Prolongation structure of the Krichever-Novikov equation

Sergei Igonine, Ruud Martini

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    11 Citations (Scopus)
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    Abstract

    We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on $u,\,u_x,\,u_{xx},\,u_{xxx}$ for the Krichever-Novikov equation $u_t=u_{xxx}-3u_{xx}^2/(2u_{x})+p(u)/u_{x}+au_{x}$ in the case when the polynomial $p(u)=4u^3-g_2u-g_3$ has distinct roots. We prove that there is a universal prolongation algebra isomorphic to the direct sum of a commutative $2$-dimensional algebra and a certain subalgebra of the tensor product of $\mathfrak{sl}_2(\mathbb{C})$ with the algebra of regular functions on an affine elliptic curve. This is achieved by identifying this prolongation algebra with the one for the anisotropic Landau-Lifshitz equation. Using these results, we find for the Krichever-Novikov equation a new zero-curvature representation, which is polynomial in the spectral parameter in contrast to the known elliptic ones.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherMeetkundig en toegepast-algebraisch onderzoek
    Number of pages12
    ISBN (Print)0169-2690
    Publication statusPublished - 2002

    Publication series

    NameMemorandum Faculty Mathematical Sciences
    PublisherUniversity of Twente, Department of Applied Mathematics
    No.1638
    ISSN (Print)0169-2690

    Keywords

    • MSC-37K30
    • MSC-35Q53
    • METIS-208635
    • MSC-37K10
    • EWI-3458
    • IR-65825

    Cite this

    Igonine, S., & Martini, R. (2002). Prolongation structure of the Krichever-Novikov equation. (Memorandum Faculty Mathematical Sciences; No. 1638). Enschede: Meetkundig en toegepast-algebraisch onderzoek.