ON THE STANDING WAVES FOR THE X-RAY FREE ELECTRON LASER SCHRODINGER EQUATION

doi:10.3934/dcds.2022139

	ON THE STANDING WAVES FOR THE X-RAY FREE ELECTRON LASER SCHRODINGER EQUATION
	Cao, Daomin 1,3; Feng, Binhua 2; Luo, Tingjian 1
刊名	DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
	2022-10-01
页码	41
关键词	XFEL Schr?dinger equation nonlinear elliptic equations normalized solutions orbital stability strong instability
ISSN号	1078-0947
DOI	10.3934/dcds.2022139
英文摘要	In this paper, we are concerned with the standing waves for the following nonlinear Schrodinger equation i partial differential t psi =-Delta psi+b2(x21+x22)psi+ lambda 1 \|x\|psi+lambda 2(\|center dot\|-1*\|psi \|2)psi-lambda 3\|psi\|p psi, (t, x) is an element of R+xR3, where 0 < p < 4. This equation arises as an effective single particle model in X-ray Free Electron Lasers. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved (i.e. b = 0 in the equation), and the second one is that b =6 0. To be precise, in the first case, when p is an element of [43 , 4), by considering a minimization problem on a suitable Pohozaev manifold, we prove the existence of radial ground states, and show further that the corresponding standing waves are strongly unstable by blow-up in a finite time. Moreover, by making use of the ideas of these proofs, we are able to prove the existence of normalized solutions, whose proof seems to be new, compared with the studies of normalized solutions in the existing literature. This study also indicates that there is a close connection between the study of the strong instability and the one of the existence of normalized solutions. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for p is an element of (0, 4), where solutions are obtained as global minimizers when p is an element of (0, 43], and as local minimizers when p is an element of [43 , 4). In the mass-critical and supercritical cases p is an element of [43 , 4), we also establish the variational characterization of ground state solutions on a new manifold which is neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on the coefficients, we prove that the ground state standing waves are strongly unstable.
资助项目	National Natural Science Foundation of China[11501137] ; Outstanding Youth Science Fund of Gansu Province[20JR10RA111] ; Department of Education of Gansu Province: Youth Doctoral Fund Project[2022QB-031] ; Guangdong Basic and Applied Basic Research Foundation[2016A030310258] ; Guangdong Basic and Applied Basic Research Foundation[2020A1515011019] ; [NWNU-LKZD202203]
WOS研究方向	Mathematics
语种	英语
出版者	AMER INST MATHEMATICAL SCIENCES-AIMS
WOS记录号	WOS:000875659900001
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/60801]
专题	中国科学院数学与系统科学研究院
通讯作者	Feng, Binhua; Luo, Tingjian
作者单位	1.Guangzhou Univ, Sch Math & Informat Sci, Guangzhou 510006, Peoples R China 2.Northwest Normal Univ, Dept Math, Lanzhou 730070, Peoples R China 3.Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Cao, Daomin,Feng, Binhua,Luo, Tingjian. ON THE STANDING WAVES FOR THE X-RAY FREE ELECTRON LASER SCHRODINGER EQUATION[J]. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2022:41.
APA	Cao, Daomin,Feng, Binhua,&Luo, Tingjian.(2022).ON THE STANDING WAVES FOR THE X-RAY FREE ELECTRON LASER SCHRODINGER EQUATION.DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,41.
MLA	Cao, Daomin,et al."ON THE STANDING WAVES FOR THE X-RAY FREE ELECTRON LASER SCHRODINGER EQUATION".DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS (2022):41.