Two super extensions of a coupled Korteweg–de Vries equation

Abstract For the coupled Korteweg–de Vries (KdV) equation known as the Drinfeld–Sokolov–Satsuma–Hirota system, we propose two super extensions that incorporate a fermionic field in addition to two classical bosonic fields. The integrability of both systems is confirmed through the existence of Lax equations, where the associated Lax operators are of the constrained Kadomtsev–Petviashvili (KP) type. Furthermore, the two super coupled KdV equations are shown to possess bi-Hamiltonian formulations, obtained by applying appropriate Dirac reductions to the Hamiltonian structures of a third-order flow generated from a generic fourth-order Lax operator of the constrained KP type. Nonlocal infinitesimal symmetries depending quadratically on eigenfunctions of the corresponding spectral problems are constructed for these two super coupled KdV equations and are further prolonged to enlarged systems in each case. Symmetry transformations of finite form are explicitly derived and used to generate non-trivial exact solutions, as well as Bäcklund transformations for the proposed two super coupled KdV equations.