A method of analysing soliton equations by bilinearization

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    Recently, a class of new solutions have been derived for a number of soliton equations using Hirota's bilinear forms of these soliton equations (S. Oishi: J. Phys. Soc. Jpn. 47 (1979) 1341). These solutions express solitons in a background of ripples, and are named generalized soliton solutions. In this paper, it is shown that the generalized soliton solutions for the Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marchenko integral equation. Using this result, relationship between Hirota's method and the inverse spectral method isclarified. Moreover, it is also shown that the initial value problems for these two equations can be solved using their generalized soliton solutions.

    Original languageEnglish
    Pages (from-to)639-646
    Number of pages8
    JournalJournal of the Physical Society of Japan
    Volume48
    Issue number2
    Publication statusPublished - 1980 Feb

    Fingerprint

    solitary waves
    spectral methods
    ripples
    determinants
    boundary value problems
    integral equations
    gels

    ASJC Scopus subject areas

    • Physics and Astronomy(all)

    Cite this

    A method of analysing soliton equations by bilinearization. / Oishi, Shinichi.

    In: Journal of the Physical Society of Japan, Vol. 48, No. 2, 02.1980, p. 639-646.

    Research output: Contribution to journalArticle

    @article{60997db3de0d4f9cb4885e5dd0cb4560,
    title = "A method of analysing soliton equations by bilinearization",
    abstract = "Recently, a class of new solutions have been derived for a number of soliton equations using Hirota's bilinear forms of these soliton equations (S. Oishi: J. Phys. Soc. Jpn. 47 (1979) 1341). These solutions express solitons in a background of ripples, and are named generalized soliton solutions. In this paper, it is shown that the generalized soliton solutions for the Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marchenko integral equation. Using this result, relationship between Hirota's method and the inverse spectral method isclarified. Moreover, it is also shown that the initial value problems for these two equations can be solved using their generalized soliton solutions.",
    author = "Shinichi Oishi",
    year = "1980",
    month = "2",
    language = "English",
    volume = "48",
    pages = "639--646",
    journal = "Journal of the Physical Society of Japan",
    issn = "0031-9015",
    publisher = "Physical Society of Japan",
    number = "2",

    }

    TY - JOUR

    T1 - A method of analysing soliton equations by bilinearization

    AU - Oishi, Shinichi

    PY - 1980/2

    Y1 - 1980/2

    N2 - Recently, a class of new solutions have been derived for a number of soliton equations using Hirota's bilinear forms of these soliton equations (S. Oishi: J. Phys. Soc. Jpn. 47 (1979) 1341). These solutions express solitons in a background of ripples, and are named generalized soliton solutions. In this paper, it is shown that the generalized soliton solutions for the Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marchenko integral equation. Using this result, relationship between Hirota's method and the inverse spectral method isclarified. Moreover, it is also shown that the initial value problems for these two equations can be solved using their generalized soliton solutions.

    AB - Recently, a class of new solutions have been derived for a number of soliton equations using Hirota's bilinear forms of these soliton equations (S. Oishi: J. Phys. Soc. Jpn. 47 (1979) 1341). These solutions express solitons in a background of ripples, and are named generalized soliton solutions. In this paper, it is shown that the generalized soliton solutions for the Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation can be transformed into a form of Fredholm's determinants of the Gel'fand-Levitan-Marchenko integral equation. Using this result, relationship between Hirota's method and the inverse spectral method isclarified. Moreover, it is also shown that the initial value problems for these two equations can be solved using their generalized soliton solutions.

    UR - http://www.scopus.com/inward/record.url?scp=77953053386&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=77953053386&partnerID=8YFLogxK

    M3 - Article

    VL - 48

    SP - 639

    EP - 646

    JO - Journal of the Physical Society of Japan

    JF - Journal of the Physical Society of Japan

    SN - 0031-9015

    IS - 2

    ER -