Abstract:
We consider the Hamiltonian theory for the multi-component KP hierarchy. We show that the second Hamiltonian structures constructed by Sidorenko and Strampp[J. Math. Phys. {\bf 34}, 1429(1993)] are not Hamiltonian. A candidate for the second Hamiltonian Structures is proposed and is proved to lead to hereditary operators.

Abstract:
\hspace{.2in}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. Several B\"acklund transformations are derived for the MRM sKdV and MR sKdV systems.

Abstract:
We consider a modified Boussinesq type equation. The Painlev\'{e} test of the WTC method is performed for this equation and it shows that the equation has weak Painlev\'{e} property. Some exact solutions are constructed.

Abstract:
A supersymmetric version is proposed for the well known Harry Dym system. A general class super Lax operator which leads to consistent equations is considered.

Abstract:
A super Hill operator with energy dependent potentials is proposed and the associated integrable hierarchy is constructed explicitly. It is shown that in the general case, the resulted hierarchy is multi-Hamiltonian system. The Miura type transformations and modified hierarchies are also presented.

Abstract:
A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with N=2 super $W_n$ algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulted hierarchy contains the N=2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric Boussinesq hierarchies.

Abstract:
The constrained Modified KP hierarchy is considered from the viewpoint of modification. It is shown that its second Poisson bracket, which has a rather complicated form, is associated to a vastly simpler bracket via Miura-type map. The similar results are established for a natural reduction of MKP.

Abstract:
We consider the Miura map between the lattice KP 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.

Abstract:
In this paper, we consider the two component derivative nonlinear Schr\"{o}dinger equation and present a simple Darboux transformation for it. By iterating this Darboux transformation, we construct a compact representation for the $N-$soliton solutions.

Abstract:
In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots.