see whether this is true in the present case, ~ e the existence of h~gher order conserved quantities for the <~ = - 2 N = 2 super Bousslnesq equation can be traced to its bl-Hamlltonlan nature The p r o o f of Integrablllty of the N = 2 super Boussinesq equation for the remaining value of a = 5/2 is an open problem For this case we were not able to find neither a Lax pair nor a first Hamfltonlan structure

6. Conclusion In this paper we have presented the results of our study o f the mtegrablhty properties o f the N = 2 supersymmetric Boussinesq equation (2 10) with the N = 2 super-W3 algebra as the underlying second Hamfitonlan structure We have found that the lntegrable N = 2 super Boussmesq hierarchies can exist only for the three special values (3 3) of the free parameter (x The lntegrablhty in the case
Acknowledgement E I and S K thank Physlkahsches Instltut in Bonn and L N F - I N F N in Frascati for hospitality during the course of this work We are grateful to Z Popowlcz for his critical remarks which helped us to remove some doubtful statements in the final version of the paper This investigation has been supported in part by the Russian Foundation of Fundamental Research, grant 93-02-3821 469

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Note added After this paper was submitted for pubhcatlon, we became aware o f a preprmt by Yung [21 ] where the existence o f the three values o f the parameter (~ is also estabhshed and the relevant conserved quantmes, up to H6, are presented

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