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 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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