Volume 153, number4,5 PHYSICSLETTERSA 4 March 1991

Thezerocurvatureformulationof theBoussinesqequation

Ashok Das, Wen-JuiHuangand Shibaji RoyDepartmentofPhysicsandAstronomy,UniversityofRochester,Rochester,NY14627,USA

Received30 July 1990;revisedmanuscriptreceived11 December1990; acceptedforpublication 18 December1990Communicatedby A.R. Bishop

We derivetheBoussinesqequationfrom thezerocurvatureconditionassociatedwith thegroupSL(3, P). Wealsopresentaconnectionbetweenthisformulationandtheonebasedonacombinedconformalandspin-3flow.

Integrablemodels[1—8]havebeenof considerableinterestin recentyears.ThesearenonlinearHamiltonianmodels— bothfinite dimensionalaswell ascontinuumsystems— which canbesolvedexactly. It is quitewellknownthat theseintegrablemodelshaveverycloseconnectionwith the two-dimensional(2-d) conformalfieldtheory[9]. The energy—momentumtensorin a conformalfield theorycanbe identifiedwith theKdV variablewhich satisfiestheVirasoroalgebra.Thisconnectionbetweenthe Virasoroalgebraandthe KdV equationwasfirst noticedby Gervais [10]. Theintroductionof higherspinobjectsin a conformallyinvariantfield theorygivesriseto additionalsymmetriesin thesystem.In fact,Zamolodchikov[11] pointedoutthat thereare in-finite additionalsymmetriesassociatedwith 2-dconformalfield theory.Forexample,the introductionofspin-3 objectgives riseto Z3 extendedconformalalgebra[11,12] or spin-3 algebraandthis is thefirst non-trivialbosonicextensionoftheVirasoroalgebra.Justlike theVirasoroalgebrais relatedwith theKdV equation,spin-3 algebrais relatedwith anothernonlinearintegrableequationcalledthe Boussinesqequation [13]. Thus aconformal field theorycanhavesymmetriesin additionto the Virasorosymmetryandit is quite importantto studytheseadditional symmetriesto obtain a full classificationof suchtheories.

Integrablemodelshavebeenstudiedboth from thegeometrical [14—17]point of view as well as from thegrouptheoretical[18—20]pointof view. Thefundamentalconceptin thegrouptheoreticalapproachis the for-mulationof the dynamicalequationasa zerocurvaturecondition [8,21,22] associatedwith somesymmetrygroup.Thus, for example,it is well knownthat mostof the (1 + 1)-dimensionalintegrablemodelssuchastheKdV equation,the mKdV equation,the sine-Gordonequation,the sinh-Gordonequationandthe nonlinearSchrodingerequation[8] canbeformulatedasa zerocurvatureconditionassociatedwith thegroup SL( 2, P).Sucha formulationnaturallybringsout theentirehierarchyof equationsassociatedwith a givensystem.Fur-thermore,it alsoleadsto the recursionrelationbetweentheconservedchargesof thesystemwhich is essentialin proving the integrabilityof the system.

TheBoussinesqequation[13,23,24] hasbeenstudiedquiteextensivelyin thepast.AlthoughtheBoussinesqequationandits relationtothe SL(3, P)groupis implicitly contained[25—281in the2-d periodicTodalatticeequationin the three-componentform, an explicit connectioncomesout in the zero curvatureformulation.In a recentpaper [29], we showedhow the Boussinesqhierarchycanbe obtainedfrom a combinedflow ofconformalandspin-3 transformations.This analysissuggestedthat the symmetrygroup associatedwith thezerocurvatureformulation of theBoussinesqequationis more likely to bethe groupSL(3,P). In thisLetter,weshowthat theentireBoussinesqhierarchyaswell astheassociatedrecursionrelationbetweentheconservedchargescan,indeed,beobtainedfrom the zero curvatureconditionassociatedwith thegroup SL( 3,P). Thisshowsa direct relationshipbetweenthe group SL(3, P) andthe W3 algebraof Zamolodchikov.

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Volume153, number4,5 PHYSICSLETTERSA 4 March 1991

TheBoussinesqequationcanbedescribedasa setof two first orderequations[19] in termsof thedynamicalvariablesq(x, t) and W(x, t) as

q1=~W~,~ (1)

Hereandin whatfollows, the subscriptsx and I denotedifferentiationwith respectto thosevariables.Let ussimply note herethat the scalingdimensionsof variousquantitiesin eq. (1) canbe determinedto be

[x] = 1, [I] =2= [q], [W] =3. (2)

Letusnextconsiderapairof gaugepotentialsbelongingto thegroupSL( 3, P). Becausex andI havedifferentscalingdimensions,differentcomponentsof thegaugepotentialswould correspondinglycarrydifferentscalingdimensions.Exploiting this,we parameterizethetwo gaugepotentialsbelongingto SL(3, P) as

/D(x,I) E(x,t) F(x,t)

A0(x,I)= ( G(x,1) H(x,t) I(x,t) J (3)\J(x, I) K(x, I) — [D(x, I) +H(x, I)J/

and

/2 Q(x,I) R(x,t)\A1(x,t)=(4 A _612 I (4)

—221

WehaveintroducedheretheconstantspectralparameterA whosescalingdimensionisunity. We alsonote herethat the scalingdimensionsof the matrix elementsincreaseacrossthe rowsanddecreasealongthe columnsby a unit of one.Wewould like to mentionherethatsix of theeightcomponentsof thegaugefieldsA1 (x) arefixed sothat the remaininggaugefreedomproducesthediffeomorphismandspin-3transformations.This be-comesclear from the equationsof motion of the dynamicalvariablesin eq. (10).

The curvatureassociatedwith the two potentialsin eqs.(3) and(4) cannow be constructedfrom (thereis only onecomponentof thecurvaturein 1+ 1 dimensions)

F01 =A11—A0~—[A0,A~]. (5)

Requiringthe curvatureto vanishwould leadto the constraintequations

—D~—4E+GQ+JR=0, —G~—4(H—D)—622J=0,

—H~+4E--3I—GQ—6A2K=0, —I~+32I+622(D+2H)—GR+4F=0,

—J~—W—4K+~G=0, —K~—31K+~(D+2H)—JQ=0 (6)

andthe two dynamicalequations

Q, =E. + (D—H)Q+ IF—KR, R,=F~— 32F—622E+(2D+H)R—JQ. (7)

The constraintequationsof (6) canbe solvedin termsof the two dynamicalvariablesQ andR andtwootherfunctionsJ andG,

D= ~

~

~ ~22J~+ ~

H= —~J~—L1~—222J+~2G+~JQ,

~

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Volume 153,number4,5 PHYSICSLETTERSA 4 March 1991

K=—~J,~—~Af+~G. (8)

If we now identify

C(x, t)=~(G—J~—4AJ), B(x, t)=~J, q(x, t)=—Q, W(x,t)=2(Q~—R—4AQ), (9)

then it canbeshownin a straightforwardmannerfrom eq.(7) that theevolution equationssatisfiedby q andW are

~

~ ~ (10)

This canalso be written in matrix form as

(~)=(K_313M)(~), (11)

wherethe 2x 2 matricesK andM havethe form

l8~ 0 8

K= ~ 2 8~ 8~ 3W~-+2W~ a3W~—+W~ ~

(12)

and

M—( 0 8/Ox‘~8/0x 0

Thisshowsthat the newly definedquantitiesproducethe correctevolutionequationfor the Boussinesqhi-erarchy.Let us note herethat if we choose

C=0, B=~, (13)

theneq. (10) or eq. (11) will give the Boussinesqequationsof eq. (1). In general,however,if we make apowerseriesexpansionof the form (n is a given positive integer)

C(x, t) = ~ A3~’~~C~•(q,W), B(x, t) = A3~~~B1(q,W) (14)

andidentify

C— B— (15)‘ 8q(x,I)’ ‘ 6W(x,t)’

whereH1 is the Hamiltoniancorrespondingto thejth flow in the Boussinesqhierarchy,theneq. (11) gives[29]~ —0 1 n—i (16

\oJI~./SW(x,I)) — k,8H~+1/~W(x,t))’ ~ ‘ ~

and

(qt \ —K( öH~/öq(x,t)’\ (17)

— ~8H~/8W(x, t),1188

Volume 153, number4,5 PHYSICSLETTERSA 4 March 1991

Note herethat eq. (16) definesthe recursionrelationsbetweenthe conservedquantitiesof the Boüssinesqhierarchywhereaseq. (17)givesthe nthflow of thehierarchy.KandM arethetwo Poissonbracketstructuresassociatedwith this system.

In writing eq. (10) we haveusedthe notationof ref. [29], whereit wasshown that the functionsC(x, I)andB(x, I) are relatedto the one-parameterspin-2 andspin-3 flows respectively.We are now in a positionto establisha connectionbetweenthe zero-curvatureformulation of the Boussinesqhierarchybasedon thegroup SL( 3, P) and the formulation basedon the spin-2 and spin-3 flows. Namely,we note that given thedynamicalvariablesq(x, t) and W(x, 1) as well as the flow functions C(x, I) andB(x, I), we canconstructthe gaugepotentialsof SL(3, P) (seeeqs.(3) and (4)) with the matrix elements

D=C~+AC+~

~

F= —1C~~~+ ~ACXX— ~CW—Cq~+4ACq+ ~ +AB~~~+2A2B~~+6A3B~—3B~W—2BW~+2ABW

~ +~Bq~+ 8AB~q+42Bq~+ 8A2Bq,

G=4(C+2B~+8AB), ~

1= ~

J=8B, K=~C—B~—2AB, Q=—q, R=—~W—q~+4Aq, (18)

suchthat requiringthe curvatureto vanishwould leadtothe equationsof theBoussinesqhierarchyaswell asthe recursionrelationbetweenthe conservedquantitiesof the system.

This showsthat the Boussinesqhierarchyas well as the relevantrecursionrelationbetweenthe conservedquantitiescanbe obtainedfromthe zerocurvatureconditionassociatedwith thegroupSL(3, P). Wehavealsogivena connectionbetweenthis formulation andthe onebasedon spin-2 andspin-3 flows.

Finally, weconcludewith the following observation.Recently,Polyakov[30] hasshownhow to get adif-feomorphismfrom the restrictedSL(2,P) gaugegroup (the sameas the gaugefixing conditionin the zerocurvatureformulation). Thisclarifies theorigin of the SL( 2, P) groupin 2-d gravity. Similarly, the zerocur-vature formulation of theBoussinesqequation (or higherorderequation)associatedwith theSL(3, R) (orhigher) group would clarify the relationbetweenthe integrablemodelsandW-gravity[31,32]. Anotherin-terestingmotivation of studyingthesesystemsis to understand2-d gravity in c (wherec is the centralchargeof the conformal field theory) greaterthanoneregime.Thesequestionsare presentlyunderstudy.

This work wassupportedin partby U.S. DOE contractno. DE-ACO2-76ER13065.

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