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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]

Abstract: A new supersymmetric Harry Dym equation is constructed by applying supersymmetric reciprocal transformation to a trivial supersymmetric Harry Dym equation, and its recursion operator and Lax formulation are also obtained. Within the framework of symmetry approach, a class of 3rd order supersymmetric equations of Harry Dym type are considered. In addition to five known integrable equations, a new supersymmetric equation, admitting 5th order generalized symmetry, is shown to be linearizable through supersymmetric reciprocal transformation. Furthermore, its Lax representation and recursion operator are given so that the integrability of this new equation is confirmed. [Copyright &y& Elsevier]

For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of the Korteweg-de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N = 2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N = 2 supersymmetric KdV equations through the superconformal transformation. The bosonic limit of this new N = 2 supersymmetric Harry Dym equation is related to a coupled system of KdV-MKdV equations. [ABSTRACT FROM AUTHOR]

Abstract: A new supersymmetric equation is proposed for the Sawada–Kotera equation. The integrability of this equation is shown by the existence of Lax representation and infinite conserved quantities and a recursion operator. [Copyright &y& Elsevier]