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)


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.


Spectral Problem Hamiltonian Operator Hamiltonian Structure Recursion Operator MKdV Equation 
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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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