dispersionless stb

17 July 2000

Ž .Physics Letters A 272 2000 65–73www.elsevier.nlrlocaterpla

Dispersionless sTB

Ashok Das a, Ziemowit Popowicz b,)

a Department of Physics and Astronomy, UniÕersity of Rochester, Rochester, NY 14627-0171, USAb Institute of Theoretical Physics, UniÕersity of Wrocław, 50-205 Wrocław, Poland

Received 25 February 2000; received in revised form 10 May 2000; accepted 1 June 2000Communicated by A.P. Fordy

Abstract

Ž . Ž .We analyze the dispersionless limits of the SUSY TB-B sTB-B and the SUSY TB sTB hierarchies. We present theLax description for each of these models, as well as the Ns2 sTB hierarchy and bring out various properties associatedwith them. We also discuss open questions that need to be addressed in connection with these models. q 2000 ElsevierScience B.V. All rights reserved.

1. Introduction

In recent years, dispersionless integrable modelshave received a lot of attention. They involve equa-

w xtions of hydrodynamic type 1–5 and include suchw xsystems as the Riemann equation 5–7 , the the

w xpolytropic gas dynamics 5,8 , the chaplygin gas andw xthe Born-Infeld equation 9,10 . These are models

which can be obtained from a ‘‘classical’’ limitw x3,11–16 of integrable models where the dispersiveterms are absent. They have many interesting proper-ties including the fact that, unlike their dispersivecounterparts, each of them can be described by a Laxequation which involves a Lax function in the classi-cal phase space and a classical Poisson bracket rela-tion. Even more interesting and more difficult are thesupersymmetric dispersionless models. In a recent

w xLetter 17 , we gave, for the first time, the Laxdescription for the dispersionless supersymmetric

) Corresponding author.Ž .E-mail address: [email protected] Z. Popowicz .

w xKdV equation 18–20 as well as the dispersionlessw xKupershmidt equation 21 . Unlike the bosonic mod-

els, the Lax functions for the dispersionless super-symmetric models do not follow trivially from theLax operator of the dispersive counterpart. Further-more, while a lot of the interesting properties followfrom the Lax description of the model, we alsopointed out several open questions that arise. In thisLetter, we follow up on our earlier investigation anddescribe the Lax formulation for the dispersionless

Ž .supersymmetric two boson TB hierarchy.w xThe TB hierarchy 22–24 as well as its super-

w xsymmetric counterpart 25,26 are known to yieldvarious other integrable models upon appropriatereduction. In this sense, the supersymmetric TB hier-archy is a more interesting model to study. In partic-

w xular, it has extended Ns2 supersymmetry 25–30and its dispersionless limit would lead to the firstNs2 supersymmetric model of its kind. There are,in fact, two distinct supersymmetric generalizationsof the TB hierarchy. The first is known as the sTB-B

w xhierarchy 31 , so-called because it leads, upon re-duction, to the supersymmetric KdV equation con-

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 00 00392-3

( )A. Das, Z. PopowiczrPhysics Letters A 272 2000 65–7366

w xsidered by the Beckers 32 . The second supersym-metric generalization, on the other hand, leads uponreduction to the supersymmetric KdV equation con-

w xsidered by Manin and Radul 18–20 . We call thisw x Žthe sTB hierarchy 25,26 although, a more appro-

.priate name may be sTB-MR along the same lines .In Section 2, we give the Lax description for the

dispersionless sTB-B hierarchy, which is fairlystraightforward, and bring out its properties as wellas the open questions associated with this model. InSection 3, we give the Lax description for the disper-sionless sTB hierarchy. This is quite nontrivial andnaturally reduces to the dispersionless sKdV equa-tion upon appropriate restriction. However, unlikethe sTB model, it does not lead to the dispersionlesssupersymmetric non-linear Schrodinger equation. In¨fact, even the dispersionless bosonic TB model doesnot quite give the dispersionless non-linear Schrodi-¨

Žnger equation at least, we have not succeeded in.finding a field redefinition which would do this . We

bring out various properties of the dispersionlesssTB model as well as the open questions associatedwith this system. In Section 4, we describe thedispersionless sTB system in a manifestly Ns2supersymmetric formulation. Finally, we make somebrief observations in Section 5. We have used RE-

w xDUCE 33 and the special supersymmetry packagew xin Reduce 34 extensively in some of the algebraic

calculations.

2. Dispersionless limit of sTB-B equation

w xThe sTB hierarchy 25,26 , like the TB hierarchy,is an integrable system in 1q1 dimensions. Thebasic dynamical variables for this system are the twofermionic superfields

F t , x ,u sc qu J ,Ž .0 0 0

F t , x ,u sc qu J , 1Ž . Ž .1 1 1

where u represents a Grassmann coordinate and weare suppressing the space-time dependence on the

right hand side for simplicity. The sTB-B hierarchyw xis described by the non-standard Lax equation 31

E Lns L, L , 2Ž . Ž .G1

E tn

where ns1,2,... and the Lax operator has the form

LsD2 y DF qDy2 DF , 3Ž . Ž . Ž .0 1

with the super-covariant derivative defined to be

E E E2Ds qu , D s . 4Ž .

Eu E x E x

Explicitly, the third flow of the sTB-B hierarchyhas the form

EF 0syF yD 6 DF DFŽ . Ž .Ž0 x x x 0 1E t

3y3 DF DF q DF ,Ž . Ž . Ž . .0 0 x 0

EF 1 2 2syF y3D DF q DF DFŽ . Ž . Ž .Ž1 x x x 1 1 0E t

q DF DF y2 DF DF .Ž . Ž . Ž . Ž . .1 x 0 1 0 x

5Ž .

The last two equations lead, under the reductionF s0, to the supersymmetric KdV equation con-0

w x Žsidered by the Beckers 32 the so-called sKdV-B.equation .

EF 1 2syF y3D DF . 6Ž . Ž .Ž .1 x x x 1E t

The Lax description for the dispersionless limit ofthe sTB-B hierarchy is quite straightforward, muchlike the dispersionless limit of the sKdV-B hierarchyw x17 . Consider the Lax function

Lspy DF qpy1 DF , 7Ž . Ž . Ž .0 1

where p is the momentum variable of the classicalphase space, satisfying the canonical PB relations

� 4 � 4 � 4x , p s1, x , x s0s p , p . 8Ž .Then, it is easily seen, with the standard canonical

Ž� 4 .Poisson bracket relations p, f syd f d x , thatthe Lax equation

E Lns L , L 9� 4Ž . Ž .G1

E tn

( )A. Das, Z. PopowiczrPhysics Letters A 272 2000 65–73 67

leads to the dispersionless sTB-B hierarchy. Explic-itly, the third flow of this hierarchy is

EF 0 3sD y DF y6 DF DF ,Ž . Ž . Ž .Ž .0 1 0E t

EF 1 2 2sy3D DF q DF DF . 10Ž . Ž . Ž . Ž .Ž .1 1 0E t

Ž . Ž .Eq. 10 allows the reduction F s0 to the disper-0

sionless supersymmetric KdV-B equation

EF 1 2sy3D DF . 11Ž . Ž .Ž .1E t

whose properties we have studied earlier in detailw x17 .

From the Lax description, the conserved quanti-ties of the hierarchy can be easily obtained. In fact,the normalized conserved quantities can be writtenas

nq1y1Ž .nH s d z Res LHn n

n1n nyms C CÝ nym mq1n ms0

=ny2 my1 mq1d z DF DF , 12Ž . Ž . Ž .H 0 1

where d z s d xdu represents the integration overthe superspace and nC represent the standard gener-k

alizations of the combinatoric factors defined to be

n ny1 PPP nykq1Ž . Ž .n nC s , C s1.k 0k!

Explicitly, the first few conserved quantities are

H s d z DF s0,Ž .H1 1

H s d z DF DF ,Ž . Ž .H2 0 1

2H s d z DF q DF DF ,Ž . Ž . Ž .H3 0 1 1

and so on. As it stands, it is clear that these con-served quantities are fermionic. This is a peculiarity

Ž .of the sTB-B system for that matter any -B systemthat the Hamiltonians are fermionic. Correspond-

ingly, the Hamiltonian structures are odd and wenote the first two structures for completeness, namely,

0 1DD s ,1 ž /1 0

Ž . y12 D DF D0DD s .2 y1 y1 y1ž /Ž . Ž . Ž .D DF D D DF Dq D DF D0 1 1

13Ž .

While the Hamiltonian structure DD can be obtained1

from the Lax function as the standard Gelfand-Dikiibracket, we do not know how to obtain the secondstructure from the Lax function. Furthermore, theJacobi identity for this structure is complicated andneeds to be checked. However, these two Hamilto-nian structures do lead to the recursion operator

Rs DDy1 DD1 2

y1 Ž . y1 Ž . Ž . y1D DF D D DF Dq D DF D0 1 1s , 14Ž .y1ž /Ž .2 D DF D0

which can be easily checked to connect the succes-sive Hamiltonians of the hierarchy.

Supersymmetric integrable systems have con-w xserved non-local charges 35,36 and the dispersion-

less sTB-B hierarchy also has conserved non-localcharges. For example, it can be checked that

nq1y1Ž .y1 nQ s d z D Res LŽ .Hn n

n1n nyms C CÝ nym mq1n ms0

=ny2 my1 mq1y1d z D DF DF ,Ž . Ž .Ž .H ž /0 1

15Ž .

with the first few charges

Q s d zF ,H1 1

Q s d z Dy1 DF DF ,Ž . Ž .Ž .Ž .H2 0 1

2 2y1Q s d z D DF DF q DF ,Ž . Ž . Ž .Ž .H ž /3 0 1 1

16Ž .


and so on, are conserved under the flow of thesystem. These are bosonic charges and, interestinglyenough, these non-local charges are also related toone another by the same recursion operator R in Eq.Ž .14 , namely,

d Q dF d Q dFnq1 0 n 0sR . 17Ž .ž / ž /d Q dF d Q dFnq1 1 n 1

It is also nice to see explicitly that if we set F s00

and F sF , then, all the even charges vanish and1

the odd charges, namely, H and Q coincide2 nq1 2 nq1

with the corresponding charges of the sKdV-B sys-w xtem 17 , as they should.

3. Dispersionless limit of sTB equation

In terms of the same basic variables, F and F ,0 1

the sTB hierarchy is described by the Lax operatorw x25,26

LsD2 y DF qDy1F , 18Ž . Ž .0 1

and the non-standard Lax equation

E Lns L, L . 19Ž . Ž .G1

E tn

Explicitly, the third flow has the form

EF 0 6sy D F qD 3F F y6 DF DFŽ . Ž .Ž . Ž0 1 0 x 1 0E t

3y DF q3 DF DF ,Ž . Ž . Ž . .0 0 0 x

EF 1 26 2sy D F y3D F DFŽ .Ž . Ž1 1 0E t

q DF F qF DF . 20Ž . Ž . Ž ..0 1 x 1 1

Ž .The last equation allows the reduction F s0 to0

the supersymmetric KdV equation considered byw xManin and Radul 18–20 , namely,

EF 1 6 2sy D F y3D F DF . 21Ž . Ž .Ž .Ž .1 1 1E t

The dispersionless limit of this Lax operator is notas straightforward. However, with some work, it canbe determined that the Lax function

Lspy DF qpy1 DF ypy2F FŽ . Ž .0 1 0 x 1

qpy3F F 22Ž .1 x 1

and the classical Lax equation

E Lns L , L 23� 4Ž . Ž .G1

E tn

give the dispersionless sTB hierarchy whose thirdflow is

EF 0sD 3F F y6 DF DFŽ . Ž .Ž 1 0 x 1 0E t

3y DF ,Ž . .0

EF 1 22sy3D F DF qF DF . 24Ž . Ž . Ž .Ž .1 0 1 1E t

There are several things to note from here. First,Ž .Eq. 24 , upon setting F s0 and F syF , gives0 1

w xthe dispersionless sKdV equation 17 , namely,

EF2s3D F DF . 25Ž . Ž .Ž .

E t

This, therefore, gives the non-standard representationŽof the dispersionless sKdV equation as opposed to

w x.the standard representation given in 17 , and isanalogous to the reduction of the sTB hierarchy tothe sKdV hierarchy. Second, the normalized con-served quantities of this system can be easily deter-mined to be

nq1y1Ž .y1 nH s d z D Res LŽ .Hn n

mmax nym !Ž .s Ý

m! my1 ! ny2mq1 !Ž . Ž .ms0

=ny2 mq1 my1d z DF DF F , 26Ž . Ž . Ž .H 0 1 1


nwhere the upper limit m s if n is even, butmax 2Ž .n q 1w xm s if n is odd. The first few of thesemax 2

conserved charges have the explicit forms

H s d zF ,H1 1

H s d z DF F ,Ž .H2 0 1

2H s d z DF q DF F , 27Ž . Ž . Ž .Ž .H3 0 1 1

and so on. These conserved charges are all bosonicand it is clear that if we set F s0 and F sF , all0 1

the even charges vanish while the odd charges coin-cide with those of the dispersionless sKdV hierarchyw x17 as they should.

We have not been able to derive the Hamiltonianstructures for this system from the Gelfand-Dikii

Žformalism as is the case in the dispersionless sKdVw x.also 17 . However, the first Hamiltonian structure

can be easily checked to be

0 DDD s 28Ž .1 ž /D 0

and this trivially satisfies the Jacobi identity. Noticethat this Hamiltonian operator defines a closed skewsymmetric two-form

V DD a,b s d z a Db qa Db , 29Ž . Ž . Ž . Ž . Ž .Ž .H1 1 2 2 1

Žwhere a, b are arbitrary, two component column.matrix bosonic superfields. We have also checked

that it is impossible to construct the second Hamilto-nian operator out of local operators alone. However,there is a possibility, which we have not checked, toconstruct such an operator out of local as well asnon-local operators.

Let us now discuss some of the outstanding ques-tions associated with such a system. First, we havenot been able to construct the non-local charges fromthis Lax function. By brute force, we have checkedthat charges, such as

ny1Q s d z D F , ns1,2,3, PPPŽ .Hn 1

2X 3 1y1Q s d z D F q F FŽ .H2 1 0 12 2

y1y D DF F , 30Ž . Ž .Ž .Ž .0 1

and so on, are conserved under the flow. Further-more, under the substitution, F s0 and F sF ,0 1

these reduce to the appropriate non-local charges ofw x Žthe dispersionless sKdV 17 In this limit, Q2

2 X .s Q . However, for lack of a systematic proce-23

dure for constructing these charges, we do not have ageneral expression for the nth charge of this set interms of the Lax function. Furthermore, we do knowthat the dispersionless sKdV has a second set of

w xnon-local charges 17 and since this system can beobtained from the dispersionless sTB, it would benatural to expect the dispersionless sTB also to havea second set of non-local charges. However, we donot know if such a set exists. This emphasizes theneed for a systematic understanding of the non-localcharges in such systems, described a classical Laxfunction.

It is also worth recalling that the sTB equationŽyields the sNLS equation supersymmetric non-linear

. w xSchrodinger equation under the redefinition 25,26¨

y1F sy Dln DQ q D QQ ,Ž .Ž . Ž .Ž .0

F syQ DQ . 31Ž . Ž .1

However, we would like to note here that we havenot been able to find a redefinition of fields whichwould take the dispersionless sTB equation to thedispersionless sNLS equation. In fact, it is worthpointing out that it is not a difficulty only for thesupersymmetric system. Even the dispersionless TB

Ž .equation bosonic does not appear to yield the dis-persionless NLS equation under the correspondingredefinition. This question certainly needs furtherstudy.

4. Ns2 formulation for the dispersionless sTBequation

The sTB hierarchy described by the Lax operatorŽ .18 has a hidden extended Ns2 supersymmetry

w xalso 25–30 . However, this extended supersymmetryis not manifest at the level of the Lax operator. Inorder to have manifest Ns2 supersymmetry, wehave to define the basic variables of the theoryappropriately. Let us recall that in the conventional


Ždescription of the system as given in the previous.section , the basic variables were two fermionic su-

perfields which depended on the usual bosonic coor-Ž .dinates, x,t , but they also depended on an addi-

tional anti-commuting Grassmann variable u . TheTaylor expansion of such superfields in the Grass-mann coordinate is simple and has been given in Eq.Ž .1 . In the description where Ns2 supersymmetry

Ž .is manifest, the superfields depend on x,t as wellas two anti-commuting variables u and u . Expand-1 2

Ž .ing the superfield F x,t,u ,u in a Taylor series in1 2

the anti-commuting variables, we obtain

Fsf qu x qu x qu u f , 32Ž .1 1 1 2 2 2 1 2

Ž .where f ,f are bosonic functions or fermionic1 2Ž .and x ,x are fermionic or bosonic if F is a1 2

Ž .bosonic or fermionic superfield. In this case, thesuper-covariant derivatives on the Ns2 superspaceare defined as

E E E ED s qu , D s qu ,1 1 2 2Eu E x Eu E x1 2

E2 2D sD sEs , D D syD D . 33Ž .1 2 2 1 1 2E x

Recently several methods have been proposed towobtain Ns2 supersymmetric soliton equations 37–

x42 . In this Letter, we consider two different super-symmetric Lax operators which generate two distinctsTB hierarchies. The first is connected with the

w xN s 2,a s 1 37,38,40,41 supersymmetric KdVŽequation namely, it reduces to this model upon

.appropriate redefinition and is defined by the Laxoperator

˜ y1 ˜LsEqF qE D D F , 34Ž .0 1 2 1

˜ ˜ Žwhere F and F are two bosonic superfields and0 1

we have used a tilde to avoid confusion with the.superfields of the earlier sections . This Lax operator

is related to the supersymmetric Lax operator consid-w xered in Ref. 40,41 , through the gauge transforma-

tion

Lseyg Le g , 35Ž .

where gsHd xdu du GsHdZ G with G a bosonic1 2

superfield.

It is straightforward to check that the Lax operatorŽ .34 leads to consistent dynamical quations throughtthe nonstandard Lax relationE L

ns L, L . 36Ž . Ž .G1E tn

˜The third flow leads, upon the reduction F s0, to0w xthe Ns2,as1 37,38 supersymmetric KdV equa-

tionWe can also construct a second Lax operator of

the formy1LsD qF qE D F , 37Ž .1 0 2 1

where F is a fermionic superfield while F is a0 1

bosonic superfield. Note that the Lax operator, in theŽ .present case, is fermionic while that in Eq. 34 was

bosonic. Nonetheless, as in the previous case, it iseasy to check that this Lax operator is gauge equiva-

w xlent to the one considered in 42 . This Lax operatorleads to dynamical equations of the non-standardform

E L2 ns L, L . 38Ž . Ž .G1

E t2 n

The third flow leads, upon the reduction F s0, to0w xthe Ns2,asy2 37,38 supersymmetric KdV

equation.Unfortunately the manifestly Ns2 supersymmet-

ric Lax operator, which gives rise to the sTB hierar-w xchy that contains the Ns2,as4 37,38 supersym-

metric KdV equation is not known as yet. Further-more, since fermionic Lax operators do not leadeasily to a dispersionless limit, we will not discussthis system any further.

In order to obtain the dispersionless sTB hierar-chy, we now have to introduce the concept of thefermionic momenta on the Ns2 superspace. Therewill be two such fermionic momenta defined byw x17,43

P sy p qu p ,Ž .1 u 11

P sy p qu p . 39Ž .Ž .2 u 22

We can now assume the ‘‘commutation’’ rules forthe functions P asi

P ,P sy2 pd . 40� 4 Ž .i j i j

Note that the P ’s generate covariant differentiationiŽ .through the PB relation for any superfield A

� 4P , A s D A , is1,2. 41Ž . Ž .i i


In trying to obtain the dispersionless Ns2 sTBhierarchy, we start from the Lax operator for the

Ž .Ns2 sTB hierarchy in Eq. 34 , and assume the Laxfunction for the dispersionless system to have thegeneral form

˜Lspqk P P qF1 1 2 0

3ys s s s sq p F qP F qP F qP P F ,Ž .Ý 0 1 1 2 2 1 2 3

ss1

42Ž .

where k is an arbitrary coefficient and F s, ks1 k˜0,1,2,3, ss1,2,3 are arbitrary functions of F and0

F̃ .1

We have checked that the classical analogue ofŽ .Eq. 36 , namely,

E Lns L , L , 43� 4Ž . Ž .G1

E tn

with the projection G1 defined as

`

s s s s 3p F qP F qP F qP P FŽ .Ý 0 1 1 2 2 1 2 3ž /ssy` G1

sP F s qP F s qP P F s1 1 2 2 1 2 3

`

s s s s 3q p F qP F qP F qP P FŽ .Ý 0 1 1 2 2 1 2 3ss1

44Ž .

leads to two possible solutions with k s0 in either1

case.The first Lax function that leads to consistent

equations has the form

˜ y1 ˜LspqF qp P P F . 45Ž .0 1 2 1

However, this hierarchy appears trivial since theequations do not have any explicit dependence onsupersymmetric covariant derivatives. Therefore, wewill not consider this hierarchy any further.

The second Lax function does not contain thefermionic functions P and has the formi

˜ y1 ˜ y2 ˜ ˜LspqF qp D D F yp D F D FŽ . Ž . Ž .0 1 2 1 2 1 1 0

y3 ˜ ˜qp D F D F . 46Ž .Ž . Ž .2 1 x 2 1

This, on the other hand, produces an interestingsupersymmetric hierarchy whose first three flowshave the forms

˜ ˜EF EF0 1˜ ˜syF , syF , 47Ž .0 x 1 xE t E t

˜EF 0 2˜ ˜sE 2 D D F qF ,Ž .ž /1 2 1 0E t

˜EF 1 ˜ ˜s2 D D F F , 48Ž .Ž .ž /2 2 1 0E t

˜EF 0 3˜ ˜ ˜sE F q6 D D F FŽ .ž 0 1 2 1 0E t

˜ ˜y3 D F D F , 49Ž .Ž . Ž . /2 1 1 0

˜EF 1 2˜ ˜ ˜ ˜s3D D F F q D F D D F .Ž . Ž . Ž .ž /2 2 1 0 2 1 1 2 1E t50Ž .

˜It is interesting to note that, when F s0, the last0

equation becomes

˜EF 1 ˜ ˜s3D D F D D F , 51Ž .Ž . Ž .ž /2 2 1 1 2 1E t

Ž .which, in fact, reduces to Eq. 25 with the substitu-˜ Ž .tion D F syF . Therefore, Eq. 51 canŽ .2 1 1u s02

be considered as the Ns2 generalization of thedispersionless N s 1 supersymmetric KdV-MRequation.

Ž .Let us also note that the Lax function in Eq. 46as well as the resulting hierarchy coincide with thoseof the previous section with the identifications

˜D F syF ,Ž .2 1 1u s02

F̃ s D F sy DF . 52Ž . Ž . Ž .0 1 0 0u s02

Namely, the redefinition also involves u ™yu or1

equivalently, D ™yD. Notice that these transfor-1

mations are highly nontrivial reductions. Conse-quently, we do not expect the conserved quantities of

Ž .Eqs. 26 to define conserved quantities of the Ns2supersymmetric hierarchy and we have verified this.On the other hand, we can construct conserved quan-tities for the Ns2 hierarchy directly and we have


found two such series of conserved quantities, bybrute force, using the computer.

The first set consists of bosonic conserved chargesof the form

n˜H s dZ F , ns1,2,3,4, . . . ,Ž .Hn 1

˜ ˜H s dZF ,H1 0

˜ ˜ ˜H s dZF F , 53Ž .H2 1 0

where dZ s d xdu du . The second series of con-1 2

served charges is fermionic of the form

˜ ˜H s dZ D F F ,Ž .H5r2 2 1 0

˜ ˜ 2 ˜H s dZ D F F q D D F ,Ž . Ž .H ž /7r2 2 1 0 1 2 1

˜ ˜ ˜ ˜ 3H s dZ D F 3 D D F F qF ,Ž . Ž .H ž /9r2 2 1 1 2 1 0 0

H11r2

˜ ˜ ˜ ˜s dZ 4 D F F F y4 D FŽ . Ž .H ž 2 1 x 1 x 1 2 1

˜ ˜ ˜ ˜ ˜=F F q12 D F D D F FŽ . Ž .1 x x 1 1 1 x 1 2 1 1

˜ ˜ ˜ 2 ˜ ˜ 4q18 D F D D F F q3 D F F .Ž . Ž . Ž . /2 1 1 2 1 0 2 1 0

54Ž .

Here, we have labelled the conserved quantities by˜ ˜Žw x w xthe weights of the integrand F s1s F and0 1

1w x w x.D s s D . Note that the second set of charges1 22

Ž .in Eq. 54 follows from the Lax function, up tonon-essential normalization, as

H s dZ Dy1 Res Ln . 55Ž .Ž .Hnq1 2 1

However, we do not know how to obtain the first setŽ .except for the lowest one from the Lax function,nor is it clear that these exhaust all the conservedcharges of the system.

Comparing with the discussion of the previoussection and particularly from the form of the non-lo-

Ž .cal conserved charges in Eq. 30 , we see that we

can write non-local conserved charges for the Ns2system as

ny1˜ ˜Q s dZ D D F , ns1,2,3,4 PPP 56Ž .Ž .Hn 1 2 1

It is, in fact, quite straightforward to check that theyare conserved. Similarly, we note that

˜X y1 ˜ ˜Q s dZ D F D FŽ .H ž /ž2 1 0 2 1

1 3y1 2˜ ˜ ˜y D F D F y F , 57Ž .Ž . Ž . /1 0 2 1 12 2

Žalso represents a conserved charge compare with theŽ ..second of the charges in Eq. 30 . However, we do

not know how to obtain these from the Lax functiondirectly.

The first Hamiltonian structure for the dispersion-Ž . Ž .less Ns2 hierarchy of Eqs. 47 – 56 is easily seen

to be

0 D2D̃D s 58Ž .1 ž /yD 02

and is trivially seen to satisfy the Jacobi identity.This Hamiltonian operator defines the closed skewsymmetric two-form

˜V DD a,b s dZ a Db ya Db , 59Ž . Ž . Ž . Ž .Ž .Ž . H1 1 2 2 1

where in contrast to the Ns1 case a and b arearbitrary, two component fermionic superfields.

5. Conclusion

In this Letter, we have studied the dispersionlesslimits of the sTB-B, the sTB as well as the Ns2sTB hierarchies in detail. We have obtained the Laxdescriptions in terms of classical Lax functions, ob-tained conserved local as well as some of the non-lo-cal charges and brought out various other featuresassociated with such systems. We have also tried topoint out various open questions associated withsuch systems, the most pressing of which is a sys-tematic understanding of the construction of non-lo-cal charges for such systems starting from the Laxdescription as well as a generalization of theGelfand-Dikii procedure for construction of Hamilto-nian structures for such systems.


Acknowledgements

A.D. acknowledges support in part by the U.S.Dept. of Energy Grant DE-FG 02-91ER40685 whileZ.P. is supported in part by the Polish KBN Grant 2P0 3B 136 16.

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dispersionless stb

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