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.
To be published in the proceedings of the IMA Workshop, “Applications of Solitons”, eds. P.J. Olver and D.H. Sattinger.
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Fordy, A.P. (1990). Isospectral Flows: Their Hamiltonian Structures, Miura Maps and Master Symmetries. In: Olver, P.J., Sattinger, D.H. (eds) Solitons in Physics, Mathematics, and Nonlinear Optics. The IMA Volumes in Mathematics and Its Applications, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9033-6_6
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