一类具有位势的非线性薛定谔方程组的定性分析.pdf

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2、½5©ÛQualitativeanalysisforasystemofnonlinearSchr¨odingerequationwithpotentialÆ;µA^êÆïÄ)µ4==µykuBÇU9ÆnÆ"c8ÕM5(²<(²¤¥ÆØ©´<3e?1ïÄóÚïĤJ§Ø©¥AO±I5Ú?§Ø©¥Ø¹Ù¦<®²uL½>LïĤJ§Ø¹¼U9ƽ٦ÅƽyÖ¦^Lá.·ÓóÓéïĤ?Ûzþ®3Ø©¥²(`²¿L«¿.ÆØ©ö¶µiFÏ:cFÆØ©¦^ÇÖÆØ©ö

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6、ЯK,§Xeµiu=−1∆u+V(x)u−µ

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10、p1−2

11、u

12、q1u,1t211111121iu=−1∆u+V(x)u−γ

13、u

14、pu−β

15、u

16、p1

17、u

18、q1−2u;2t222222122u1(x,0)=u10,u2(x,0)=u20,x∈RN,t>0.·Ì'%þã§)Û35±9k»,ÙÌSN´éc<óo(Úí2

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20、;³;k»;Û35AbstractSchr¨odingerequationwasputforwardbyAustrianphysicistSchr¨odinger§itisthebasicequationinquantummechanicsis.Schr¨odingerequationrevealsthemotionlawofmicroscopicparticlesandveryimportantfortheresearchofquantummechanicsis

21、.Ontheotherhand,Schr¨odingerequationisatypicalmodelofdispersionequation.ItcanbealsoappliedinthefieldofPhysics,ChemistryandOptics.Therefore,itisveryinterestingtostudythepropertiesforthesolution,meanwhile,newdirectionsandchallengesareproposedinthefieldofmathematicalphysicsequations.Howe

22、ver,itisoftentechnicallysimplerthanotherdispersiveequationlikethewaveequationorKdvequation.ByusingenergymethodandStrichartzestimate,mathematiciansstudyboththeproblemsoflocalnature(localexistenceofsolutions,uniqueness,regularity,smoothingimpact)andtheproblemsofglobalnature(globalexis

23、tence,finite-timeblowup,asymptoticbehaviorofsolutions,scattering).Sofar,aperfecttheoreticalstructureisestablished.ThepapermainlystudiestheinitialvalueproblemsofthenonlinearSchrodingerequationswithpotentialsinn-dEuclideanspace,asfollowsiu=−1∆u+V(x)u−µ

24、u

25、pu−α

26、u

27、p1−2

28、u

29、q1u,1t211111

30、121iu=−1∆u+V(x)u−γ

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