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By the nonlocal symmetry approach, Hernández-Heredero and Reyes [J. Phys. A: Math. Theor. 42, 182002 (2009)] and Bies et al. [J. Math. Phys. 53, 073710 (2012)] obtained Darboux Transformations (DTs) of the Camassa–Holm equation and the modified Camassa–Holm equation. However, the wave function does not appear in their DTs explicitly. We introduce wave functions to the DTs and show how they are related to the binary DT of the first negative flow in the Korteweg–de Vries (KdV) hierarchy. Furthermore, we connect nonlocal symmetries of the Camassa–Holm equation and the modified Camassa–Holm equation with those of the negative KdV equation and the negative modified KdV equation, respectively. [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]