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On the theory of recursion operator

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Abstract

The general structure and properties of recursion operators for Hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered. Weak and strong recursion operators are introduced. The conditions which determine weak and strong recursion operators are found.

In the theory of nonlinear waves a method for the calculation of the recursion operator, which is based on the use of expansion into a power series over the fields and the momentum representation, is proposed. Within the framework of this method a recursion operator is easily calculated via the Hamiltonian of a given equation. It is shown that only the one-dimensional nonlinear evolution equations can posses a regular recursion operator. In particular, the Kadomtsev-Petviashvili equation has no regular recursion operator.

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Authors and Affiliations

  1. L. D. Landau Institute of Theoretical Physics, Kosigin Str. 2, SU-117334, Moscow, USSR

    V. E. Zakharov

  2. Institute of Nuclear Physics, SU-630090, Novosibirsk-90, USSR

    B. G. Konopelchenko

Authors
  1. V. E. Zakharov
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  2. B. G. Konopelchenko
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Zakharov, V.E., Konopelchenko, B.G. On the theory of recursion operator. Commun.Math. Phys. 94, 483–509 (1984). https://doi.org/10.1007/BF01403883

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  • DOI: https://doi.org/10.1007/BF01403883

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