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Constrained KP hierarchy and bi-Hamiltonian structures

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Abstract

The Kakomtsev-Petviashvili (KP) hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions. Constraining the KP flows in terms of squared eigenfunctions one obtains 1+1-dimensional integrable equations with scattering problems given by pseudo-differential Lax operators. The bi-Hamiltonian nature of these systems is shown by a systematic construction of two general Poisson brackets on the algebra of associated Lax-operators. Gauge transformations provide Miura links to modified equations. These systems are constrained flows of the modified KP hierarchy, for which again a general description of their bi-Hamiltonian nature is given. The gauge transformations are shown to be Poisson maps relating the bi-Hamiltonian structures of the constrained KP hierarchy and the modified KP hierarchy. The simplest realization of this scheme yields the AKNS hierarchy and its Miura link to the Kaup-Broer hierarchy.

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Authors and Affiliations

  1. Dept. of Mathematical Sciences, University of Technology, LE11 3TU, Loughborough, U.K.

    W. Oevel

  2. FB 17-Mathematik, GH-Universität Kassel, Holländische Str. 36, D-34127, Kassel, Germany

    W. Strampp

Authors
  1. W. Oevel
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  2. W. Strampp
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Communicated by H. Araki

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Oevel, W., Strampp, W. Constrained KP hierarchy and bi-Hamiltonian structures. Commun.Math. Phys. 157, 51–81 (1993). https://doi.org/10.1007/BF02098018

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  • DOI: https://doi.org/10.1007/BF02098018

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