Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy



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Liu, Q. P. and Mañas Baena, Manuel (1999) Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy. Journal of nonlinear science, 9 (2). pp. 213-232. ISSN 0938-8974

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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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Uncontrolled Keywords:Time-dependent schrodinger; Inverse scattering transform; Rational solutions; Davey-stewartson; Gauge transformations; Jacobian varieties; Soliton-equations; Kp hierarchy; Evolution; Systems
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:32483
Deposited On:23 Jul 2015 10:46
Last Modified:10 Dec 2018 15:10

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