On the equivalence of two sKdV theories: a biHamiltonian viewpoint

From the MR review by M.Mehdi: "The authors are interested in the equivalence of the sKdV theory introduced by Manin and Radul (MR) and the so-called N=2, α = −2 sKdV theory of Laberge and Mathieu (LM). They use the approach of Inami and Kanno to obtain from a Lie superalgebraic framework the matrix and scalar Lax formalism for sKdV theories, and subsequently to infer a Hamiltonian formalism by reducing the R-matrix Poisson quadratic structure of the algebra of superpseudodifferential operators. The authors obtain directly a bi-Hamiltonian structure reducing two general Poisson structures associated to the loop superalgebras of simple Lie superalgebras. They find a transformation between the phase spaces of the MR and LM theories, setting up an equivalence between the Lax formulation and mapping the former hierarchy into the latter. They prove that the Inami-Kanno transformation preserves the bi-Hamiltonian structures corresponding to the MR and LM theories.