Abstract：Inthistalk,wegivesomerecentresultsontheexistenceofgroundstatefornonlinearSchr?dinger-Poisson-Slaterequationwithunboundedpotential.ByusingEkeland’svariationalprincipleweprovethatthereexistsagroundstatewithnegativeenergylevel.ForthespecialcaseofSchr?dinger-Poisson-Slaterequationwithharmonicpotential,weshowthatthegroundstatemustbenonradial.

Abstract:Inthistalk,wepresentsomerecentresultsonthestabilityoftravelingwavesofthreedimensionalGross-Pitaevskii(GP)equation.TheGPequationisanonlinearSchrodingerequationwithnonzeroconditionatinfinity,whichhastravelingwaveswithnon-vanishinglimitatinfinity,firstdiscoveredbyphysicists(Jones,Robertsetal.)in1980s.TheexistenceofsuchtravelingwaveshasbeenstudiedalotinrecentyearsbyBetheul,Saut,Marisandmanyothers.However,thestabilityanddynamicalbehaviorsofthesetravelingwavesarenotwellunderstood.IncollaborationwithprofessorsZhiwuLinandChongchunZengatGeorgiaTech,weprovedanonlinearstabilitycriterionfor3DtravelingwavesofGPequationasconjecturedinthephysicalliterature.