We consider an energy-dependent version of the third-order scalar Lax operator, thus extending our previous results. Unlike the second-order case it is no longer possible to expand the potentials as arbitrary polynomials in b. We prove that there are exactly four cases. We present Hamiltonian operators, Hamiltonian Miura maps and modifications for two new examples.

We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of Manin-Radul-Mathieu sKdV and Manin-Radul sKdV. Sevetral Backlund transformations are derived for the MRM sKdV and MR sKdV systems.

Summary. We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n-th Gel’fand Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above-mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy, we get the line soliton, the lump solution, and the Johnson-Thompson lump, and the corresponding determinant formulae for the nonlinear superposition of several of them. For Kadomtsev-Petviashvili set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel’fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases.

We consider the Miura map between the lattice Kadomtsev–Petviashvili hierarchy and the lattice modified KP hierarchy and prove that the map is canonical not only between the first Hamiltonian structures, but also between the second Hamiltonian structures. © 2001 American Institute of Physics. [DOI: 10.1063/1.1359416]

The Hamiltonian theory for the two-component AKNS hierarchy and Yajima–Oikawa hierarchy is considered from the viewpoint of reduction. We show that the second Hamiltonian structures of the former is a Dirac reduction of the sl (3) current algebra, while the latter is related to the classical W4 ~3! algebra. © 1996 American Institute of Physics. [S0022-2488(96)01905-8]