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

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 Jk. The operators Jk have a particular form for each specific spectral problem. Examples include spectral dependent versions of the Schr8dinger 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.