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Abstract An extended equation of dispersive water waves, proposed by Kupershmidt, is considered. By an ansatz on eigenfunctions of its recursion operator, a linear spectral problem, which turns to be type of the energy dependent Schrödinger, is constructed for the system. As by-products, modified systems are presented. Furthermore, a new bi-Hamiltonian system is obtained and shown to be related to a three component generalization of the Camassa–Holm equation under a Miura-type transformation. [ABSTRACT FROM AUTHOR]

Abstract: The changing rule of super Hamiltonian operators under general superconformal transformations is established by means of investigating behavior of supersymmetric Euler derivatives under the same kind of changes of variables. In the two particular yet frequent cases such as supersymmetric Miura-type transformations and reciprocal transformations, the results are detailed and applied to construct bi-Hamiltonian structures of some supersymmetric evolution equations. As an interesting example, one of supersymmetric Harry Dym equations is shown to be a bi-Hamiltonian system through its reciprocal link to the classical Harry Dym equation. [Copyright &y& Elsevier]