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Liu, Q. P. Mañas Baena, Manuel Enrique 2023-06-20T20:09:10Z 2023-06-20T20:09:10Z 1999-03 0938-8974 10.1007/s003329900070 https://hdl.handle.net/20.500.14352/59691 http://dx.doi.org/10.1007/s003329900070 http://link.springer.com http://arxiv.org/abs/solv-int/9705012 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 II apart from the line solitons, we get singular rational 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. eng open access Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy journal article