On a system of nonlinear Schrödinger equations with quadratic interaction

    Research output: Contribution to journalArticle

    27 Citations (Scopus)

    Abstract

    We study a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n≤6. The Cauchy problem is studied in L 2, in H1, and in the weighted L2 space 〈 x〉-1L2=F(H1) under mass resonance condition, where 〈x〉=( 1+|x|2)1/2 and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.

    Original languageEnglish
    Pages (from-to)661-690
    Number of pages30
    JournalAnnales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
    Volume30
    Issue number4
    DOIs
    Publication statusPublished - 2013 Jul

    Fingerprint

    System of Nonlinear Equations
    Nonlinear equations
    Ground state
    Ground State
    Conformal Transformation
    Blow-up Solution
    Weighted Spaces
    Invariance
    Interaction
    Variational Methods
    Fourier transform
    Cauchy Problem
    Fourier transforms
    Form

    ASJC Scopus subject areas

    • Analysis
    • Mathematical Physics

    Cite this

    @article{b0b6951d21fc40f18b2dee7b3b4c631d,
    title = "On a system of nonlinear Schr{\"o}dinger equations with quadratic interaction",
    abstract = "We study a system of nonlinear Schr{\"o}dinger equations with quadratic interaction in space dimension n≤6. The Cauchy problem is studied in L 2, in H1, and in the weighted L2 space 〈 x〉-1L2=F(H1) under mass resonance condition, where 〈x〉=( 1+|x|2)1/2 and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.",
    author = "Nakao Hayashi and Tohru Ozawa and Kazunaga Tanaka",
    year = "2013",
    month = "7",
    doi = "10.1016/j.anihpc.2012.10.007",
    language = "English",
    volume = "30",
    pages = "661--690",
    journal = "Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis",
    issn = "0294-1449",
    publisher = "Elsevier Masson SAS",
    number = "4",

    }

    TY - JOUR

    T1 - On a system of nonlinear Schrödinger equations with quadratic interaction

    AU - Hayashi, Nakao

    AU - Ozawa, Tohru

    AU - Tanaka, Kazunaga

    PY - 2013/7

    Y1 - 2013/7

    N2 - We study a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n≤6. The Cauchy problem is studied in L 2, in H1, and in the weighted L2 space 〈 x〉-1L2=F(H1) under mass resonance condition, where 〈x〉=( 1+|x|2)1/2 and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.

    AB - We study a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n≤6. The Cauchy problem is studied in L 2, in H1, and in the weighted L2 space 〈 x〉-1L2=F(H1) under mass resonance condition, where 〈x〉=( 1+|x|2)1/2 and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.

    UR - http://www.scopus.com/inward/record.url?scp=84881164350&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84881164350&partnerID=8YFLogxK

    U2 - 10.1016/j.anihpc.2012.10.007

    DO - 10.1016/j.anihpc.2012.10.007

    M3 - Article

    AN - SCOPUS:84881164350

    VL - 30

    SP - 661

    EP - 690

    JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

    JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

    SN - 0294-1449

    IS - 4

    ER -