An elementary construction of complex patterns in nonlinear schrödinger equations

Manuel Del Pino, Patrido Felmer, Kazunaga Tanaka

    Research output: Contribution to journalArticle

    20 Citations (Scopus)

    Abstract

    We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

    Original languageEnglish
    Pages (from-to)1653-1671
    Number of pages19
    JournalNonlinearity
    Volume15
    Issue number5
    DOIs
    Publication statusPublished - 2002 Sep

    Fingerprint

    Spike
    spikes
    Nonlinear equations
    nonlinear equations
    Nonlinear Equations
    concentrating
    Standing Wave
    standing waves
    Small Parameter
    Set of points
    integers
    Geodesic
    Finite Set
    Integer
    Arbitrary
    Energy
    energy

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics
    • Mathematical Physics
    • Statistical and Nonlinear Physics

    Cite this

    An elementary construction of complex patterns in nonlinear schrödinger equations. / Del Pino, Manuel; Felmer, Patrido; Tanaka, Kazunaga.

    In: Nonlinearity, Vol. 15, No. 5, 09.2002, p. 1653-1671.

    Research output: Contribution to journalArticle

    Del Pino, Manuel ; Felmer, Patrido ; Tanaka, Kazunaga. / An elementary construction of complex patterns in nonlinear schrödinger equations. In: Nonlinearity. 2002 ; Vol. 15, No. 5. pp. 1653-1671.
    @article{ff2538d6cc42439dbfb9b1d0b018595a,
    title = "An elementary construction of complex patterns in nonlinear schr{\"o}dinger equations",
    abstract = "We consider the problem of finding standing waves to a nonlinear Schr{\"o}dinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.",
    author = "{Del Pino}, Manuel and Patrido Felmer and Kazunaga Tanaka",
    year = "2002",
    month = "9",
    doi = "10.1088/0951-7715/15/5/315",
    language = "English",
    volume = "15",
    pages = "1653--1671",
    journal = "Nonlinearity",
    issn = "0951-7715",
    publisher = "IOP Publishing Ltd.",
    number = "5",

    }

    TY - JOUR

    T1 - An elementary construction of complex patterns in nonlinear schrödinger equations

    AU - Del Pino, Manuel

    AU - Felmer, Patrido

    AU - Tanaka, Kazunaga

    PY - 2002/9

    Y1 - 2002/9

    N2 - We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

    AB - We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

    UR - http://www.scopus.com/inward/record.url?scp=0041972657&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0041972657&partnerID=8YFLogxK

    U2 - 10.1088/0951-7715/15/5/315

    DO - 10.1088/0951-7715/15/5/315

    M3 - Article

    AN - SCOPUS:0041972657

    VL - 15

    SP - 1653

    EP - 1671

    JO - Nonlinearity

    JF - Nonlinearity

    SN - 0951-7715

    IS - 5

    ER -