Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy

We consider the vectorial approach to the binary Darboux transformations fo r 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 hier archy introducing spectral operators and obtaining similar results. We redu ce the above-mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In partic ular for the Kadomtsev-Petviashvili I hierarchy, we get the line soliton, t he lump solution, and the Johnson-Thompson lump, and the corresponding dete rminant formulae for the nonlinear superposition of several of them. For Ka domtsev-Petviashvili II apart from the line solitons, we get singular ratio nal solutions with its singularity 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.