A super mKdV equation: bi-Hamiltonian structures and Darboux transformations

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.