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 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.
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Received June 4, 1997; final revision received March 6, 1998; accepted March 23, 1998
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Liu, Q., Mañas, M. Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy. J. Nonlinear Sci. 9, 213–232 (1999). https://doi.org/10.1007/s003329900070
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DOI: https://doi.org/10.1007/s003329900070