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For a -component Camassa–Holm equation, as well as a -component generalization of the modified Camassa–Holm equation, nonlocal infinitesimal symmetries quadratically dependent on eigenfunctions of linear spectral problems are constructed from functional gradients of spectral parameters. With appropriate pseudopotentials, these nonlocal infinitesimal symmetries are prolonged to enlarged systems, and then explicitly integrated to generate symmetry transformations in finite form for the enlarged systems. As implementations of these finite symmetry transformations, some kinds of nontrivial solutions and Bäcklund transformations are derived for both equations. [ABSTRACT FROM AUTHOR]

Under an appropriate Miura-type transformation, a super modified Korteweg–de Vries (mKdV) equation, introduced by Hu in 1997, is shown to be a modification of the generalised super Korteweg–de Vries (KdV) equation. By correspondingly changing bi-Hamiltonian formulations of the latter, bi-Hamiltonian structures are established for the super mKdV equation. Moreover, from an ansatz about gauge transformations of its linear spectral problem, elementary Darboux transformations are constructed, and suitably composing them yields a binary Darboux transformation, as well as the fourth one. As a reduction of the fourth Darboux transformation, a proper Darboux transformation is obtained for Kupershmidt's super mKdV equation. [ABSTRACT FROM AUTHOR]

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]

Two coupled systems involving both bosonic and fermionic fields are proposed as super generalizations of the equation . Linear spectral problems are presented to certify their integrability and lead to infinitely many conservation laws. Based on natural conservation laws, reciprocal transformations are defined that map one super equation to Kupershmidt's super modified Korteweg–de Vries (mKdV) equation, and the other super equation to the supersymmetric mKdV equation. By means of these connections, bi-Hamiltonian formulations are established for the super equations. [ABSTRACT FROM AUTHOR]

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]