novikov s.p. solitons and geometry (lezioni fermiane, cambridge univ. press, 1994, web draft, 1993)(50s)_pd_

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1、Revisiondate:4/11/93SolitonsandGeometryS.P.Novikov1Lecture1Introduction.Planofthelectures.Poissonstructures.ThetheoryofSolitons(solitarywaves")dealswiththepropagationofnon-linearwavesincontinuummedia.Theirfamousdiscoveryhasbeendoneintheperiod1965-1968(

2、byM.KruscalandN.Zabuski,1965;G.Gardner,I.Green,M.KruscalandR.Miura,1967;P.Lax,1968

3、seethesurvey[8]orthebook[22]).ThefamiliarKdV(Korteweg-deVries)non-linearequationwasfoundtobeexactlysolvableinsomeprofoundnontrivialsensebytheso-calledInverseScatteringTr

4、ansform"(IST)atleastfortheclassofrapidlydecreasinginitialdata©(x):©t=6©©x¡©xxx[KdV]:Someotherfamoussystemsarealsosolvablebyanalogousprocedures.Thefollowingareexamplesof1+1-systems:©´³=sin©[SG]2i©t=¡©xx§j©j©[NS§]¢¢¢:Themostinterestingintegrable2+1-system

5、sarethefollowing:8

6、³=sin©appearedforthe¯rsttimeinaproblemof3-dimensionalgeometry:itdescribeslocallytheisometricimbed-dingsoftheLobatchevski2-planeL2(i.e.thesurfacewithconstantnegativeGaussiancurvature)intheEuclidean3-spaceR3.Here©istheanglebe-tweentwoasymptoticdirections(

7、´;³)onthesurfacealongwhichthesecond2(curvature)formiszero.IthasbeenusedbyBianchi,LieandBacklundfortheconstructionofnewimbeddings(Backlundtransformations",discoveredbyBianchi).b)Theellipticequation4©=sinh©appearedrecentlyforthedescriptionofgenus1surface

8、s(topologicaltori")inR3withconstantmeancurvature(H.Wente,1986;R.Walter,1987).Startingfrom1989F.Hitchin,U.Pinkall,N.Ercolani,H.Knorrer,E.Trubowitz,A.Bobenkousedinthis¯eldthetechniqueoftheperiodicIST"

9、see[5].Solitonsandalgebraicgeometry.a)Thereisafamous

10、connectionofSolitontheorywithalgebraicgeometry.Itappearedin1974-1975.ThesolutionoftheperiodicproblemsofSolitontheoryledtobeautifulanalyticalconstructionsinvolvingRiemannsurfacesandtheirJacobianvarieties,£-functionsandlateralsoPry