Lagrangian and Hamiltonian structures in an integrable hierarchy and space-time duality

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Avan, J., Caudrelier, V., Doikou, A. & Kundu, A. (2016). Lagrangian and

Hamiltonian structures in an integrable hierarchy and space-time duality. Nuclear Physics B,

902, pp. 415-439. doi: 10.1016/j.nuclphysb.2015.11.024

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Nuclear Physics B 902 (2016) 415–439

www.elsevier.com/locate/nuclphysb

Lagrangian

and

Hamiltonian

structures

in

an

integrable

hierarchy

and

space–time

duality

Jean Avan

_,

_{Vincent Caudrelier}

_,

_{Anastasia Doikou}

_,

_{Anjan Kundu}

F-95302 Cergy-Pontoise,France

b_{DepartmentofMathematics,CityUniversityLondon,NorthamptonSquare,EC1V0HBLondon,UnitedKingdom} c_{DepartmentofMathematics,Heriot-WattUniversity,EH144AS,Edinburgh,UnitedKingdom}

d_{SahaInstituteofNuclearPhysics,TheoryDivision,Kolkata,India}

Received 9October2015;receivedinrevisedform 23November2015;accepted 25November2015 Availableonline 27November2015

HubertSaleur

Abstract

Wedefineandillustratethenovelnotionofdualintegrablehierarchies,ontheexampleofthenonlinear Schrödinger(NLS)hierarchy.Foreachintegrablenonlinearevolutionequation(NLEE)inthehierarchy, dualintegrablestructuresarecharacterizedbythefactthatthezero-curvaturerepresentationoftheNLEE canberealizedby twoHamiltonianformulationsstemmingfromtwodistinctchoicesofthe configura-tionspace,yieldingtwoinequivalentPoissonstructuresonthecorrespondingphasespaceandtwodistinct Hamiltonians.Thisisfundamentallydifferentfrom thestandardbi-Hamiltonian orgenerallymultitime structure.Thefirstformulationchoosespurelyspace-dependentfieldsasconfigurationspace;ityieldsthe standardPoissonstructureforNLS.Theotheroneisnew:itchoosespurelytime-dependentfieldsas con-figurationspaceandyieldsadifferentPoissonstructureateachlevelofthehierarchy.Thecorresponding NLEEbecomesaspaceevolutionequation.WeemphasizetheroleoftheLagrangianformulationasa uni-fyingframeworkforderivingbothPoissonstructures,usingideasfromcovariantfieldtheory.Oneofour mainresultsistoshowthatthetwomatricesoftheLaxpairsatisfythesameformofultralocalPoisson algebra(uptoasign)characterizedbyanr-matrixstructure,whereastraditionallyonlyoneofthemis in-volvedintheclassicalr-matrixmethod.WeconstructexplicitdualhierarchiesofHamiltonians,andLax representationsofthetriggereddynamics,fromthemonodromymatricesofeitherLaxmatrix.An

appeal-* _{Correspondingauthor.}

E-mailaddresses:Jean.Avan@u-cergy.fr(J. Avan), v.caudrelier@city.ac.uk(V. Caudrelier), A.Doikou@hw.ac.uk (A. Doikou), Anjan.Kundu@saha.ac.in(A. Kundu).

http://dx.doi.org/10.1016/j.nuclphysb.2015.11.024

ingproceduretobuildamulti-dimensionallatticeofLaxpair,throughsuccessiveusesofthedualPoisson structures,isbrieflyintroduced.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Thediscoveryoftheintegrabilityofcertainnonlinear evolutionequations(NLEE),starting with the pioneering work of Gardner et al. [1] and Zakharov–Shabat [2], and based on the fundamental notionof aLax pair [3],has markedacrucialturninseveral majordomainsof mathematicalphysicsofthelastcentury.The influenceisstill stronglyfelttoday.Many com-plementaryandinterrelatedpointsofviewandtheorieshavebeendevelopedaroundthissingle theme,tonamebutafew:

• theanalyticapproachculminatingintheinversescatteringapproach(seee.g. [4]),

• thegeometricapproachculminatinginthePoisson–Liegroupformulationofintegrable hi-erarchiesviatheclassicalr-matrixformalism,e.g. [5–8],andtherelatedalgebraicapproach tointegrablehierarchiesviaKac–Moodyalgebras(seee.g. [9]andreferencestherein),

• thealgebro-geometricapproachdealingwithperiodicsolutions ofintegrableNLEE,with stronglinkstoRiemannsurfacetheory,e.g. [10–12],

• thepseudo-differential operatorapproach [13,14]applicableinparticularto higher-dimen-sionalintegrablesystemssuchasKPhierarchy.

Thereaderwillhavetoexcuseouroversimplifiedlistoftopicsandreferenceswhichcannotmake justicetothehugeamountofliteratureaccumulatedoverdecadesonthistheme.Severalbooks canbeconsultedforamuchbetterandfairerdescriptionoftheseareas,e.g. [15–17].

Despiteallthesefacets,afewkeyingredientsarealwayspresentandprovideaunifiedbasis forsuchtreatments:aLaxpairformulationandtheassociatedzerocurvaturerepresentationof theNLEE.Thevariousapproachesmentionedabove(andthemanymorehiddenwithinthem) aremotivated by differentgoals: for instance,the Hamiltonian description of aNLEE is not necessaryifoneisonlyinterestedinviewingitasaPDEandfindingitssolutions.However,it becomesofimportanceifoneinsistsonprovingLiouvilleintegrabilityofthisNLEEviewedas aninfinitedimensionaldynamicalsystemorifonehasinminditsquantization(oneofthekey motivationsbehindtheclassicalr-matrixapproach).

Wewouldliketomakeonesimplebutstrikingobservation:mostoftheseapproaches,and mostnoticeablythe(Liouville)Hamiltonianconceptofintegrability,alwayssetemphasisonthe x-part of theLax pair as basicobjectandviewthe t-partas auxiliary.In asense, thisisnot at allsurprisingsincethe very phrase“nonlinearevolution equations”suggeststhat timeisa distinguishedvariableas theparameterof theevolutionflow underinvestigation.Adeparture fromthispointofviewalreadyariseswhenagivenNLEEisviewedasamemberofanentire integrable hierarchy describing the commuting flows for an infinite numberof times tn.The

independent variablex isthensimplyoneofthe“times”. However,evenwiththispicture,as soonasonesinglesoutagivenLaxpairinthehierarchyinviewofstudyingthecorresponding

evolutionequation,thebiasmentionedaboveisrestoredandtheequationisseenasanevolution

Inparticular,consideringNLEEformulatedbyanAKNSauxiliaryproblem

∂x=U , (1.1)

∂t=V , (1.2)

andtherelatedzerocurvaturecondition,

∂tU−∂xV+

U,V

=0, (1.3)

andformulatingthesysteminthefamousclassicalr-matrixapproachtointegrableNLEE,the fundamentalresultdealswiththePoissonalgebrasatisfiedbythe“fundamental”,x-relatedLax matrixU.Thet-relatedhalfoftheLaxpair,V,playsasecondaryroleinthatitisdeterminedby ther-matrixandthechoiceofHamiltonianinthehierarchydeducedfromthemonodromyofU. Thispointofviewisinsharpcontrastwiththenaturalsymmetricrolesplayedaprioribyxand tintheAKNSformulation.

Thissymmetrybetweenxandtnaturallysuggestsanotionofdualityintheprocessof choos-ingxortastheevolutionparameterdefiningtheintegrablepicture.Notealsothattheapproach todiscretizationofconformalintegrablequantumtheories,e.g.àladeVega–Destri [18]already usesasfundamentalevolutionvariablesthelightconecoordinatesx±t,departingthusfromthe x,tformulation.Onemayevenimagineresortingtosomemoregeneral(linear)functionofx,t astheadequateevolutionparameter,dependingonthephysicalinterpretationofthedynamical problem.Inotherwordsonemayhaveseveralrelevantchoicesofclassicalconfigurationspace, determiningthe phasespaceandtheHamiltonianformulationofthe dynamicalproblem.The issueisthenwhethersomeotherchoicesthantheusualone(herethespacesliceoffields sym-bolicallydenoted{φ(x)})alsoyieldLiouville-integrableHamiltoniansystems.Thisisthecore issueofourinvestigations.

Theneedtoexploitthissimpleobservationwasfeltwhendealingwithinitial-boundaryvalue problemsasopposedtoinitial-valueproblems.Itwasusedtoitsfullextentinthedevelopmentof theso-calledunifiedmethodofFokasforinitial-boundaryvalueproblems [19].Thismethodfalls intotheanalyticcategorymentionedabove.Morerecently,stillintherealmofinitial-boundary valueproblems, twoof thepresent authorsintroducedsomeideasfromcovariantfieldtheory intotheHamiltoniantheoryofintegrablesystemstotacklethequestionofLiouvilleintegrability foraNLEEwithanintegrabledefect [20].Thisapproachfallsintothegeometric/Hamiltonian approachmentionedabove.Ithadbeenmotivatedbyanapparentconceptualgapbetweenearlier studiesonclassicalintegrabilitywithdefects,seee.g. [21–23]andreferencestherein.Themain ideawas totreatthe two independentvariablesonequal footing whendeveloping aclassical r-matrix approachtointegrable defects.This putsome of the pointsmade in [24] ona firm basis.Asaconsequence,ontheexampleofthenonlinearSchrödinger(NLS)equation,several importantobservationsweremade:

• InadditiontotheusualPoissonstructure{,}SusedfortheHamiltoniandescriptionofNLS

asanevolutionequationintime,onecanconstructanotherPoissonstructure{,}T which

multitimeHamiltonianprocedurewherethesamephasespaceisendowedwithahierarchy ofmutuallycompatiblePoissonstructures.Inthiswell-knownframeworkthex-evolutionis triggeredbythechoiceofthemomentumPasHamiltonian,andthesamePoissonstructure. Inanutshell:hierarchiesofPoissonbracketstructuresaremutuallycompatibleonthesame phasespace; dualPoissonstructures liveondifferent phasespaces arisingfrom different choicesofconfigurationspace,albeitforthesameoriginal“dynamical”system.

• ThesecondPoissonstructure{, }T givesrisetoanultralocalPoissonalgebraforthet-part

VoftheLaxpair,withthesameclassicalr-matrix(uptoasign)astheultralocalPoisson algebrasatisfiedbythe x-partUwithrespecttothe usualPoissonbracket{, }S.We

in-sistthatthissecondPoissonstructureisnotacompatiblePoissonstructureinthesenseof bi-Hamiltoniantheory [25].

• TheLagrangianformulationbecomesthemostfundamentaldescriptioninthiscontext. In-deed,bothPoissonstructurescanbederivedfromit,togetherwiththecorrespondingspace andtimeHamiltonians,onceachoiceofgeneralcoordinatesorconfigurationspaceis stipu-lated.TheideaofgoingbacktoaLagrangianformulationofintegrableequationsofmotion, encapsulatingaso-calledmultisymplecticform,goesbackatleasttotheworksin [14]:at thattimeitwasundertakeninthecontextofmultitimecompatibleHamiltonianstructures. Herehowever,asalreadyemphasized,weusetheLagrangianformulationtoextracttwo in-equivalentchoicesofconfigurationspace,inducingtheconstructionofdifferentphasespaces andassociatedPoissonstructuresforourdynamicalsystem.

Inouropinion,thelastobservationaboveisthemainconceptualbasisforanunderstandingof space–timeduality(andpossiblymoregeneraldualities)inintegrablehierarchies.Lagrangians naturallycontaininformationonspace–timesymmetriesanddonotdiscriminatebetweenspace andtime.WhenonegoesovertothetraditionalHamiltonianpictureforatwo-dimensionalfield theory,adistinctionbetweentimeandspaceimmediatelyarisesonceonechooseswhattocall generalizedcoordinatesandtheirgeneralvelocities.

Note thatthe distinctionalsooccurs,evenbeforeHamiltonianformulation,as soonas one is leadtospecify“boundary conditions”for instancedistinguishingbetweenx=cstandt = cst boundaries. The validityof thisscheme andof theseobservations wasalso provedinthe context ofdefecttheory for thesine-Gordon modelby oneofthe authors [26].Althoughthe secondPoissonstructureanditsconsequenceswereoriginallyintroducedtosolvetheproblem ofLiouvilleintegrabilityinthepresenceofadefectraisedin [27],itbecameclearthattheabove observationsareofamorefundamentalnature.

Letusnowspecifyhowweplantousethisscheme.WeshallusebothLagrangianand Hamil-tonianformalismsjointlyandsuccessively.Inafirststep,thewell-known(Hamiltonian-based) constructionofahierarchyofHamiltoniansprovidesuswithahierarchyofsimultaneously solv-ableNLEE.ThederivationofadualPoissonstructurethenrequirestoidentifytheLagrangian densityyieldingeachNLEE.WerecoverthatthePoissonstructuredescribingthetn evolution

is theinitialPoissonstructure, asit shouldbeby firstprinciplesof Hamiltonianintegrability. ThedualPoissonstructuredescribingthexevolutionatlevelnmustbeconstructedasasecond stepbyapplicationofthecanonicalprocedurewithrespecttoxderivativesofthefields,as dic-tatedbythecovariantapproachtofieldtheoryandthedualchoiceofphasespaceandsymplectic structure.Onceitisconstructed,theexistenceof anr-matrixstructureforthePoissonalgebra ofthetn shiftoperatorV(n),naturallyidentifiedastheLax operatorinthisalternativechoice,

theprocedurejustdescribed,inanintegrablehierarchy.Weshallparticularizeourapproachto thenonlinearSchrödingerequationasasimplebutenlighteningexample.Butitisclearthatour ideascanbeadaptedtootherAKNS-describedhierarchies,withappropriatechanges.

Thispaperisorganisedasfollows.InSection2,wegivesomeusefulnotationsand conven-tions.Section3isdevotedtotheintroductionofthemainconceptsof thispaper.Wedescribe thedualHamiltonianformulationoftheNLShierarchy,theexplicitconstructionbeingdonefor level2(NLS)andlevel3(complexMKdV).TheroleoftheLagrangianformulationcombined withideasfromcovariantfieldtheoryisemphasizedinordertoderivethetwoPoissonstructures forthedynamicalfields.Thelastpartofthissectionexplainshowtheresultscanbegeneralized toallthelevelsinthehierarchy.InSection4,wethenbuildther-matrixformulationassociated withthisdualpicture.OntopoftheusualultralocalPoissonalgebraforthespaceLaxmatrixof NLS,wederivethesamePoissonalgebraforthehierarchyoftimeLaxmatrices(uptoasign). Consequencesfor conservedquantitiesandintegrabilityarediscussed.Section5isconcerned withsomeconclusions,remarks anddefinitionsbased onthe resultsobtainedinthe previous sections.Aphysicalapplicationandanexcitingnewperspectiveoffuturedevelopmentsarealso given.Technicalissuesregardingtheconstructionofdualhierarchiesaregivenin Appendices A andB.

2. Notationsandconventions

InthestandardapproachtotheNLShierarchyin1+1 dimensions,thedependentvariables (fields)ψ,ψ¯ areconsideredtobefunctionsofaninfinitesequenceoftimestn,n≥0,withthe

spacevariablexbeingt0forinstance.Wewillkeepitexplicitly asxtomakecontactwithsome

well-knownequationsintheNLShierarchy.EachHamiltonianH_S(n)atlevelninthehierarchy describestheevolutionofthesystemwithrespecttoagiventimetninsuchawaythat

ψtn= {H

(n)

S , ψ}, ψ¯tn= {H

(n)

S ,ψ¯}, (2.1)

withappropriatelychosenPoissonbrackets,isequivalenttothezerocurvatureconditionforthe Laxpair(U,V(n))atleveln

∂tnU−∂xV

(n)_{+ [U,V}(n)_{] =0. (2.2)}

Inthecourseofourstudy,thenotionofdualdescriptionwillemergewhereby,theequationsof motioncapturedby (2.2)canbeequivalentlyrewrittenasthex evolutionoffieldsgovernedby HamiltoniansthataredualtoH_S(n)inaprecisesense.ThisiswhyweintroducedthesubscriptS in (2.1)toindicatethattheHamiltonianisaxintegralofalocalHamiltoniandensityinthefields andtheirx derivatives.Thisisofcoursecompletelyredundantinthetraditionalapproachsince thisisalways assumedtobe thecase. But, inourtheory,the dualHamiltoniansH_T(n) appear naturallyfromaLagrangianpictureandare nowintegrals overtheformerlyidentified“time” variablestnoflocalHamiltoniandensitiesinthefieldsandtheirtnderivatives.

Thefieldsandtheirderivativeswithrespecttoagivensubsetoftheindependentvariablesare denotedasfollows

ψ (x,t) , ψ(x,¯ t) , ∂xtj₁...jqψ (x,t)=

∂q+1ψ (x,t) ∂x∂tj1. . . ∂tjq

, etc. (2.3)

confusionarises,weomittheargumentsaltogether.Whenonlysomeoftheindependentvariables areinvolved,typicallyx andonetn orpossiblytwotn’s,wemaydisplayexplicitlyonlythose

variablesthatplayarole,forinstancebywritingexpressionslike

ψ (x, t2) , ∂xt5ψ=ψxt5, etc. (2.4)

Wewillusethesamerulesforalltheobjectsbuiltfromthefieldsandtheirderivatives (La-grangians,Hamiltonians, Laxpairs).Finally,thenotionof equal-timePoissonbracket{, }(n)_S andequal-spacePoissonbrackets{, }(n)_T willplayanimportantroleinourtheory.Thesubscript SisusedtoemphasizethatthePoissonbracketdependsonlyonthespacevariable(equal-time) whileT istheretoremindusthatthePoissondependsonlyonthetimevariable(equal-space). The superscriptencodesthe levelof the hierarchythat we arelookingat.Whenwritingsuch Poissonbrackets,onlytheindependentvariablethatareinvolvedwillbeshown.Forinstance,an expressionlike

{ψ (x),ψ(y)¯ }(_S2)=iδ(x−y) (2.5)

involvingψ (x,t)andψ(y,¯ t)isshortfor

{ψ (x, t2),ψ (y, t¯ 2)}S(2)=iδ(x−y) (2.6)

andmeansthatweworkatequaltime(indicatedbythesubscriptS)andforn=2 (indicatedby thesuperscript(2))i.e.t2=t2.TheunderstandingisthatsuchaPoissonbracketwillcontrolthe

evolutionwithrespecttot2ofasystemdescribedbyaHamiltoniandensitydependingonx.

Inthispaper,weconsiderasboundaryconditionstheeasy-to-handlechoiceofperiodic bound-aryconditionsonafiniteinterval,bothinspaceandtimegivenourpurposes.Werewetoconsider fieldtheoriesoninfiniteintervalswewouldintroducethestandardfast-decayconditionsonthe fields. More specificboundary conditionson the intervalmaybe consideredfor specific pur-poses,interplayingwithbulkintegrabilitypropertiesusingsuitableformulationsdrawnfromthe well-knownCherednik–Sklyaninapproaches.

Finally, it isquiteprofitable tousea notionof scaling dimensionto keeptrackof various quantitiesinthehierarchy.Forinstance,weattachascalingdimension1 tothefieldsψ,ψ,¯ to thespectralparameterλ,aswellastothederivativewithrespecttox,andascalingdimension ofntothetimetn.Aswewillseeinthefollowing,thesuperscriptinV(n)capturesthisnotionin

thatV(n)_{isapolynomialinλoftheform}

V(n)_{(x, λ)=λ}n_V(n)_(x)+λn−1_V(n−1)_{(x)+ · · · +V}(0)_{(x) (2.7)}

whereV(j )isofscalingdimensionn−j andcontainsonlythefieldsandtheirx-derivativesin combinationsofscalingn−j.Forinstance,inV(2),V(2)=i₂σ3,V(1)= −√κQ,with

Q=

0 ψ¯

ψ 0

, (2.8)

andV(0)=iκQ2−iσ3Qx.HigherV(j )’scanbedeterminedrecursively.Thescalingdimension

3. TheNLShierarchyrevisited:LagrangiananddualHamiltonianpictures

3.1. LagrangiananddualHamiltonianpictureforNLS

Wereviewtheresultsfirstobtainedin [20]inlightofthepresentcontextandnotations.ALax pairforNLSreads

U(1)_{= −i}λ

2σ3+

√

κQ=

λ

2i

√

κψ¯

√

κψ −₂λ_i

, (3.1)

V(2)_=iλ2

2 σ3−λ

√

κQ(x)+iκQ2−iσ3Qx

=

−λ2

2i +iκ|ψ|

2 _−λ√_κψ¯ _−i√_κψ¯

x

−λ√κψ+i√κψx λ 2

2i −iκ|ψ|

2

. (3.2)

Intherestofthispaper,wedropthesuperscript(1)onU(1)asthisUwillbetheonlyoneinvolved here.Thezerocurvaturecondition

∂t2U−∂xV

(2)_{+ [U,V}(2)_{] =}

0, (3.3)

yieldstheNLSequation,

iψt2+ψxx=2κ|ψ|

2_{ψ (and complex conjugate). (3.4)}

ALagrangian densityforNLSis

L(2)₌ i

2(ψ ψ¯ t2−ψψ¯t2)− ¯ψxψx−κ(ψ ψ )¯

2_.

(3.5)

Wenow derivethetwoPoissonbrackets{, }_S(2) (whichisthestandardoneused e.g.in [15]) and{, }(_T2),thenewequal-spacePoissonbracketfirstintroducedin [20].Wewillalsofindthe correspondingHamiltoniansH_S(2)andH_T(2).Forconvenience,letusdenoteφ1=ψandφ2= ¯ψ

whichare treatedas independentfields. Withinthissection, sincewe workat thefixedlevel n=2, it isconvenient todrop the subscripton t2 andwe simplyuse t.Finally, whetherwe

discusstheequal-timepictureortheequal-spacepicture,oneofthetwoindependentvariables willbefixed.Hence,wecanaffordtouseclassicalmechanicsnotationsinsteadoffieldtheoretic ones(wewillthenalsorestrictourattentiontodensities).Bythiswemeanthatinsteadofwriting anexpressionlike

{j(x, t), φk(y, t)} =δj kδ(x−y) , (3.6)

wewillsimplywrite

{j, φk} =δj k. (3.7)

Similarly,aquantitylike ∂tφj will bewritten asφ˙j and∂xφj asφ_j.Thecontextwill always

beclearsothisabuseofnotationshouldnotleadtoanyconfusion.Withthisinmind,theusual canonicalconjugatetoφj isdefinedby

1=

∂L(2) ∂φ˙1

= i

2φ2, 2= ∂L(2)

∂φ˙2

= −i

Theseequationscannotbeusedtoeliminateφ˙j infavourofthemomentajwhenformingthe

Hamiltonian.Onthecontrary,weseethattheyrelatevariablesthataresupposedtobe indepen-dentintheHamiltonianpicturei.e.jandφk.SowefallwithinDirac’sschemeforconstrained

dynamics [28],theprimaryconstraintsbeing

C1=1−

i

2φ2, C2=2+ i

2φ1. (3.9)

TheenlargedHamiltoniandensity,takingtheseconstraintsintoaccount,istheusualHamiltonian plusalinearcombinationoftheconstraints,

H∗(2) S =H

(2)

S +λ1C1+λ2C2 (3.10)

=1φ˙1+2φ˙2−L(2)+λ1C1+λ2C2. (3.11)

TheinitialcanonicalPoissonbracketsaregivenasusualby

{j, φk} =δj k, (3.12)

andalltheothersarezero.TheevolutionoftheconstraintsunderH∗_S(2)andwithrespecttothese bracketsmustremainzero,whichentailstheconsistencyconditions

{H∗(2)

S , Cj} =0, j=1,2. (3.13)

Onecomputes

{C1, C2} = −i , (3.14)

whichshowsthattheseprimaryconstraintsare secondclass.Then,theconsistencyconditions yield

λ1= −i{H(S2), C2}, λ2=i{H(S2), C1}, (3.15)

whichfixesthe valueofλj andshowsthat therearenomoreconstraints.Dirac’s generalized

Hamiltoniantheory [28]teachesusthatwecaneitheruse{, }withH∗_S(2)asdeterminedabove todescribethedynamicsofthesystem,orwecanusethefamousDirac’sbracket{, }D which

allowsustosetCj=0 identically,andhenceuseHS(2)todescribethedynamics.Tothisend,we

needthematrixofPoissonbracketsofsecondclassconstraints

M=

0 {C1, C2}

{C2, C1} 0

=

0 −i

i 0

. (3.16)

TheDiracbracketisthendefinedas

{f, g}D= {f, g} −

2

j,k=1

{f, Cj}(M−1)j k{Ck, g}, (3.17)

foranyfunctionsf,gofthedynamicalvariables.Thenonzerocanonicalbracketswithrespect to{, }Darethenmodifiedto

{φ1, φ2}D=i , {j, φk}D=

1

2δj k, {1, 2}D= − i

4. (3.18)

We can seethat theyare of courseconsistent with the constraintsCj =0 andthese are the

bracketsweshouldidentifywith{, }(_S2).Withthesebrackets,wecanusetheHamiltonianH(_S2) moduloCj=0,whichreads

H(2)

Intermsoftheoriginalfieldsψ,ψ,¯ thethreebrackets (3.18)allconsistentlyreducetothe well-knownone [15]

{ψ (x),ψ (y)¯ }(_S2)=iδ(x−y) , (3.20)

andH(_S2)istheNLSHamiltoniandensity|ψx|2+κ|ψ|4.Theequationsofmotionarisefrom

ψt= {H_S(2), ψ}(_S2) (and complex conjugate), (3.21)

whereH_S(2)isthespaceintegralofH_S(2).

Followingthesameprinciple we nowturn tothe equal-spacePoissonbracket.The ideais tocompletetheLegendretransformandconsideranothersetofcanonical conjugatemomenta definedwithrespecttothexderivativeofφj,

P1=

∂L(2) ∂φ₁ = −φ

2, P1=

∂L(2) ∂φ₂ = −φ

1. (3.22)

Hence,fromthepointofviewofthex variable,weareinthestandardsituationwherewecan eliminatethe“velocities”φ_j infavourofthemomentaPj whenformingtheHamiltonian.The

analysisisthereforestraightforwardandwecanusethedefaultcanonicalPoissonbrackets

{Pj, φk} =δj k (all others zero), (3.23)

asourequal-spacebracket{, }_T(2).TheHamiltoniandensityisalsoreadilyfoundtobe H(2)

T =P1φ1+P2φ2−L(2). (3.24)

Intermsoftheoriginalfields,thisreads

H(2) T =

i

2(ψψ¯t− ¯ψψt)− ¯ψxψx+κ(ψ ψ )¯

2_,

(3.25)

togetherwith1

{ψ (t ),ψ¯x(τ )}T(2)=δ(t−τ ) , {ψ (t ),ψ (τ )¯ } (2)

T =0= {ψx(t),ψ¯x(τ )}(T2). (3.26)

Theequationsofmotionarethenconsistentlygivenby

ψx= {H_T(2), ψ}(_T2) (and complex conjugate), (3.27)

(ψx)x= {H_T(2), ψx}(_T2) (and complex conjugate), (3.28)

whereH_T(2)isthetimeintegralofH(_T2).

AsindicatedintheIntroductionwenowhavetwoequallyacceptablePoissonstructures yield-ingthe sameequationsof motion,albeit withtwodistinctconfigurationspaces andunrelated Hamiltonians.

1 _{Thedifferenceinsignscomparedtotheresultsin [20]comesfromthefactthatweusetheconvention{p,q}=1 and} ˙

3.2. LagrangiananddualHamiltonianpictureforthecomplexmKdVequation

Atlevel(3),thezerocurvatureconditionbasedonthepair(U,V(3)),with

V(3)₌

−λ3

2i +iλκ|ψ|

2_{+X −λ}2√_κψ¯_−iλ√_κψ¯

x+ ¯Y

−λ2√κψ+iλ√κψx+Y λ 3

2i −iλκ|ψ|

2_−X

(3.29)

and

X= −κ(ψ¯xψ−ψxψ),¯ Y=

√

κψxx−2κ 3

2|_ψ|2_{ψ , (3.30)}

yieldsthecomplexmodifiedKdVequation [29].

ψt3−ψxxx+6κ|ψ|

2_ψ

x=0, (and complex conjugate). (3.31)

NoticethatourV(3),asderivedfromtheformulasin Appendix A,yieldsaminussigninfrontof thethirdspacederivative,whereas(complex)MKdVisusuallypresentedwithaplussign.That isirrelevantasonecansimplysett3= −t andκ→ −κ.

Weusethesameshorthandnotationsasbefore:t3=t (asweworkatthefixedleveln=3),

˙

φj for∂tφj,φ_j for∂xφj,etc.ALagrangiandensityforthisequationis

L(3)₌i

2(ψ ψ¯ t−ψψ¯t)+ i

2(ψ¯xψxx−ψxψ¯xx)− 3iκ

2 ψψ(ψ¯ ψ¯x− ¯ψ ψx) (3.32)

=i

2(φ2φ˙1−φ1φ˙2)+ i 2(φ

2φ1−φ1φ2)−

3iκ

2 φ2φ1(φ1φ

2−φ2φ1) . (3.33)

Thederivationoftheequal-timebracket{, }_S(3) isexactlythesameas inthepreviouscase sincethepartofL(3)_{involvingthetimederivativesofthefieldsisidenticaltothatofL}(2)_.This

isunchangedandthisisofcourseconsistentwiththefactthatweuseUagainandonlychange the Vpart,from V(2) _toV(3)_{.TheLagrangianapproach presentedisthusconsistentwiththe}

traditional Hamiltonianpicturewherethe Poissonbracketisintrinsictothedynamicalsystem (againwe arenotconsideringhereMagri’smulti-Hamiltonian approach),andthechangeinV correspondstoachangeinthechoiceofcommutingHamiltonians.Italsomeansthatfromnow on,wecandropthesuperscriptontheequal-timebracket{, }S.Again,asiswellknown fromthe

usualHamiltonianapproachandfromwhatwefindgenericallyatlevel nfromtheLagrangian picture(seenextsubsection),theequal-timebracket{, }S isthesameforthewholehierarchy.

ThecorrespondingHamiltoniandensityreads,intermsoftheoriginalfields,

H(3) S = −

i

2(ψ¯xψxx−ψxψ¯xx)+ 3iκ

2 ψψ(ψ¯ ψ¯x− ¯ψψx) . (3.34)

tooursingularcase.Thisgoesasfollows.Thefirststepistoreducetheorderofthehigherorder Lagrangianbyintroducingnewfieldsϕj suchthat

ϕj=φj, j=1,2. (3.35)

TheLagrangianL(3)theneffectivelybecomesastandardfirstorderLagrangianinthe indepen-dentfieldsφj,ϕj

L(3)_=L(3)_(φ

j,φ˙j, ϕj, ϕj) . (3.36)

Theimportantrelation (3.35)ensuringequivalencewiththeoriginalLagrangianisnowdealtwith as aconstraintbyintroducing Lagrangemultipliers μj.Weintroduce thefollowing auxiliary

Lagrangian

Laux(φj,φ˙j, ϕj, ϕj, μj)=L(3)(φj,φ˙j, ϕj, ϕj)+

2

j=1

μj(ϕj−φj) . (3.37)

Forconvenience,letussplitL(3)as L(3)_(φ

j,φ˙j, ϕj, ϕj)=L0(φj, ϕj, ϕj)+L1(φj,φ˙j) , (3.38)

with

L0(φj, ϕj, ϕj)=

i 2(ϕ2ϕ

1−ϕ1ϕ2)−

3iκ

2 φ2φ1(φ1ϕ2−φ2ϕ1) , (3.39) L1(φj,φ˙j)=

i

2(φ2φ˙1−φ1φ˙2) . (3.40)

ThisismotivatedbythefactthatL1doesnotplayanactiveroleintheconstructionofH(T3)and

thecorrespondingPoissonbrackets,beingthetimepartoftheLagrangian.

WeapplythevariationalprincipletotheauxiliaryLagrangianfortheindependentfieldsφj,

ϕj andμj asusual.Ofcourse,avariationinμj enforcestheconstraints (3.35).Avariationin

φj yields

∂L0

∂φj +

μ_j+∂L1 ∂φj =

∂L˙1

∂φ˙j

. (3.41)

Avariationinϕj producesanequationfixingμj

μj= −

∂L0

∂ϕj +

∂L0

∂ϕ_j

. (3.42)

Inparticular,

μ1=iϕ2−

3iκ 2 φ

2

2φ1, (3.43)

μ2= −iϕ1+

3iκ 2 φ

2

1φ2. (3.44)

Notethat,uponinserting (3.35)and (3.42)into (3.41),weobtaintheequationsofmotion

˙

φ1−φ1+6κφ 2

2φ1=0, (3.45)

˙

φ2−φ2+6κφ 2

thatis, (3.31)anditscomplexconjugate,asrequired.Notealsothat (3.42)allowustoeliminate ϕ_j infavouroftheotherfieldsμj andφj whenformingtheHamiltonianbelow.

Wecannow proceed tothetransition tothe Hamiltonianpicture.Wedefinethe conjugate momentaofthefieldsφj,ϕj andμj,

P0

j =

∂Laux ∂φ_j , P

1

j =

∂Laux

∂ϕ_j , j= ∂Laux

∂μ_j . (3.47)

TheinitialcanonicalPoissonbracketsaretherefore

{P0

j, φk} =δj k, {Pj1, ϕk} =δj k, {j, μk} =δj k, (3.48)

withallothersequaltozero.Explicitly,theconjugatemomentaread

P0

j = −μj, j=0, (3.49)

and

P1 1=

i 2φ2, P

1 2= −

i

2φ1. (3.50)

OntheHamiltonianphasespaceparametrizedby(φj,P_j0;ϕj,P_j1;μj,j),theseequations

rep-resentprimaryconstraints.Thefirstset (3.49)isinherenttotheconstructionof [30].Thesecond set (3.50)isonlypresentinourcasebecauseourLagrangianissingular,inparticular,itislinear inthe“velocities”ϕ_j.NotethattheyareexactlyofthesamenatureastheconstraintsCj

dis-cussedintheNLScase,andtheywillhaveexactlythesameeffectonthePoissonbracketstobe derivedbelow.WealsocallthemCj here.Summarizing,wenowneedtoapplyDirac’sscheme

withtheprimaryconstraints

C1=P11−

i

2φ2, C2=P

1 2+

i

2φ1, (3.51)

C3=P10+μ1, C4=P20+μ2, C5=1, C6=2. (3.52)

WecomputethefollowingnonvanishingPoissonbracketsamongtheseconstraints,whichshow thattheyaresecondclass

{C1, C2} =i , {C3, C5} =1= {C4, C6}. (3.53)

WenowformtheHamiltonianHauxintheusualway

Haux=

2

j=1

(P_j0φ_j +P_j1ϕ_j +jμj)−Laux (3.54)

aswellastheextendedHamiltonianwithconstraintsH∗

H∗=Haux+

6

j=1

αjCj. (3.55)

TheconstraintsshouldremainequaltozerounderevolutionbyH∗andthisyieldstheconsistency conditions

Afterstraightforwardbutlengthycalculations,wefindthatthesegive

αj=0, j=1,2,5,6, αj=

∂L˙1

∂φ˙j₋2

, j=3,4. (3.57)

Thustherearenosecondaryconstraintsandthecoefficientsαjarefixed.Thelaststepistoderive

theDiracbracketsforthissystemsothatwecansettheconstraintstozeroidentically.TheDirac bracketswillalsobetheequal-spacebrackets{, }_T(3) thatwe arelookingfor. Tothisend,we needthematrixofPoissonbracketsoftheconstraints

M=({Cj, Ck})=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 i 0 0 0 0

−i 0 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 −1 0 0 0

0 0 0 −1 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. (3.58)

AsbeforetheDiracbracketforanytwofunctionsf,gofthedynamicalvariablesisdefinedby,

{f, g}D= {f, g} −

6

j,k=1

{f, Cj}(M−1)j k{Ck, g}. (3.59)

Usingthesebrackets,theconstraintscanbesettozeroidenticallyintheHamiltonianH∗andwe canworkonthereducedphase spaceconsistingofthefields(φj,Pj0,ϕj).Weobtain

Hred=P0

1ϕ1+P20ϕ2+

3iκ

2 φ2φ1(φ1ϕ2−φ2ϕ1)− i

2(φ2φ˙1−φ1φ˙2) , (3.60) withtheDiracbrackets

{φ1, φ2}D=0, {ϕ1, ϕ2}D=i , {P10,P20}D=0, (3.61)

{φj, ϕk}D=0, {Pj0, φk}D=δj k, {Pj0, ϕk}D=0. (3.62)

Restoring allthe results interms of the originalfields ψ andψ,¯ we obtain ourHamiltonian densityH(_T3)aftersimplificationas

H(3)

T =i(ψ¯xψxx−ψxψ¯xx)−

i

2(ψ ψ¯ t−ψψ¯t) . (3.63)

TheDiracbracketistheequal-spacebracket{, }_T(3)wewantand (3.61)–(3.62)translateinto

{ψ (t ),ψ (τ )¯ }(_T3)=0, (3.64)

{ψ (t ), ψx(t)}(T3)=0= {ψ (t ),ψ¯x(τ )}(T3), (3.65)

{ψx(t), ψxx(τ )}(_T3)=0= {ψx(t),ψ¯xx(τ )}(_T3), (3.66)

{ψx(t),ψ¯x(τ )}_T(3)=iδ(t−τ ) , (3.67)

{ ¯ψ (t ), ψxx(τ )}(_T3)=iδ(t−τ ) , {ψ (t ), ψxx(τ )}(_T3)=0, (3.68)

{ψxx(t),ψ¯xx(τ )}_T(3)= −6iκ|ψ|2(t)δ(t−τ ) . (3.69)

Onecannowverifythat

anditscomplexconjugateareindeedequivalenttothecomplexmKdVequationandits conju-gate (3.31).ThisconcludesourdualHamiltonianformulationofcomplexmKdV,theequation correspondingtothelevel(3)intheNLShierarchy.

Themostimportantoutcomeinviewofourdualclassicalr-matrixapproachtobedeveloped inthenextsectionisthederivationofthebrackets (3.64)–(3.69).Indeed,theseallowusto com-putethe Poissonbracketrelationsof theentriesofV(3),withthe fundamentalresultthat they obeytheultralocalPoissonalgebrawithrationalrmatrix,justlikeUdoeswithrespectto{ }(_S3), butuptoaminussign.

3.3. Generalizationtohigherlevelsinthehierarchy

The previous two explicitlevels indicate awayinwhich ourscheme willhold forhigher levels.Letusgivehereasketchofthisgeneralization.FirstofallitisclearthattheLagrangian densityatlevelnisoftheform

L(n)₌ i

2(ψ ψ¯ tn−ψψ¯tn)−H

(n)

S (3.71)

whereH(n)_S isdeterminedfromthegeneratingfunctionz(L,−L,λ)via (B.18).Thetotalnumber of x derivatives appearing under the integral in H_S(n) is equal to n, inview of the recursion relationsdefiningW(n),andalsoconsistentlywithscalingarguments.Thereforeusingintegration bypartsrepeatedly,inviewofourassumptionofperiodicboundaryconditions,wecanalways bringH(n)_S tosuchaformthatthehighestordertermisproportionalto

∂xpψ∂¯ xpψ , n=2p ,

i(∂xpψ∂¯ xp+1ψ−∂xpψ ∂xp+1ψ) , n¯ =2p+1,

(3.72)

NLSisanexampleoftheevencaseandcomplexmKdVanexampleoftheoddcase,bothwith p=1.Forhighervaluesofp,thegeneralstrategytoobtainafulldualHamiltoniandescription ofthemodelisclear.Theequal-timebracket{, }S isthesameatalllevelsandisobtainedas

previouslydescribed for NLSor complex mKdV.To obtain the equal-spacebrackets{, }(n)_T andthecorrespondingHamiltoniandensity H(n)_T ,weproceedasexplainedaboveforNLSand complexmKdV,byusingtheformalismof [30].Forevencases,theLagrangianisregularand the only constraintsappearing are the ones intrinsic tothat higher orderformalism. Forodd cases, oneobtains two moreconstraints duetothe linearityof the Lagrangianin thehighest x derivative.Irrespectiveof theirorigin, alltheconstraintsaresecondclassandnoadditional constraintsappearasaresultoftheconsistencyconditions.Onethensimplyproceedsasinthe complexmKdVcase,derivingtheDiracbracketsandthecorrespondingreducedHamiltonian. Thesearethenidentifiedwith{, }(n)_T andH(n)_T .

4. Classical r-matrixformulationinthedualpicture

4.1. Theoriginalobservation:twoultralocalPoissonalgebrasforNLS

HamiltoniandualformulationsbyconstructionofmonodromymatriceswithPoisson-commuting traces.ForNLSitself,theresultwasalreadyobtainedandusedin [20]forthespecificpurpose ofstudyingintegrabledefects.Letussimplyrestatetheresultinthepresentcontext.

Proposition4.1. Equipped withthe canonicalPoisson brackets {, }_S(2) and {, }(_T2),the Lax

matricesUandV(2)satisfythefollowingultralocalPoissonalgebras

{U1(x, λ),U2(y, μ)}S(2)=δ(x−y)[r12(λ−μ),U1(x1, λ)+U2(y1, μ)], (4.1)

{V(2)

1 (t2, λ),V

(2)

2 (τ2, μ)}

(2)

T = −δ(t2−τ2)[r12(λ−μ),V (2)

1 (t2, λ)+V

(2)

2 (τ2, μ)], (4.2)

whereristhesl(2)rationalclassicalr-matrix

r12(λ)=

κ

λP12, (4.3)

P12isthepermutationoperatoractingonC2⊗C2,andwehaveusedthetensorproductnotation

withindices1and2.

Itisquiteremarkable,andapriorinotexpectedtobeageneralfeatureofsuchdualitypictures, thatbothHamiltonianstructuresbedescribedbythesamer matrix.In anycaseanimmediate consequenceoftheexistenceofthesetwoHamiltonianstructuresisthat,inadditiontotheusual spacemonodromymatrixT_U(L,−L,λ)builtfromUandcontainingalltheinformationabout thesystem, inparticular theconserved charges intime,we maynow considerthe time mon-odromymatrixT(_V2)(T ,−T ,λ)builtfromV(2).Inparticular,itgeneratestheconservedchargesin space.OnecantalkaboutintegrabilityofthemodelwithrespecttothesetwoPoissonstructures andhence,about“dualintegrable”hierarchies,inasensetobegeneralizedintheConclusion.

4.2. HierarchyofdualultralocalPoissonalgebrasforhigherNLSsystems

Atlevel(3),wemayusethePoissonbrackets (3.64)–(3.69)andtheexplicitform (3.29)for

V(3)_{toshowdirectlythatV}(3)_{satisfiestheultralocalPoissonalgebra (4.2)withrespectto{,}}(3) T .

AsexplainedinSection3.3,onecanobtainthehigherPoissonbrackets{,}(n)_S (whicharethe sameatalllevels)and{, }(n)_T forthefieldssystematically.Itisnothardtoconvinceoneselfthat theultralocalPoissonalgebrastructuremayholdalsoforhigherV(n).Hence,weconjecture

Proposition4.2. Equipped withthe canonicalPoisson brackets{, }(n)_S and {, }(n)_T , theLax

matricesUandV(n)satisfythefollowingultralocalPoissonalgebras,foralln∈N,

{U1(x, λ),U2(y, μ)}(n)S =δ(x−y)[r12(λ−μ),U1(x, λ)+U2(y, μ)], (4.4)

{V(n)

1 (tn, λ),V

(n)

2 (τn, μ)}

(n)

T = −δ(tn−τn)[r12(λ−μ),V (n)

1 (tn, λ)+V

(n)

2 (τn, μ)]. (4.5)

Remark. the Poissonalgebra for V(n) givenin the proposition is not just aconsequence of the Poisson brackets on the dynamical fields involved in V(n) but is in fact equivalent to them. For n=3, this means that the Poisson algebra for V(3) is equivalent to the brackets

n=2[20].Atthislevel,itwasalsoknownthattheequal-spacebracketsforthefieldscouldbe

derivedfromacovariantLagrangianpicture.Oneofourmainsuccessesinthepresentpaperisto

showthatwecancompletethispicturefromtheLagrangianpointofviewalsoatthehigher lev-elsinthehierarchy,aswedidexplicitlyforn=3,henceproducing (4.5)asaniceconsequence oftheLagrangianapproach.However,this“historicalremark”showstheguidingpowerofthe algebraicapproachtoHamiltonianclassicalintegrablesystems.

4.3. AhierarchyofmatricesUforeachV(n)

Recallthatfollowing [15](see Appendix A),giventheLaxmatrixUanditsultralocalPoisson algebra (4.1),onecanderiveanexplicitexpressionforthegeneratingfunctionofallthematrices

V(n)_{,n∈NoftheNLShierarchy.Forinstance,onefindsthefirstfewas:}

V(0)_{(λ)= −}1

2i

1 0

0 −1

,

V(1)_{(x, λ)=}

−λ

2i −

√

κψ¯

−√κψ ₂λ_i

,

V(2)_{(x, λ)=}

−λ2

2i +iκ|ψ|

2 _−λ√_κψ¯ _−i√_κψ¯

x

−λ√κψ+i√κψx λ 2

2i −iκ|ψ|

2

,

V(3)_{(x, λ)=}

−λ3

2i +iλκ|ψ|2+X −λ2

√

κψ¯ −iλ√κψ¯x+ ¯Y

−λ2√κψ+iλ√κψx+Y λ 3

2i −iλκ|ψ|

2_−X

, (4.6)

where

X= −κ(ψ¯xψ−ψxψ),¯ Y=

√

κψxx−2κ 3

2|_ψ|2_{ψ . (4.7)}

In view of our results, inparticular the ultralocal Poisson algebra (4.5) satisfied by V(n) (provedfromn=1,2,3,conjecturedforhighern),itseemsnaturaltofollowthesameprocedure startingfromafixedV(n)anditsultralocalPoissonalgebrathistime.Followingtheconstruction detailed in Appendix A,we obtainahierarchy ofmatricesU(m,n),m∈N associatedtoV(n). Asanillustration,supposewepickV(2)correspondingtoξ=t2.Wefindthefirstfewmatrices U(m,2)_as

U(0,2)_(t

2, λ)=

1 2i

1 0

0 −1

,

U(1,2)_(t

2;λ)=

λ

2i +

√

κψ¯

√

κψ −₂λ_i

,

U(2,2)_(t

2;λ)=

λ2

2i −iκ|ψ|

2 _+λ√_κψ¯_+i√_κψ¯

x

λ√κψ−i√κψx −λ 2

2i +iκ|ψ|2

,

U(3,2)_(t

2;λ)=

λ3

2i −iλκ|ψ|

2_{− λ}2√_κψ¯ _+iλ√_κψ¯

x− ¯

λ2√κψ−iλ√κψx− −λ 3

2i +iλκ|ψ|

2₊

, (4.8)

wherewedefine

= −κ(ψ¯xψ−ψxψ ), ¯ = −i

√

κψt2−2κ 3

2|_ψ|2_{ψ (4.9)}

Oncethishierarchyisconstructed,wecaninprinciplerepeattheprocessstartingfromagiven

U(m,n)_{andobtainahierarchyofassociatedLaxmatricesV}(p,m,n)_{,p∈N,andsoon.}

Thisappealingidea suggeststhatonecan constructacollectionof Lax pairs(U(I ),V(J )), whereI,J aremultiindices,whichcouldbeorganisedinamultidimensionallattice,hence pro-ducingmultidimensionalintegrablePDEs.Oneisnaturallyledtodistinguishbetween“vertical” and“horizontal”indicesdependingwhethertheindexlabelsahierarchyofelementsbuiltfrom a“equal-space”or“equal-time”Poissonbracket.Quiteobviouslytheeven(resp.odd)indicesof

V(respU)arehorizontalindicesinthissense.Thedetailedinvestigationofsuchapossibility andofthecorrespondinglatticeofLaxpairsisleftforfuturework.Averyspecialexampleof thisideaappeareddifferentlyintheformofatripleofLaxmatricesin [31]andwasatthebasis ofacertain2+1-dimensionalversionofNLS.Ingeneral,itisfarfromobviousthattheprocess justdescribed“closes”inthesensethatthelatticeisfinitedimensionalandthatalltheequations obtainedbytakingthezerocurvatureassociatedtoeachpairinthelatticearemutually consis-tent.The lastissueisalready anontrivialresultinthe well-knowncaseof asinglehierarchy (a one-dimensionallatticefromourpointofview)thatcanbefounde.g.in [9].

5. Conclusionsandoutlook

Ourmainresultscanbesummarizedasfollows:

•LagrangiananddualHamiltonianformulation:ForeachLaxpair(U,V(n)_)atlevelnin

theNLShierarchy,weconstructtwoPoissonbrackets{, }(n)_S and{,}(n)_T forthecanonicalfields andtwoHamiltoniansH_S(n)andH_T(n)(recallagainthatthesearenotMagri’smulti-Hamiltonian structures).Theyare derivedfromaLagrangianwhichreproducestheequations ofmotion of thezerocurvatureconditionfor(U,V(n))throughitsEuler–Lagrangeequations.Theequal-time bracket{,}(n)_S foundinthisprocedureisthesameateachleveln,whichisconsistentwithwhat isknownfromthetraditionalHamiltonianapproachtoanintegrablehierarchy.

•Dualclassicalr-matrixformulation:ForeachLaxpair(U,V(n))atlevel nintheNLS hierarchy,thePoissonbracket{, }(n)_S (resp.{,}(n)_T )forthecanonicalfieldsinducesanultralocal PoissonalgebraforU(resp.V(n)).Aconsequenceofthisisthatthespacemonodromymatrix

TU generates integrals of motion (intime)that are ininvolutionwithrespect to{, }(n)_S (the well-knownresult)andthetimemonodromymatrixT_V(n)generatesintegralsofmotion(inspace) thatareininvolutionwithrespectto{, }_T(n)(thenew,dualresult).TheHamiltoniansH_S(n)and H_T(n)arecontainedinthesefamiliesofintegralsofmotion.

Wearenowinapositiontoformulatesomegeneralremarksandmoreprecisedefinitionsof ourprocedures.

5.1. Thenotionofdualityinanintegrablehierarchy

Wehaveusedtheterminologydualordualityratherlooselysofar,countingonthefactthatit isself-explanatoryinthevariousexamplesthatwehaveusedtoillustrateit.However,wewould nowliketoproposeanattemptatamoreprecisedefinitiontobeusedinfuturedevelopments.

ThemonodromyT_UofthisLaxmatrixthennaturallygeneratesPoissoncommuting Hamil-tonians asintegrals overx of (local)densities,andthedataof U(φ(x),λ)andr allowoneto constructanVmatrix dependingon λ,φ andpossiblythe x-derivatives of φ,for everysuch Hamiltonian,describingatimetranslationfortheauxiliarywavefunction,andsuchthatthe com-patibilityofthet- andx-evolutions(flatnessofconnection)impliestheHamiltonianequationsof motion(timeevolution)fortheinitialfieldsφ(x)nowviewedasφ(x,t ).ThePoissonstructure isherebyconstructionanequal-timePoissonbracket.

InviewoftheexplicitresultsobtainedsofaronourNLSexample,weshalldefinethenotion

ofdualformulationasfollows.Supposeonecanexhibitanother,equal-spacePoissonstructure

forthefields,nowchosentobeφ(t),parametrizinganotherchoiceofconfigurationspacesuch that:

–ThecorrespondingPoissonstructureofthematrixVnownaturallyidentifiedastherelevant Laxmatrix,hasalinearr-matrixdescriptionwithmatrixr,and,

–theλexpansionofthemonodromymatrixT_Vofthet-translationdescribedbyV(φ(t),λ) containsinparticularaHamiltonianintegralovert,suchthatitsassociatedtranslationoperator builtfromther-matrixstructurerasin Appendix AcoincideswithU.

Thenthedynamicalequationfor φ,containedinthezero-curvatureconditionforU,V,can beeitherrepresentedasaHamiltonianintegrableevolutionforφ(x)alongtorforφ(t)alongx. Thisdefinitionishereexplicitly formulatedforAKNSsystemsdescribingmutually compati-blexandt evolutions.Itmustbeemphasizedthatonemayeasilyextendthisnotiontoanypair of AKNS-type gaugeconnections over atwo-dimensional space–time, originally yielding by zero-curvaturecondition2-dimensionalnonlinearevolutionequationsforsomefields.Asabove, inthegeneralcasetheseconnectionswillberespectivelyidentifiedwiththeinfinitesimalshifts ofan auxiliarywavefunction, thistimealong twoindependent space–time coordinates,linear combinationsorevenalgebraiccombinationsofx andt:theexampleoflight-conecoordinates mentionedaboveisoneobvioussuchextension.

Wheneversuchaschemeisrealised,weshallspeakofdual(canonical)formulationsordual

(canonical)representationsof theintegrablePDE. Asisseenonthe definition we imposein

particularthatinbothcasesthe“generalizedtime-shift”elementoftheLaxconnectionwillbe identifiedas apartial trace describingan r-matrix co-adjointactiononthe differentialof the Hamiltonianderivedfromintervalmonodromiesofthe“generalizedspace-shift”element,such asderivedin [7].Thisimpliesaverynontrivial propertyonthe“initial”Laxmatrix(beitpicked asUorV),i.e.thatitmustberecoveredfromr-matrixcoadjointactiononthespectral-parameter expansion ofthe monodromymatrixofits Laxpartnerwhichisitselfobtainedfromr-matrix coadjointactiononthespectral-parameterexpansionofthefirstmatrix;averynontrivial consis-tencyoftheset(Laxmatrix/r-matrix)isthussuggested.

Inthecasewherether-matricescoincide,r=r,weshallcallthesystemaself-dual Hamil-toniansystem.Wedonothaveexamplesatthistimeofdualintegrablesystemswithdifferentr matricesforthetwodualintegrabilitystructuresbutwecannotexcludesuchsituations,hencethis particularization.NotethattheNLSequationcanthenbecalledanti-self-dualsinceweshowed thatwehaver= −rinthispaper.

Thedefinitioncanbeeasilyextendedtoawholehierarchy.Onceagain,givenaLaxmatrix

UanditsPoissonalgebracharacterisedbyr,onecanbuildawholeintegrablehierarchyofLax pairs(U,V(n)),n∈N,inwhichV=V(1)say.Then,bythenotionofdualityjustexplained,we havethedataofVanditsassociatedPoissonalgebracharacterisedbyr.Therefore,wecanbuild ahierarchyofLaxmatricesU(n),n∈N,withU(1)=U.Thehierarchies(U(1),V(n)),n∈Nand (U(n),V(1)),n∈Narecalleddualhierarchies.

Notethattheterminologyof“duality”hasalreadyappearedinthedomainofintegrable sys-tems,albeitinadifferentcontext,i.e.thefamousposition/momentumdualitybeforeHamiltonian reduction,relatingCalogero–MoserandRuijsenaars–Schneidermodelsunderthegenericname ofRuijsenaars duality [33,34],withidenticalPoissonstructure.Herebycontrastthe“duality” exchangestheroleofvariablesx andt inthedefinitionofintegrabilityitself,andthePoisson structuresaredistinct.In asense andtoborrowaphrasefromstringtheory,RSdualityis tar-getspace duality,whereasournotion ismorerelatedtoworldsheetduality, or changeof the conformalstructure.

5.2. Physicalapplication:aHamiltoniandescriptionofNLSinOptics

TheNLSequationthatservedasthemainbasistodevelopourformalismturnsouttobealso anexcellentphysicalexampleoftheinterchangebetweenspaceandtimeincertainintegrable systemswithauniversalcharacter.BythiswemeanthatNLSarisesasanapproximateequation inwavedynamicsunderfairlygeneralassumptionsbutwithdifferentinterpretationsofthe inde-pendentvariablesdependingonthecontext.Anicediscussionofthephysicalsignificanceand universalcharacterofNLScanbefoundin [35].Whenderivedinthefluidmechanicscontext forinstance,theusualformisrelevant

iφt+φxx=2κ|φ|2φ , (5.1)

wheret hasthemeaningoftimeandx themeaningofspace.Thatisbyfarthemostpopular settingandthestandardHamiltoniandescriptionisalwaysgivenforthisform.But,whenNLS isderivedinnonlinearoptics,thephysicalmeaningattachedtoxandtisswapped.Forinstance, inChapter5of [36],onefindsNLSintheform

iAz=

β2

2 Aτ τ −γ|A|

2_{A ,}

(5.2)

whereA(z,τ )istheamplitudeofthe pulseenvelope andβ2,γ areparameterscharacterizing

thephysicalpropertiesoftheopticalfibre.Herezisthespatialcoordinatealongthefibre andτ isattachedtoaframeofreferencemovingwiththepulseatgroupvelocityvg,τ=t−_vz_g.Of

coursethiscanalwaysberescaledtothestandardform (5.1)butthepointisthatxisnowatime andt aspacevariable.ToimplementtheHamiltonianapproachofthisformofNLSseenasan evolutionequationintime,onehasinfacttouseourformalismtoproduceφxx=(φx)xasthe

PoissonbracketofsomeHamiltonianwithφx.Thisisnothingbut (3.28).

bracketasjustdescribed,oneinsistsonusingthestandardHamiltonianformulationof (5.1)i.e.

(3.21),butnowinthephysicalcontextofopticalfibres.Thenoneisinfactdescribingthespace evolutionofthewave.

Thisdiscussion showsthateveninthenonrelativistic setting,it issometimesusefulnotto attachan intrinsicmeaningtowhat coordinatedescribesspaceandwhatcoordinatedescribes time.Westressonceagainthat,beitmathematicallyorphysically,treatingvariablesonanequal footinginagivenintegrablehierarchyprovidesnewinsight.

5.3. Multi-indexLaxpairs

Wehavesuggestedintheprevioussectionthatonecouldimaginesuccessiverepeatsof the dualization procedure,startingfrom the two initialhierarchies of Uand V,andsuccessively buildinghigherlevelsof“horizontal”orU-typeand“vertical”orV-typehierarchiesbystarting from alowerlevel hierarchyelement, assumingit isendowedwiththe doublesetof Poisson structures.Animmediatequestionariseswhetherthereexist connectionsbetweenUandV ob-jectsrespectivelylabelled bytheirorderedsetofalternatively“vertical”and“horizontal”indices: (U(I ),V(J ))whensomerelationexistsbetweentherespectivesetsI andJ(amoreprecise state-mentof the issueof “finite-dimensionality of the lattice” alluded tointhe previous section). Anobvioussecondquestioniswhetherthe dualPoissonstructuresapriorideducedfromthe Lagrangianformulationofhigher-levelhierarchyequationsarestillrepresentedbyr-matrix for-mulations;whethertheser-matriceswillthenremainthesame(as isthecaseforthefirsttwo levelsofhierarchyinNLS);orwhethernewr-matriceswillarise;or(worst-case)dual integra-bilitybreaksaltogetherathigherlevelsandnodualr-matrixformulationisavailable.

Acknowledgements

PreliminarystepsforthisworkwereundertakenwhileJ.A.visitedCityUniversityLondon andwhileV.C.visitedHeriot-WattUniversity.TheIndianNationalScienceAcademyiswarmly acknowledged forawardingtheDrRamalingaswamiINSAChair2015toV.C.andsupporting hisvisittotheBoseCentreandSahaInstituteinKolkata,duringwhichthemainpartofthiswork wascompleted.

Appendix A. HierarchyofLaxmatricesfromthe r-matrix

GiventhesymmetricrolesplayedbythematricesUandV(n)andinviewofthediscussionin Section4.3,wepresentinthisappendixthegeneralizationoftheconstructionof [15][Chap III, §3].Ityieldsaformula forageneratingfunctionof thehierarchy ofLax matricesV(n) given

Weneedneutralnotationstoallowforthisprocess.So,letusassumethatwehaveconstructed appropriatePoissonbrackets{, }(N )_A ,2suchthatequationsofmotionarisingfromthezero cur-vaturefortheLaxpair(X,Y)

∂ηNX(ξ, ηN, λ)−∂ξY(ξ, ηN, λ)+ [X,Y](ξ, ηN, λ)=0, (A.1) read

∂ηNX(ξ, ηN, λ)= {H

(N )_{,X(ξ, η}

N, λ)}(N )_A . (A.2)

Here,X,YmatrixpolynomialsinλofdegreeN andMrespectively,and

TrX=0 , X¯(ξ, η,λ)¯ =σX(ξ, η, λ) σ , (A.3)

whereσ=σ1ifκ >0 andσ=σ2ifκ <0.Wethenassumethattheseequationsofmotionat

thesinglelevelNareembeddedintothefollowinghierarchyofequations

∂ηnX(ξ, ηn, λ)= {H

(n)_{,X(ξ, η}

n, λ)}(N )A , n∈N, (A.4)

wheretheHamiltoniansH(n)areconstructedfromXasin Appendix B.Inaddition,assumethat

{X1(ξ, η, λ),X2(ζ, η, μ)}(N )_A =γ δ(ξ−ζ )[r12(λ−μ),X1(ξ, η, λ)+X2(ζ, η, μ)], (A.5)

whereγ= ±1 andristhestandardone-polerationalclassicalr-matrix.Thislastassumptionis motivatedbyoneofthemainresultsofthispaperwherebyUandV(n)_{satisfythesamePoisson}

algebrauptoasign,hencetheintroductionofγ.

Atthisstage, wecanrepeattheargumentsof [15][ChapIII,§3]andsimplyincorporateγ. Weobtaintheequationsofmotioninzerocurvatureformforeachlevelnas

∂ηnX(ξ, ηn, λ)−∂ξY

(n)_{(ξ, η}

n, λ)+ [X,Y(n)](ξ, ηn, λ)=0, (A.6)

wherethehierarchyofmatricesY(n)isobtainedfromtheexpansion

Y(ξ, η, λ;μ)=κ ∞

n=1

Y(n₋1)_{(ξ, η, λ)}

μn , (A.7)

where

Y(ξ, η, λ;μ)= γ κ

2i(λ−μ)(1+W (ξ, η, μ)) σ3(1+W (ξ, η, μ))

−1_{. (A.8)}

WehaveaddedtheexplicitηdependenceofW(definedin (B.4))forconsistency.Inparticular, weobtainthatYNcoincideswiththeoriginalYin (A.1),asitshould.

Ofcourse,ifweapplythisconstructiontoX=Uwithξ=x andγ=1,werecoverY(n)₌

V(n)_withη

n=tn.TheideaisthatwewanttobeabletoapplytheconstructiontoX=V(n)for

somefixednandξ=tn,toyieldahierarchy ofU(m)’s,etc.

Remark.Animportantconsequenceofformula (A.8)isthatthetwopropertiesofX,i.e. trace-lessness andsymmetry,transfertoalltheY(n)(ξ,η,λ)’s.Thisiscrucialifwewanttobeableto repeattheconstructionwithoneoftheY(n)_{’sasXsincethesepropertiesentertheargumentto}

obtainformula (A.8).

2 _{AcanstandforeitherSorT heredependingonwhetherweuseξasspaceandηastimeorviceversa.Nisafixed}

Wenowinvestigateaconsequenceofformula (A.8)thatgivesausefulrecursiveformulafor

Y(n)_intermsofY(n−1)_{andthecoefficientsW}(j )_{,j=1,. . . ,ndiscussedin Appendix B.Toour}

knowledge,thisrecursionformulawasnotknown(atleastnotgiven)before.Itreads,

Y(0)_{(ξ, η, λ)=}iγ κ

2 σ3, (A.9)

Y(n)_{(ξ, η, λ)=λY}(n−1)_{(ξ, η, λ)+iγ σ}

3

n

j=1

(−1)j

m1+···+mj=n

W(m1)_{. . . W}(mj)_{. (A.10)}

Inthelastsum,foragivenj,thesumrunsoverallpartitionofnintoj integersm1,. . . ,mj.To

proveit,simplymultiply (A.8)byλ−μandrewriteitas

_∞

n=1

1 μn

λY(n−1)−Y(n)

−Y(0)_−iγ

2σ3

= −iγ σ3(1+W )−1. (A.11)

Theresultfollows byidentifyingthetermoforderμ−noneachside.

Appendix B. Riccatiequationandconservedquantities

WekeepthenotationsintroducedinthepreviousAppendixthatallowustodiscussthe con-structionof agenerating functionof conserved quantitiesintimeor space,depending onthe startingpointusedforXandξ asexplainedabove.Again,weextendtheargumentspresented in [15][ChapI,§4]toourcase.Theensuingresultsareobtainedbydirectcalculation.Consider theauxiliaryproblem

∂ξ(ξ, η, λ)=X(ξ, η, λ)(ξ, η, λ) (B.1)

∂η(ξ, η, λ)=Y(ξ, η, λ)(ξ, η, λ) (B.2)

withX,YmatrixpolynomialsinλofdegreeNandMrespectively,and

TrX=0 , X¯(ξ, η,λ)¯ =σX(ξ, η, λ) σ , (B.3)

where σ =σ1 if κ >0 and σ =σ2 if κ <0. Consider periodic boundary conditions in ξ ∈

[−,].3 Let T(ξ,ζ,λ) bethe fundamentalsolution of (B.1)normalised toT(ξ,ξ,λ)=1. IntroducethelargeλexpansionofTasfollows

T(ξ, ζ, λ)=(1+W (ξ, λ))eZ(ξ,ζ,λ)(1+W (ζ, λ))−1, (B.4)

whereW isanoff-diagonalmatrix,Zadiagonalmatrix,andtheyhavethefollowingexpansion

W (ξ, λ)= ∞

n=1

W(n)(ξ )

λn , Z(ξ, ζ, λ)= N

n=0

Z(−n)(ξ, ζ ) λn+ ∞

n=1

Z(n)(ξ, ζ )

λn . (B.5)

Ageneratingfunctionofthereal-valuedlocalconservedquantitiesinηisnowgivenby

1

2iκTr[σ3Z(,−, λ)] (B.6)

3 _{Notethatingeneralξwillbeeitheraspacexoratimet}

nsothatweworkwithperiodicconditionsbothinspaceand