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1、AnalysisonFaddeevKnotsandSkyrmeSolitons:RecentProgressandOpenProblemsFanghuaLinandYisongYangItisourgreatpleasuretodedicatethisarticletoHaimBrezisforhisleadership,encouragement,andsupportovermanyyears.Abstract.Wereportsomerecentprogressanddiscusssomerelatedunsolvedproblemsconcerningtheexi
2、stenceoftheenergy-minimizingconfigurationsintheFaddeevquantumfieldtheorymodelgivingrisetoknottedsolitonsandintheSkyrmemodelmodelingelementaryparticles.ManyissuesrelatedtothecorrespondingevolutionarysystemsfortheFaddeevknotsandSkyrmesolitons,however,remainuntouched.Theseareratherunusualandc
3、hallengingsystemsofnonlinearpartialdifferentialequationsofhyperbolictype.Wealsodescribesomesimplebutfundamentalmathematicalissuesconcerningthesemodels.1.IntroductionTheareaofquantumfieldtheoryisfascinatingforanalystsbecauseitinvolvesalltypesofanalysisproblemsatboththeclassicallevelsandtheq
4、uantumlevelsofdiversesubjectsoffundamentalimportance.Accordingtoquantumfieldtheory,elementaryparticlesandtheirinteractionsareelegantlydescribedbycontinuousfieldsdefinedoverspacetimeandhostedwithinavectorbundle.Spacetimesym-metryisrelatedtospecialrelativityandgravityandvectorbundlesymmetryis
5、relatedtoallotherfundamentalinter-particleforces.Therearemanyinterest-ingandimportantmathematicalissuesinquantumfieldtheoryworthyofpursuit,whichprovidechallengesaswellasopportunitiesformathematicians.Forexample,topologicalsolitonsknownasdomainwalls,vortices,monopoles,andinstantonsintheirs
6、implestcasesareoftenrelatedtointegrablesystemsandtheconstructionofthesesolutionsleadstofurtherdevelopmentofthesolutionmethodsincludingin-versescatteringmethod[1,34,48,60,95],theB¨acklundtransformations[1,72],thePenrosetwistormethod[8,86]andADHM[6],duality[45],grouprepresen-tations[46];wh
7、enthesystemsencounteredarenotintegrable,functionalanalysismethodsareneededinordertounderstandthestructureofthesolutions[41,93].2000MathematicsSubjectClassification.35J20,35Q51,58Z05,81T45,81V35.Keywordsandphrases.Quantumfieldtheory,topologicalinvariants,Faddeevmodel,Skyrmem
