On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations

Jacopo Bellazzini, Vladimir Simeonov Gueorguiev, Enno Lenzmann, Nicola Visciglia

    研究成果: Article

    抄録

    We consider nonlinear half-wave equations with focusing power-type nonlinearityi∂tu=-Δu-|u|p-1u,with(t,x)∈R×Rdwith exponents 1 < p< ∞ for d = 1 and 1 < p< (d+ 1) / (d- 1) for d ≥ 2. We study traveling solitary waves of the formu(t, x) = e i ω t Q v (x- vt) with frequency ω∈ R, velocity v∈ R d , and some finite-energy profile Q v ∈ H 1 / 2 (R d ) , Q v ≢ 0. We prove that traveling solitary waves for speeds | v| ≥ 1 do not exist. Furthermore, we generalize the non-existence result to the square root Klein–Gordon operator -Δ+m2 and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds | v| < 1. Finally, we discuss the energy-critical case when p= (d+ 1) / (d- 1) in dimensions d ≥ 2.

    元の言語English
    ジャーナルCommunications in Mathematical Physics
    DOI
    出版物ステータスPublished - 2019 1 1

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    Solitary Waves
    Traveling Wave
    wave equations
    Wave equation
    solitary waves
    Scattering
    scattering
    Critical Case
    Energy
    Square root
    Nonexistence
    nonlinearity
    Exponent
    exponents
    Nonlinearity
    operators
    Generalise
    energy
    profiles
    Operator

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematical Physics

    これを引用

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    title = "On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations",
    abstract = "We consider nonlinear half-wave equations with focusing power-type nonlinearityi∂tu=-Δu-|u|p-1u,with(t,x)∈R×Rdwith exponents 1 < p< ∞ for d = 1 and 1 < p< (d+ 1) / (d- 1) for d ≥ 2. We study traveling solitary waves of the formu(t, x) = e i ω t Q v (x- vt) with frequency ω∈ R, velocity v∈ R d , and some finite-energy profile Q v ∈ H 1 / 2 (R d ) , Q v ≢ 0. We prove that traveling solitary waves for speeds | v| ≥ 1 do not exist. Furthermore, we generalize the non-existence result to the square root Klein–Gordon operator -Δ+m2 and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds | v| < 1. Finally, we discuss the energy-critical case when p= (d+ 1) / (d- 1) in dimensions d ≥ 2.",
    author = "Jacopo Bellazzini and Gueorguiev, {Vladimir Simeonov} and Enno Lenzmann and Nicola Visciglia",
    year = "2019",
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    doi = "10.1007/s00220-019-03374-y",
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    AU - Bellazzini, Jacopo

    AU - Gueorguiev, Vladimir Simeonov

    AU - Lenzmann, Enno

    AU - Visciglia, Nicola

    PY - 2019/1/1

    Y1 - 2019/1/1

    N2 - We consider nonlinear half-wave equations with focusing power-type nonlinearityi∂tu=-Δu-|u|p-1u,with(t,x)∈R×Rdwith exponents 1 < p< ∞ for d = 1 and 1 < p< (d+ 1) / (d- 1) for d ≥ 2. We study traveling solitary waves of the formu(t, x) = e i ω t Q v (x- vt) with frequency ω∈ R, velocity v∈ R d , and some finite-energy profile Q v ∈ H 1 / 2 (R d ) , Q v ≢ 0. We prove that traveling solitary waves for speeds | v| ≥ 1 do not exist. Furthermore, we generalize the non-existence result to the square root Klein–Gordon operator -Δ+m2 and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds | v| < 1. Finally, we discuss the energy-critical case when p= (d+ 1) / (d- 1) in dimensions d ≥ 2.

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