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Based on a 4 × 4 matrix spectral problem, a super Degasperis–Procesi (DP) equation is proposed. We show that under a reciprocal transformation, the super DP equation is related to the first negative flow of a super Kaup–Kupershmidt (KK) hierarchy, which turns out to be a particular reduction of a super Boussinesq hierarchy. The bi-Hamiltonian structure of the super Boussinesq hierarchy is established and subsequently produces a Hamiltonian structure, as well as a conjectured symplectic formulation of the super KK hierarchy via suitable reductions. With the help of the reciprocal transformation, the bi-Hamiltonian representation of the super DP equation is constructed from that of the super KK hierarchy. We also calculate a positive flow of the super DP hierarchy and explain its relations with the super KK equation. Infinitely many conservation laws are derived for the super DP equation, as well as its positive flow. [ABSTRACT FROM AUTHOR]

Reciprocal transformations are introduced for two super Camassa–Holm (CH) equations. Under these transformations and appropriate changes of dependent variables, the super CH equation, proposed by Geng et al. [Stud. Appl. Math. 130, 1 (2013)], is converted to a negative member of the super Korteweg–de Vries (KdV) hierarchy studied by Geng and Wu in 2010 [Appl. Math. Lett. 23, 716 (2010)], while the other super CH equation, due to Zhang and Zuo [J. Math. Phys. 52, 073503 (2011)], is related to a new super KdV hierarchy. In the latter case, algebraic properties of this new super KdV hierarchy are established, including Hamiltonian operators, a recursion operator, and conserved quantities. [ABSTRACT FROM AUTHOR]