Coverings and the fundamental group for partial differential equations

S. Igonin

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    Abstract

    Following I. S. Krasilshchik and A. M. Vinogradov, we regard systems of PDEs as manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and B\"acklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a discrete group, but a certain system of Lie groups. From this we deduce an algebraic necessary condition for two PDEs to be connected by a B\"acklund transformation. For the KdV equation and the nonsingular Krichever-Novikov equation these systems of Lie groups are determined by certain infinite-dimensional Lie algebras of Kac-Moody type. We prove that these two equations are not connected by any B\"acklund transformation. To achieve this, for a wide class of Lie algebras $\mathfrak{g}$ we prove that any subalgebra of $\mathfrak{g}$ of finite codimension contains an ideal of $\mathfrak{g}$ of finite codimension.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Publication statusPublished - 2003

    Publication series

    NameMemorandum
    PublisherDepartment of Applied Mathematics, University of Twente
    No.1678
    ISSN (Print)0169-2690

    Keywords

    • IR-65863
    • EWI-3498
    • MSC-17B65
    • MSC-37K30

    Cite this

    Igonin, S. (2003). Coverings and the fundamental group for partial differential equations. (Memorandum; No. 1678). Enschede: University of Twente, Department of Applied Mathematics.