Relationship between Hirota's method and the inverse spectral method - The Korteweg-de Vries equation's case

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    Abstract

    Recently, it is shown that for a number of soliton equations, their solutions expressing multiple solitons in a background of ripples, which may be called generalized soliton solutions, can be constructed using Hirota's bilinear forms of these soliton equations (S. OISHI: submitted to J. Phy. Soc. Jpn.). In this letter, taking the KdV equation as an example, relationship between Hirota's method and the inverse spectral method is clarified by showing that its generalized soliton solutions can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marčenko integral equation.

    Original languageEnglish
    Pages (from-to)1037-1038
    Number of pages2
    JournalJournal of the Physical Society of Japan
    Volume47
    Issue number3
    Publication statusPublished - 1979 Sep

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    spectral methods
    solitary waves
    ripples
    determinants
    integral equations
    gels

    ASJC Scopus subject areas

    • Physics and Astronomy(all)

    Cite this

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    title = "Relationship between Hirota's method and the inverse spectral method - The Korteweg-de Vries equation's case",
    abstract = "Recently, it is shown that for a number of soliton equations, their solutions expressing multiple solitons in a background of ripples, which may be called generalized soliton solutions, can be constructed using Hirota's bilinear forms of these soliton equations (S. OISHI: submitted to J. Phy. Soc. Jpn.). In this letter, taking the KdV equation as an example, relationship between Hirota's method and the inverse spectral method is clarified by showing that its generalized soliton solutions can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marčenko integral equation.",
    author = "Shinichi Oishi",
    year = "1979",
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    T1 - Relationship between Hirota's method and the inverse spectral method - The Korteweg-de Vries equation's case

    AU - Oishi, Shinichi

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    N2 - Recently, it is shown that for a number of soliton equations, their solutions expressing multiple solitons in a background of ripples, which may be called generalized soliton solutions, can be constructed using Hirota's bilinear forms of these soliton equations (S. OISHI: submitted to J. Phy. Soc. Jpn.). In this letter, taking the KdV equation as an example, relationship between Hirota's method and the inverse spectral method is clarified by showing that its generalized soliton solutions can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marčenko integral equation.

    AB - Recently, it is shown that for a number of soliton equations, their solutions expressing multiple solitons in a background of ripples, which may be called generalized soliton solutions, can be constructed using Hirota's bilinear forms of these soliton equations (S. OISHI: submitted to J. Phy. Soc. Jpn.). In this letter, taking the KdV equation as an example, relationship between Hirota's method and the inverse spectral method is clarified by showing that its generalized soliton solutions can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marčenko integral equation.

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