On the classification of the spectrally stable standing waves of the Hartree problem

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial dimension. We construct variationally the “normalized” solutions for the corresponding Choquard–Pekar model—in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz–Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the “normalized” solutions (which exist only in a portion of the range) are spectrally stable.

    Original languageEnglish
    Pages (from-to)29-39
    Number of pages11
    JournalPhysica D: Nonlinear Phenomena
    Volume370
    DOIs
    Publication statusPublished - 2018 May 1

    Fingerprint

    standing waves
    solitary waves
    nonlinearity
    bells

    Keywords

    • Ground states
    • Hartree equation
    • Semilinear

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Condensed Matter Physics

    Cite this

    On the classification of the spectrally stable standing waves of the Hartree problem. / Gueorguiev, Vladimir Simeonov; Stefanov, Atanas.

    In: Physica D: Nonlinear Phenomena, Vol. 370, 01.05.2018, p. 29-39.

    Research output: Contribution to journalArticle

    @article{9341e08186864b84b42f9cd1c0dedae8,
    title = "On the classification of the spectrally stable standing waves of the Hartree problem",
    abstract = "We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial dimension. We construct variationally the “normalized” solutions for the corresponding Choquard–Pekar model—in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz–Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the “normalized” solutions (which exist only in a portion of the range) are spectrally stable.",
    keywords = "Ground states, Hartree equation, Semilinear",
    author = "Gueorguiev, {Vladimir Simeonov} and Atanas Stefanov",
    year = "2018",
    month = "5",
    day = "1",
    doi = "10.1016/j.physd.2018.01.002",
    language = "English",
    volume = "370",
    pages = "29--39",
    journal = "Physica D: Nonlinear Phenomena",
    issn = "0167-2789",
    publisher = "Elsevier",

    }

    TY - JOUR

    T1 - On the classification of the spectrally stable standing waves of the Hartree problem

    AU - Gueorguiev, Vladimir Simeonov

    AU - Stefanov, Atanas

    PY - 2018/5/1

    Y1 - 2018/5/1

    N2 - We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial dimension. We construct variationally the “normalized” solutions for the corresponding Choquard–Pekar model—in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz–Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the “normalized” solutions (which exist only in a portion of the range) are spectrally stable.

    AB - We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial dimension. We construct variationally the “normalized” solutions for the corresponding Choquard–Pekar model—in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz–Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the “normalized” solutions (which exist only in a portion of the range) are spectrally stable.

    KW - Ground states

    KW - Hartree equation

    KW - Semilinear

    UR - http://www.scopus.com/inward/record.url?scp=85043370144&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85043370144&partnerID=8YFLogxK

    U2 - 10.1016/j.physd.2018.01.002

    DO - 10.1016/j.physd.2018.01.002

    M3 - Article

    VL - 370

    SP - 29

    EP - 39

    JO - Physica D: Nonlinear Phenomena

    JF - Physica D: Nonlinear Phenomena

    SN - 0167-2789

    ER -