Physics Letters A 302 (2002) 87–93

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Alternative dispersionless limit ofN = 2 supersymmetricKdV-type hierarchies

Ashok Dasa, Sergey Krivonosb, Ziemowit Popowiczc,∗

a Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171, USAb Bogoliubov Laboratory of Theoretical Physics, JINR 141980, Dubna, Moscow region, Russia

c Institute of Theoretical Physics, University of Wrocław pl. M. Born 9, 50-204 Wrocław, Poland

Received 30 April 2002; received in revised form 30 July 2002; accepted 1 August 2002

Communicated by A.P. Fordy

Abstract

We present a systematic procedure for obtaining the dispersionless limit of a class ofN = 1 supersymmetric systems startingfrom the Lax description of their dispersive counterparts. This is achieved by starting with anN = 2 supersymmetric systemand scaling the fields in an alternative manner so as to maintainN = 1 supersymmetry. We illustrate our method by workingout explicitly the examples of the dispersionless supersymmetric two boson hierarchy and the dispersionless supersymmetricBoussinesq hierarchy. 2002 Published by Elsevier Science B.V.

1. Introduction

Dispersionless integrable systems [1] have beenstudied, in recent years, from various points of view.While the properties of such bosonic models are quitewell understood, much remains to be learnt about theirsupersymmetric counterparts. For example, the Laxdescription of only a handful of supersymmetric dis-persionless systems have been constructed, to date,by brute force [2,3] and there is no systematic proce-dure for obtaining them starting from the correspond-ing supersymmetric dispersive Lax descriptions [4–7].Similarly, supersymmetric dispersionless systems of-ten have more conserved charges than their dispersive

* Corresponding author.E-mail address:ziemek@ift.uni.wroc.pl (Z. Popowicz).

counterparts and we do not yet know how to relate thenew charges to the Lax function itself.

In this Letter, we take a modest step and show howone can obtain, systematically, a Lax description for aselect class of supersymmetric dispersionless systemsstarting from their dispersive counterparts. This ances-tor N = 2 supersymmetric systems are described interms of the Lax operators that involve only “boson-ic” operators (∂). We define in Section 2 the basicprocedure for taking the dispersionless limit in theLax description itself, within the context of bosonicmodels. We work out some known examples to illus-trate the procedure and present the Lax description ofsome new bosonic models. In Section 3, we extend thismethod to a class of supersymmetric models. We workout explicitly the example of dispersionless supersym-metric two boson hierarchy starting from theN = 2

0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0375-9601(02)01087-3

88 A. Das et al. / Physics Letters A 302 (2002) 87–93

supersymmetric KdV hierarchy. In Section 4, we ex-tend the analysis and discuss the dispersionless super-symmetric Boussinesq hierarchy and present a briefconclusion in Section 5.

2. Bosonic dispersionless systems

In this section, we describe the basic procedureof taking the dispersionless limit of bosonic systemswithin the framework of the Lax description itself. Letus consider a general Lax operator of the form

(1)L= ∂n +∞∑m=1

Am∂n−m

where the coefficients,Am’s, are functions of thedynamical variables of the system which depend onthe coordinates(x, t). Let us assume that the Laxequation

(2)∂L

∂tk= [(

Lk)�s ,L

], s = 0,1,2,

describes the dynamical system of equations. Here( )�s represents the projection with respect to thepowers of∂ . In going to the dispersionless limit, firstof all, we replace∂ → p. Then, we scalep → αp

and all the basic dynamical variables asJi → (α)iJi ,whereJi represents the basic dynamical variables ofthe system with the respective dimensionsi. The Laxfunction which describes the dispersionless system ofequations is obtained to be [1]

(3)L = limα→∞

1

αnLα

whereLα denotes the scaled Lax function. The dis-persionless equations are then obtained from the Laxequation

(4)∂L∂tk

= −{(Lk)�s,L}where

(5){A,B} = ∂A

∂x

∂B

∂p− ∂A

∂p

∂B

∂x

represents the Poisson bracket on the classical phasespace.

Let us illustrate this procedure with a few exam-ples. The reduction of the KdV equation to its disper-sionless limit is well-known and, therefore, we will not

repeat it here. Rather, let us look at the two boson hi-erarchy [8] described by the Lax operator

(6)L= ∂ − J + ∂−1T

with the nonstandard Lax equation given by

(7)∂L

∂tk= −[(Lk)�1,L

].

Here J and T are dynamical field variables withdimensions one and two respectively. Therefore, underthe scaling discussed earlier, we have for the presentcase,p→ αp, J → αJ , T → α2T and it follows thatin the dispersionless limit, the Lax operator goes into

(8)L= p− J + Tp−1

and the Lax equation

(9)∂L∂tk

= {(Lk)�1,L

}describes the dispersionless system of equations.

The two boson hierarchy [8] can also be alterna-tively described in terms of the gauge equivalent Laxoperator [9]

(10)L= ∂ − 1

∂ + J

(T

2

)and the standard Lax equation

(11)∂L

∂tk= [(

Lk)�0,L

].

This description of the system of equations is moreconvenient from the point of view of our subsequentdiscussions. We note that the second flow

(12)Jt2 = (

Jx + T − J 2)x, Tt2 = −Txx − 2(JT )x

is the two boson equation which is also related to thenonlinear Schrödinger equation (NLS) [9]. The thirdflow, on the other hand, is obtained to be

Jt3 = Jxxx + (J 3 − 3JT − 3JJx

)x,

(13)Tt3 = Txxx + 3

(J 2T − 1

2T 2 + JTx

)x

and this coincides with the bosonic sector of theN = 2supersymmetric KdV equation witha = 4 [4] after thetransformations

J → 2J, T → −2(T + Jx),

(14)x → ix, t → it.

A. Das et al. / Physics Letters A 302 (2002) 87–93 89

In the present case, it is easy to check that, in thedispersionless limit, the Lax function becomes

(15)L = p− 1

p+ J

(T

2

).

The second and the third flow equations followingfrom the Lax equation

(16)∂L∂tk

= −{(Lk)�0,L}

are given by

(17)Jt2 = (T − J 2)

x, Tt2 = −2(JT )x,

(18)

Jt3 = (J 3 − 3JT

)x, Tt3 = 3

(J 2T − 1

2T 2)x

.

These are indeed the correct dispersionless limits ofthe two boson hierarchy.

We would like to conclude this section by present-ing a new system of dispersionless equations. Let usconsider a general Lax operator of the form [6]

(19)L= ∂ − 1

∂m +∑mi=1Ji∂

m−i

(m∑i=1

J̄i∂m−i

).

HereJi, J̄i represent fields of dimensioni. It can bechecked that

(20)∂L

∂tk= [(

Lk)�0,L

]leads to a consistent set of dynamical equations. If wenow follow the earlier discussion, it is easy to checkthat, in the dispersionless limit, the Lax function

(21)L = p− 1

pm +∑mi=1Jip

m−i

(m∑i=1

J̄ipm−i

)

leads to the system of dispersionless equations ob-tained from the Lax equation

(22)∂L∂tk

= −{(Lk)�0,L}.

3. Supersymmetric KdV and two boson hierarchy

In this section, we will generalize the ideas of theprevious section to construct the Lax description fora class of dispersionless supersymmetric systems. In

particular, we will work out in detail the case of dis-persionlessN = 1 supersymmetric two boson hierar-chy starting from the Lax description of theN = 2supersymmetric KdV hierarchy witha = 4 [6] whosebosonic sector we have studied in the last section. Letus first note that the few dispersionless supersymmet-ric systems [2] whose Lax descriptions have been con-structed by brute force show that although the Laxfunction is defined in terms of superfields, it involvesonly bosonic momenta and that the conserved chargesare obtained from the bosonic residues of powers ofthe Lax function. Furthermore, in the dispersionlesslimit, we know that∂ → p. However, the reduction ofthe fermionic covariant derivative in the dispersionlesslimit is not well understood. It is also already noted [2]that a scaling of the fermionic covariant derivative isessential in order to preserve supersymmetry in go-ing to the dispersionless limit. In view of the abovementioned difficulties, our strategy, as a first step, is tolook at supersymmetric systems which are describedin terms of Lax operators that involve only bosonic∂operators.

For one of the three known families of inte-grable supersymmetric hierarchies withN = 2,Wn su-peralgebra as the second Hamiltonian structure, theLax operators contain only bosonic operators of theforms [6]:

(23)

Ls = ∂ −[D

1

∂s +∑si=1 Ji∂

s−i D(

s∑i=1

Ji∂s−i)].

Here,s = 0,1,2, . . . andJi are bosonicN = 2 super-fields of dimensionsi. Furthermore, the square brack-ets stand for the fact that theN = 2 supersymmetricfermionic covariant derivativesD andD defined to be

(24)D = ∂

∂θ− θ̄

2∂, D = ∂

∂θ̄− θ

2∂

act only on the superfields inside the brackets.Let us consider the conventional dispersionless

limit in the simplest case ofN = 2 supersymmetricKdV hierarchy for whichs = 1. The Lax operator, inthis case, has the form

(25)L= ∂ −[D

1

∂ + JDJ].

90 A. Das et al. / Physics Letters A 302 (2002) 87–93

The second flow, following from this Lax operator,reads

(26)Jt2 = ([D, D ]J − J 2)

x.

Under the rescaling∂t → λ∂t , ∂ → λ∂ , (D, D) →λ1/2(D, D), this equation will reduce to (asλ→ 0)

(27)Jt2 = −(J 2)x.

However, this equation, despite beingN = 2 super-symmetric, is not very interesting.

A different possibility is to rescale in a standardway all the fields together with the fermionic deriva-tives in the Lax operator (25). This leads to the fol-lowing Lax function in the dispersionless limit:

(28)L = p−12T

p+J −12ψ1ψ2

(p+J )2 ,

where we introduced the component fields:

J = J |, ψ1 = (D + D )J |, ψ2 = (

D− D )J |,(29)T = [

D, D ]J |,and the restriction| stands for keeping the(θ = θ̄ = 0)term. One can check, that the second flow equations

Jt2 = (T −J 2)

x, Tt2 = −2(JT −ψ1ψ2)x,

(30)(ψ1)t2 = −2(Jψ1)x, (ψ2)t2 = −2(Jψ2)x,

do not possess any supersymmetry at all. They breakeven theN = 1 supersymmetry. The same is also truefor higher flows.

Therefore we propose the following alternative ap-proach to taking the dispersionless limit. The mainidea is to rescale the fermionic components of the su-perfieldsJi differently from the conventional methodas

Ji | → αiJi |,(D + D )Ji | → αi

(D + D )Ji |,(

D − D )Ji | → αi+1(D − D )Ji |,(31)

[D, D ]Ji | → αi+1[D, D ]Ji |.

It is clear that this unconventional, alternative rescal-ing (31) will explicitly break theN = 2 supersymme-try. However, a subset ofN = 1 supersymmetry, gen-erated by(D+ D) will survive and, in the dispersion-less limit, we will have a Lax description for anN = 1supersymmetric system of equations.

Let us demonstrate in detail how all these work forthe Lax operator (25). According to our alternative

procedure, the first step will consist of representing1

∂+J as

(32)

1

∂ + J≡ ∂−1 +A2∂

−2 +A3∂−3 +A4∂

−4 + · · · ,

where all the functionsAn can be recursively calcu-lated and the first few have the explicit forms

A2 = −J, A3 = J 2 + Jx,

(33)A4 = −J 3 − 3JJx − Jxx, . . . .

Thus, our Lax operator can also be written as

(34)

L= ∂ − [D(∂−1DJ +A2∂

−2 DJ +A3∂−3DJ

+A4∂−4DJ + · · ·)],

and we should move the partial derivatives to the rightin (34).

The first non trivial term on the right hand sideof (34) generates an infinite series of terms when thederivative is moved to the right, namely,

∂−1[DDJ ]≡ 1

2(T −Jx)∂−1 − 1

2(T −Jx)x∂−2

(35)+ 1

2(T −Jx)xx∂−3 + · · · .

We may now replace∂ → p in the r.h.s. of (35) andrescale

p→ αp, J → αJ , ψ1 → αψ1,

ψ2 → α2ψ2, T → α2T .Then, it is easy to see that the only term from amongthose in (35) that will contribute to limα→∞ 1

αLα is

1

2T p−1.

The second term inside the square bracket in the righthand side of (34) needs some more work:

(DA2)∂−2(DJ )+A2∂

−2(DDJ )≡ (DA2)

(DJ )∂−2 − 2(DA2)(DJ )

x∂−3

(36)+A2

(DDJ )∂−2 − 2A2

(DDJ )

x∂−3 + · · · ,

where the dots stand for terms with∂−4 and higher. Inthe scaling limit, only the following terms will survive

(37)1

2ψ1ψ2p

−2 − 1

2ψ2(ψ2)xp

−3 − 1

2J T p−2.

A. Das et al. / Physics Letters A 302 (2002) 87–93 91

Continuing in a similar manner, we find the Laxoperator in the dispersionless limit to be

(38)L = p−12T

p+J −12ψ1ψ2

(p+J )2 +14ψ2(ψ2)x

(p+J )3 .

The second flow, following from the Lax equation, hasthe form

Jt2 = (T −J 2)

x, Tt2 = −2(JT −ψ1ψ2)x,

(39)(ψ1)t2 = (

(ψ2)x − 2Jψ1)x, (ψ2)t2 = −2(Jψ2)x .

These equations can also be easily rewritten in termsof N = 1 superfields,

(40)j = J + θψ1, ψ = ψ2 − θT

as

(41)jt2 = −(Dψ + j2)x, ψt2 = −2(jψ)x,

where

D = ∂

∂θ− θ∂, D2 = −∂.

Let us note here that the Lax operator (38) is notnew and is gauge equivalent to the one which has beenconstructed earlier by brute force in [3]. However, wesee that it can be systematically obtained from thealternative dispersionless limit of the simplest of theLax operators in the family (23).

4. Supersymmetric Boussinesq hierarchy

As a second example of our method, in this section,we will work out the dispersionless limit starting fromtheN = 2 supersymmetric Boussinesq hierarchy withα = 5

2 [7], which is described by the Lax operator (23)with s = 2 [6]. The Lax operator (23), in this case, hasthe explicit form

(42)L= ∂ −[D

1

∂2 + J1∂ + J2

D(J1∂ + J2)

].

Following our procedure, we will first rewrite

(43)1

∂2 + J1∂ + J2= ∂−2 +A1∂

−3 +A2∂−4 + · · · ,

where all theAn’s can be easily calculated,

A1 = −J1, A2 = −J2 + 2(J1)x + J 21 ,

(44)

A3 =(

2J2 − 3(J1)x − 5

2J 2

1

)x

+ 2J1J2 − J 31 , . . . .

With this, the first few terms in the square bracketin (42) have the form[D

1

∂2 + J1∂ + J2

D(J1∂ + J2)

]

(45)

= [D∂−2 D(J1∂ + J2)

]+ [DA1∂

−3D(J1∂ + J2)]

+ [DA2∂

−4D(J1∂ + J2)]

+ [DA3∂

−5D(J1∂ + J2)]+ · · · .

Let us next introduce the components

J1 = J1|, ψ1 = (D + D )J1|,

ψ2 = (D − D )J1|, T1 = [

D, D ]J1|,T2 = J2|, ξ1 = (

D+ D )J2|,(46)ξ2 = (

D − D )J2|, W = [D, D ]J2|,

which have the scaling behaviors:

(J1,ψ1)→ α(J1,ψ1),

(ψ2,T1,T2, ξ1)→ α2(ψ2,T1,T2, ξ1),

(47)(ξ2,W)→ α3(ξ2,W).

We are now ready to find a Lax function in thedispersionless limit.

We can now have the fermionic derivatives act onthe fields in (45), move the partial derivatives to theright and replace∂ → p. After this, it is easy to seethat there will be three types of terms that may survivein the limit (3):

L≡ p−A−B − C,

(48)

A ≡ 1

2(T1p+W)

(p−2 +A1p

−3 +A2p−4 + · · ·),

B ≡ ((DA1)p

−3 + (DA2)p−4 + · · ·)

(49)× ((DJ1)p+ (DJ2

)),

C ≡ (−3(DA1)p−4 − 4(DA2)p

−5 + · · ·)(50)× ((DJ1

)p+ (DJ2

))x.

Note that the expressions in the parenthesis forA,B,Ccontain terms with and without derivatives (see

92 A. Das et al. / Physics Letters A 302 (2002) 87–93

Eq. (44)). For terms of the typesA and C, there isno problem, since in the dispersionless limit (scalinglimit), only terms without derivatives inAn (44) con-tribute. In this case, we have:

A =12(T1p+W)

p2 +J1p+ T2,

(51)

C = −[D

2p+ J1

(p2 + J1p+ J2)2

]((DJ1)p+ (DJ2

))x.

However, for terms of the typeB, the scaling requireus to keep also the terms linear in the first derivativesin all theAn’s. This leads to

B =[D

(1

p2 + J1p+ J2+ (2p+ J1)(J1p+ J2)x

(p2 + J1p+ J2)3

)](52)× ((DJ1

)p+ (DJ2

)).

In the dispersionless limit, the Lax function nowbecomes

L= p−12(T1p+W)

p2 +J1p+ T2−

14ψ2(ψ2p+ ξ2)x

(p2 +J1p+ T2)2

−12(ψ1p+ ξ1)(ψ2p+ ξ2)

(p2 +J1p+ T2)2

+14(2p+J1)(ψ2p+ ξ2)(ψ2p+ ξ2)x

(p2 +J1p+ T2)3

(53)+14(J1p+ T2)xψ2ξ2

(p2 +J1p+ T2)3.

It is now easy to check that the Lax equation (22)leads to the dispersionless supersymmetric Boussinesqhierarchy. Explicitly, the second flow of this hierarchyis given by

(J1)t2 = (2T1 + 2T2 −J 2

1

)x,

(T1)t2 = 2(W −J1T1 +ψ1ψ2)x,

(T2)t2 = −2(J1)xT2 +J1(T1)x,

(W)t2 = −2(J1)xW − 2(T1)xT2 + T1(T1)x

+ψ2(ψ2)xx + 2(ξ1(ψ2)x − ξ2(ψ1)x

),

(ψ1)t2 = 2(ξ1 + (ψ2)x −J1ψ1

)x,

(ψ2)t2 = 2(ξ2 −J1ψ2)x,

(ξ1)t2 = −2(J1)xξ1 − 2T2(ψ1)x + (T1)xψ1

+J1(ψ2)xx,

(54)(ξ2)t2 = −2(J1)ξ2 − 2T2(ψ2)x + (T1)xψ2.

This system of equations can also be rewritten interms ofN = 1 superfields

j1 = J1 + θψ1, η1 =ψ2 − θT1,

(55)j2 = T2 + θξ1, η2 = ξ2 − θW

as

(j1)t2 = (−2Dη1 + 2j2 − j21

)x,

(η1)t2 = 2(η2 − j1η1)x,

(j2)t2 = −2(j1)xj2 − j1(Dη1)x,

(56)(η2)t2 = −2(j1)xη2 − 2j2(η1)x − (Dη1)xη1.

Thus, we explicitly demonstrate that our system pos-sessesN = 1 supersymmetry.

5. Conclusion

In this Letter, we have given a systematic derivationof the dispersionless limit of a class ofN = 1 super-symmetric models starting from the Lax descriptionof their dispersive counterparts. This is achieved bystarting with theN = 2 systems and making an alter-native scaling of the field variables which maintainsonly anN = 1 supersymmetry. This approach is moti-vated by the structure of the dispersionless limit of thepure bosonic sectors of theseN = 2 systems whichcannot be extended toN = 2 supersymmetry withoutintroducing the additional bosonic fields. We discussour proposal explicitly within the context of the su-persymmetric two boson hierarchy where our startingpoint is theN = 2 supersymmetric KdV hierarchy. Asa second example, we also work out the dispersion-less limit of the supersymmetric Boussinesq hierarchystarting from theN = 2 supersymmetric system.

Acknowledgements

S.K. and Z.P. would like to thank the Departmentof Physics at the University of Rochester for hospi-tality during the course of this work. This work wassupported in part by US DOE grant no. DE-FG-02-91ER40685, NSF-INT-0089589, INTAS grant No. 00-00254, grant DFG 436 RUS 113/669 as well as RFBR-CNRS grant No. 01-02-22005.

A. Das et al. / Physics Letters A 302 (2002) 87–93 93

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