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1、ANEWGEOMETRICSETTINGFORLAXEQUATIONS~JFCarinenayandEMartnezzyDepto.deFsicaTeorica,UniversidaddeZaragoza50009-Zaragoza,SpainzDepto.deMatematicaAplicada,CPSI,UniversidaddeZaragozaMaradeLuna3,50015-Zaragoza,SpainAbstract.{WeshowtheexistenceofLaxequationswhichdoesnotcorre-spondtothevani
2、shingoftheLieorcovariantderivativeofa(1,1)-typetensor eldwithrespecttoavector eld,buttheyhaveadi erentorigin,andweprovideamoregeneralsettingforthiskindofequations.PACS:02.30.+m,03.20.+i.1991MSC:34A26,34C20,58F07.Keywords:Laxequations,secondorderdi erentialequations,derivations.1.Introducti
3、onLaxequations[1]wereintroducedwhenstudyingisospectralproblems.Moreespeci caly,ifL(t)isafamilyoflinearoperatorsdependingonaparam-etertandtheevolutionequationintcanbewrittenasL=[M;L],thentheteigenvalues(t)ofL(t)areconstantandtheireigenvectors,i.e.L(t)=,evolveaccordingto=M.InthatwaytheKort
4、ewegdeVriesequationistassociatedwithSchrodingerequationandtheexistenceofanin nitenumberofconstantsofmotionininvolutionisrelatedwiththepossibilityofsuchdescrip-tionasaLaxequation.Whenitwasdiscoveredthatmanyinterestingphysicalsystemsdescribedbypartialdi erentialequationscanbedealtwithinthefr
5、ameworkofin nite-dimensionalHamiltoniansystems,theinterestinlookingforageometricalsettingfortheabovementionedLaxequationswasevidentandseveralpapers[2,3]proposedageometricalinterpretationascorrespondingtothevanishingoftheLiederivativeofatype(1;1)tensor eldwrtavector eldX,L=0.Thisfactisver
6、yimportantbecausethereexistmanywaysXofconstructionofsuchtensor eldsinvariantunderavector eldX.ThisisforinstancethecasewhenXisalocally-Hamiltonianvector eldinasymplectic0manifold(M;!)andthereexistsanotherX-invarianttwoform!.Suchaninvariantformcanbefoundwhenwehaveasymmetry,eithera nite'orani
7、n nitesimaloneY,ofXwhichisnotasymmetryof!.Infact,itsucesto1~2CarinenaandMartnezconsidertherelation'L!=L'!,or[LL?LL]!=0,inXXYYX'(X)00ordertoseethat!='!inthe rstcaseor!=L!inthesecondoneisY?10suchinvariant2-form.Then,the(1,1)tensor eld:!^!^obtainedbyV1
