Bi-Hamiltonian systems on the dual of the Lie alge

arXiv:math-ph/0610001v130Sep2006BI-HAMILTONIANSYSTEMSONTHEDUALOFTHELIEALGEBRAOFVECTORFIELDSOFTHECIRCLEANDPERIODICSHALLOWWATEREQUATIONSBORISKOLEVAbstrat.Thispaperisasurveyartileonbi-HamiltoniansystemsonthedualoftheLiealgebraofvetoreldsontheirle.WeinvestigatethespeialasewhereoneofthestruturesistheanonialLie-Poissonstrutureandtheseondoneisonstant.ThesestruturesalledaneormodiedLie-PoissonstruturesareinvolvedintheintegrabilityofertainEulerequationsthatariseasmodelsofshallowwaterwaves.1.IntrodutionInthelastfortyyearsorso,theKorteweg-deVriesequationhasreeivedmuhatten-tioninthemathematialphysisliterature.SomesigniantontributionsweremadeinpartiularbyGardner,Green,Kruskal,Miura(see[46℄foraompletebibliographyandahistorialreview).Itisthroughthesestudies,thatemergedthetheoryofsolitonsaswellastheinversesatteringmethod.OneremarkablepropertyofKorteweg-deVriesequation,highlightedatthisoasion,istheexisteneofaninnitenumberofrstintegrals.Themehanism,bywhihtheseonservedquantitiesweregenerated,isattheoriginofanalgorithmalledtheLenardreursionshemeorbi-Hamiltonianformalism[18,36℄.Itisrepresentativeofinnite-dimensionalsystemsknownasformallyintegrable,inreminiseneofnite-dimensional,lassialintegrablesystems(inthesenseofLiouville).Otherexamplesofbi-HamiltoniansystemsaretheCamassa-Holmequation[16,4,6,14,21℄andtheBurgersequation.Oneommonfeatureofallthesesystemsisthattheyanbedesribedasthegeodesiowofsomeright-invariantmetrionthedieomorphismgroupoftheirleoronaentralrealextensionofit,theVirasorogroup.Eahleft(orright)invariantmetrionaLiegroupindues,byaredutionproess,aanonialowonthedualofitsLiealgebra.Theorrespondingevolutionequation,knownastheEulerequation,isHamiltonianrelativelytosomeanonialPoissonstruture.ItgeneralizestheEulerequationofthefreemotionofarigidbody1.Inafamousartile[1℄,Arnoldpointedoutthatthisformalismouldbeappliedtothegroupofvolume-preservingdieomorphismstodesribethemotionofanDate:16aoßt2006.2000MathematisSubjetClassiation.35Q35,35Q53,37K10,37K65.Keywordsandphrases.Bi-Hamiltonianformalism,Dieomorphismsgroupoftheirle,Lenardsheme,Camassa-Holmequation.Thispaperwaswrittenduringtheauthor’svisittotheMittag-LeerInstituteinOtober,2005,inonjuntionwiththeProgramonWaveMotion.TheauthorwishestoextendhisthankstotheInstituteforitsgeneroussponsorshipoftheprogram,aswellastotheorganizersfortheirwork.TheauthorexpressesalsohisgratitudetoDavidSattingerforseveralremarksthathelpedtoimprovethispaper.1Inthatase,thegroupisjusttherotationgroup,SO(3).12B.KOLEVidealuid2.Thereafter,itbeamelearthatmanyequationsfrommathematialphysisouldbeinterpretedthesameway.In[19℄(seealso[44℄),DorfmanandGelfandshowedthatKorteweg-deVries[27℄equationanbeobtainedasthegeodesiequation,ontheVirasorogroup,oftheright-invariantmetridenedontheLiealgebrabytheL2innerprodut.In[41℄,MisiolekhasshownthatCamassa-Holmequation[4℄whihisalsoaonedimensionalmodelforshallowwaterwaves,anbeobtainedasthegeodesiowontheVirasorogroupfortheH1-metri.WhileboththeKorteweg-deVriesandtheCamassa-Holmequationhaveageometriderivationandbotharemodelsforthepropagationofshallowwaterwaves,thetwoequa-tionshavequitedierentstruturalproperties.Forexample,whileallsmoothperiodiinitialdatafortheKorteweg-deVriesequationdevelopintoperiodiwavesthatexistforalltimes[48℄,smoothperiodiinitialdatafortheCamassa-Holmequationdevelopeitherintoglobalsolutionsorintobreakingwaves(seethepapers[5,8,9,39℄).Inthispaper,westudytheaseofright-invariantmetrisonthedieomorphismgroupoftheirle,Diff(S1).Notiehoweverthatasimilartheoryislikelywithouttheperiodiityondition(inwhihase,someweightedspaesexpresshowlosethedieomorphismsofthelinearetotheidentity[7℄).Eahright-invariantmetrionDiff(S1)isdenedbyaninnerprodutaontheLiealgebraofthegroup,Vect(S1)=C∞(S1).Ifthisinnerprodutisloal,itisgivenbytheexpressiona(u,v)=ZS1uA(v)dxu,v∈C∞(S1),whereAisaninvertible,symmetri,lineardierentialoperator.TothisinnerprodutonVect(S1),orrespondsaquadratifuntional(theenergyfuntional)HA(m)=12ZS1mA−1(m),onthe(regular)dualVect∗(S1).ItsorrespondingHamiltonianvetoreldXAgeneratestheEulerequationdmdt=XA(m).AmongEulerequationsofthatkind,wehavethewell-knowninvisidBurgersequationut+3uux=0,andCamassa-Holm[4,16℄shallowwaterequationut+uux+∂x(1−∂2x)−1u2+12u2x=0.Indeed,theinvisidBurgersequationorrespondstoA=I(L2innerprodut),whereastheCamassa-HolmequationorrespondstoA=I−D2(H1innerprodut)(see[10,11℄).Burgers,Korteweg-deVriesandofCamassa-Holmequationsarepreiselybi-Hamiltonianrelativelytosomeseondane(afterSouriau[47℄)ompatiblePoissonstruture3(see2However,thisformalismseemstohavebeenextendedtohydrodynamisbeforeArnoldbyMoreau[42℄.3TheanestrutureontheVirasoroalgebrawhihmakesKorteweg-deVriesequationabi-HamiltoniansystemseemstohavebeenrstdisoveredbyGardner[17℄andforthisreason,someauthorsallittheGardnerbraket(seealso[15℄.BI-HAMILTONIANSYSTEMS3[14,32,37℄).SinetheseequationsarespeialasesofEulerequationsinduedbyHk-metri,itisnaturaltoaskwhether,ingeneral,theseequationshavesimilarpropertiesforanyvalueofk.In[12℄,itwasshownthatthiswasnotthease.Therearenoanestru-tureonVect∗(S1)whihmak