Gelfand-Dickey Realizations of the supersymmetric classical W-algebras for gl (n+1|n) and gl (n|n)

In this paper we realize the supersymmetric classical W-algebras W (gl (n+1|n) ) and W (gl (n|n) ) as differential algebras generated by the coefficients of a monic superdifferential operator L. In the case of W (gl (n|n) ) (resp. W (gl (n+1|n) ) ) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator L. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra W (gl (n+1|n) ) one needs to introduce a new concept of even supersymmetric Poisson vertex algebras.