简介：本文研究了Hilbert空间L^2（R^2）上由势函数V（x）（V≥0，连续）给出的一类Schrǒdinger算子H=-△＋V的谱。本文的主要结果：（1）H的谱σ（H）不会出现本性谱与离散谱交替出现的情况，其谱要么是离散的，要么从infσcos（H）开始全是本性谱；（2）lim‖x‖→∞V（x）=∞是σcos（H）=φ的充要条件。（3）借助于讨论H的Zhis-lin谱，在一定的条件下。lim‖x‖→∞V（x）=0是σcos（H）=[∞，0）的充要条件。我们还提出了几个没有解决的问题。

简介：WediscussseparabilityofsolutionstoaSchr?dingerequationthatdescribesacompositequantumsystemandgivesomekindsofHamiltoniansH(t)suchthatthesolutiontoSchr?dingerequationinducedbyH(t)isseparableatanytimeprovidedthatitisseparableatt=0.Forexample,weprovethatiftheHamiltonianHistime-independentandequalstotheproductPA■PBoftwoprojectionsonthesubsystemsKAandKB,respectively,thenthestate|ψ(t)ofthecompositesystemstartingfromaseparableinitial|ψ(0)=|ψA■|ψBisseparableforallt∈[0,T]ifandonlyifeither|ψAisaneigenstateofPA,or|ψBisaneigenstateofPB.

简介：VariationalmethodsareusedtostudythenonlinearSchrdinger-Poissontypeequationswhichmodeltheelectromagneticwavepropagatingintheplasmainphysics.ByanalyzingtheHamiltonianpropertytoconstructaconstrainedvariationalproblem,theexistenceofthegroundstateofthesystemisobtained.Furthermore,itisshownthatthegroundstateisorbitallystable.

简介：Inthispaper,weinvestigatethesemiclassicallimitofthegeneralizednonlinearSchr?dingerequationforinitialdatawithSobolevregularity.Also,wewillanalyzethestructureofthefluiddynamicalsystemwithquantumeffectcorrespondingtothesemiclassicallimitofthegeneralizednonlinearSchr?dingerequation.

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简介：Weprovetheexistenceandtheorbitalstabilityofstandingwavesforthenonau-tonomousSchrdingersystemiut+ua(x)u+(|u|2p+b(x)|u|p-1|v|p+1)u=0,x∈RN,ivt+va(x)v+|v|2p+b(x)|v|p-1|u|p+1v=0,x∈RNundersuitableconditionsonthecoefficientfunctionsaandb.Wefollowtheideaofanalyzingthecompactnessofminimizingsequenceoftheconstrainedminimizationproblems.

简介：Westudythedynamicsofthecubic–quinticnonlinearSchr?dingerequationbythesymplecticmethod.Thebehaviorsoftheequationarediscussedwithharmonicallymodulatedinitialconditions,andthecontributionsfromthequintictermarediscussed.Weobservetheellipticorbit,homoclinicorbitcrossing,quasirecurrence,andstochasticmotionwithdifferentnonlinearparametersinthissystem.Numericalsimulationsshowthatthechangingprocessesofthemotionofthesystemandthetrajectoriesinthephasespacearevariousfordifferentcubicnonlinearparameterswiththeincreaseofthequinticnonlinearparameter.

简介：Thispaperconsiderstheone-dimensionaldissipativecubicnonlinearSchrdingerequationwithzeroDirichletboundaryconditionsonaboundeddomain.Theequationisdiscretizedintimebyalinearimplicitthree-levelcentraldifferencescheme,whichhasanalogousdiscreteconservationlawsofchargeandenergy.Theconvergencewithtwoordersandthestabilityoftheschemeareanalysedusingaprioriestimates.Numericaltestsshowthatthethree-levelschemeismoreefficient.

简介：LetL=-?+VbeaSchrdingeroperatoractingonL2(Rn),n≥1,whereV≡0isanonnegativelocallyintegrablefunctiononRn.Inthisarticle,wewillintropduceweightedHardyspacesHL(w)associatedwithLbymeansofthesquarefunctionandthenstudytheiratomicdecompositiontheory.WewillalsoshowthattheRiesztransform?L-1/2associatedwithLisboundedfromournewspaceHpL(w)totheclassicalweightedHardyspaceHp(w)whenn/(n+1)

简介：利用二阶微商的四阶精度紧致差分逼近公式，给出解Schroedinger方程的精度为O((1-2θ)τ+τ^2h^4)的一个新的加权差分格式，当1／2≤θ≤1时格式绝对稳定．特别地，当θ=1／2时，文章所给出的差分格式可高达四阶精度，数值结果与理论分析相一致．

简介：Inthispaper,severalnewconstant-amplitudeandvariable-amplitudewavesolutions(namely,travelingwavesolutions)ofageneralizednonlinearSchrdingerequationareinvestigatedbyusingtheextendedhomogeneousbalancemethod,wherethebalancemethodisappliedtosolvetheRiccatiequationandthereducednonlinearordinarydifferentialequation,respectively.Inaddition,stabilityanalysisofthosesolutionsarealsoconductedbyregularphaseplanetechnique.

简介：WestudythegeneralizedDarbouxtransformationtothethree-componentcouplednonlinearSchr¨odingerequation.First-andsecond-orderlocalizedwavesareobtainedbythistechnique.Infirst-orderlocalizedwave,wegettheinteractionalsolutionsbetweenfirst-orderroguewaveandone-dark,one-brightsolitonrespectively.Meanwhile,theinteractionalsolutionsbetweenone-breatherandfirst-orderroguewavearealsogiven.Insecond-orderlocalizedwave,one-dark-one-brightsolitontogetherwithsecond-orderroguewaveispresentedinthefirstcomponent,andtwo-brightsolitontogetherwithsecond-orderroguewavearegainedrespectivelyintheothertwocomponents.Besides,weobservesecond-orderroguewavetogetherwithone-breatherinthreecomponents.Moreover,byincreasingtheabsolutevaluesoftwofreeparameters,thenonlinearwavesmergewitheachotherdistinctly.Theseresultsfurtherrevealtheinterestingdynamicstructuresoflocalizedwavesinthethree-componentcoupledsystem.

简介：ThefollowingcoupledSchrdingersystemwithasmallperturbationuxx+u-u3+βuv2+f(,u,ux,v,vx)=0inR,vxx-v+v3+βu2v+g(,u,ux,v,vx)=0inRisconsidered,whereβandaresmallparameters.ThewholesystemhasaperiodicsolutionwiththeaidofaFourierseriesexpansiontechnique,anditsdominantsystemhasaheteroclinicsolution.Thenadjustingsomeappropriateconstantsandapplyingthefixedpointtheoremandtheperturbationmethodyieldthatthisheteroclinicsolutiondeformstoaheteroclinicsolutionexponentiallyapproachingtheobtainedperiodicsolution(calledthegeneralizedheteroclinicsolutionthereafter).

简介：TheprimarygoalofthispaperistopresentacomprehensivestudyofthenonlinearSchrdingerequationswithcombinednonlinearitiesofthepower-typeandHartreetype.Undercertainstructuralconditions,theauthorsareabletoprovideacompletepictureofhowthenonlinearSchrdingerequationswithcombinednonlinearitiesinteractinthegivenenergyspace.ThemethodusedinthepaperisbasedupontheMorawetzestimatesandperturbationprinciples.

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简介：Byusingthemethodofdynamicalsystem,theexacttravellingwavesolutionsofthecouplednonlinearSchrdinger-KdVequationsarestudied.Basedonthismethod,allphaseportraitsofthesystemintheparametricspacearegiven.Allpossibleboundedtravellingwavesolutionssuchassolitarywavesolutionsandperiodictravellingwavesolutionsareobtained.WiththeaidofMaplesoftware,thenumericalsimulationsareconductedforsolitarywavesolutionsandperiodictravellingwavesolutionstothecouplednonlinearSchrdinger-KdVequations.Theresultsshowthatthepresentedfindingsimprovetherelatedpreviousconclusions.

简介：WeinvestigatethePainlevéintegrabilityofnonautonomousnonlinearSchrdinger(NLS)equationswithbothspace-andtime-dependentdispersion,nonlinearity,andexternalpotentials.ThePainlevéanalysisiscarriedoutwithoutusingtheKruskal'ssimplification,whichresultsinmoregeneralizedformofinhomogeneousequations.TheobtainedequationsareshowntobereducibletothestandardNLSequationbyusingapointtransformation.WealsoconstructthecorrespondingLaxpairandcarryoutitsKundu-typereductiontothestandardLaxpair.SpecialcasesofequationsfromchoosinglimitedformofcoefficientscoincidewiththeequationsfromthepreviousPainlevéanalysesand/orbecomeunknownnewequations.

简介：本文运用一种变量代换将非线性Sdhrodinger方程转变为半线性椭圆型方程,再利用山路引理,Lion集中紧引理,Soblev嵌入不等式证明一类Schrodinger方程孤子解的存在性.

简介：Thispaperdisccussestheinversescatteringproblemforone-dimensionalSchrodingeroperatorsrelatedtothegeneralStarkeffect.WeprovideaganeralframeworkwhichcanbeappliedbothtotheStark-effectoperatorandtheordinarySchrodingeroperator.

简介：LetL=L0+VbethehigherorderSchrdigertypeoperatorwhereL0isahomogeneousellipticoperatoroforder2mindivergenceformwithboundedcoefficientsandVisarealmeasurablefunctionasmultiplicationoperator(e.g.,including(-?)m+V(m∈N)asspecialexamples).Inthispaper,assumethatVsatisfiesastronglysubcriticalformconditionassociatedwithL0,theauthorsattempttoestablishatheoryofHardyspaceHpL(Rn)(0