Communications in Mathematical Physics

, Volume 94, Issue 4, pp 483–509 | Cite as

On the theory of recursion operator

  • V. E. Zakharov
  • B. G. Konopelchenko


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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics General Structure 
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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • V. E. Zakharov
    • 1
  • B. G. Konopelchenko
    • 2
  1. 1.L. D. Landau Institute of Theoretical PhysicsMoscowUSSR
  2. 2.Institute of Nuclear PhysicsNovosibirsk-90USSR

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