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Isospectral Flows: Their Hamiltonian Structures, Miura Maps and Master Symmetries

  • Allan P. Fordy
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 25)

Abstract

We consider a variety of spectral problems, polynomially dependent upon the spectral parameter. When the polynomial is of degree N, there are (generically) (N+1) locally defined, compatible Hamiltonian structures which have a universal form, involving some operator J k . The operators J k have a particular form for each specific spectral problem.

Examples include spectral dependent versions of the Schrödinger operator and its super-extensions and of generalised Zakharov—Shabat problems. Associated equations include the KdV, DWW, Ito, SIT and the Heisenberg FM equations.

A simple shift in the spectral parameter induces a transformation of the variables, corresponding to a particularly simple master symmetry. This gives a simple proof of compatibility of the Hamiltonian structures.

A remarkable sequence of Miura maps can be presented for many of these equations. The modified equations are also (multi-) Hamiltonian.

Keywords

Spectral Problem Hamiltonian Operator Hamiltonian Structure Recursion Operator MKdV Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Allan P. Fordy
    • 1
  1. 1.Department of Applied Mathematical Studies and CNLSUniversity of LeedsLeedsUK

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