Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy |
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Authors: | Q P Liu M Mañas |
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Institution: | (1) Departamento de Fisica Teórica II, Universidad Complutense, E28040-Madrid, Spain, ES;(2) Departamento de Matemática Aplicada y Estadistica, Escuela Universitaria de Ingenieria Técnica Areonaútica, Universidad Politécnica de Madrid, E28040-Madrid, Spain, ES |
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Abstract: | 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.
Received June 4, 1997; final revision received March 6, 1998; accepted March 23, 1998 |
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