# Elementary Darboux transformations for the $n$-component $KP$-hierarchy

G.F. Helminck, J.W. van de Leur, J.W. van de Leur

Research output: Book/ReportReportProfessional

### Abstract

In this paper a purely algebraic setting is described in which a characterization of the dual wavefunctions of the multicomponent $KP$-hierarchy and an interpretation of the bilinear form of this system of nonlinear equations can be given. The framework enables the construction of solutions starting from a matrix version of the Sato Grassmannian and the expression in formal power series determinants, the so-called $\tau$-functions. This leads to a geometric description of the elementary Darboux transformations for the $n$-component $KP$-hierarchy and one concludes with showing how to construct them, both at the differential operator level as at the $\tau$-function level.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 22 Published - 2002

### Publication series

Name Memorandum Faculty Mathematical Sciences University of Twente, Department of Applied Mathematics 1645 0169-2690

• METIS-208643
• MSC-22E65
• MSC-35Q58
• EWI-3465
• MSC-58B25
• IR-65831
• MSC-22E70

### Cite this

Helminck, G. F., van de Leur, J. W., & van de Leur, J. W. (2002). Elementary Darboux transformations for the $n$-component $KP$-hierarchy. (Memorandum Faculty Mathematical Sciences; No. 1645). Enschede: University of Twente, Department of Applied Mathematics.
Helminck, G.F. ; van de Leur, J.W. ; van de Leur, J.W. / Elementary Darboux transformations for the $n$-component $KP$-hierarchy. Enschede : University of Twente, Department of Applied Mathematics, 2002. 22 p. (Memorandum Faculty Mathematical Sciences; 1645).
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abstract = "In this paper a purely algebraic setting is described in which a characterization of the dual wavefunctions of the multicomponent $KP$-hierarchy and an interpretation of the bilinear form of this system of nonlinear equations can be given. The framework enables the construction of solutions starting from a matrix version of the Sato Grassmannian and the expression in formal power series determinants, the so-called $\tau$-functions. This leads to a geometric description of the elementary Darboux transformations for the $n$-component $KP$-hierarchy and one concludes with showing how to construct them, both at the differential operator level as at the $\tau$-function level.",
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author = "G.F. Helminck and {van de Leur}, J.W. and {van de Leur}, J.W.",
note = "Imported from MEMORANDA",
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Helminck, GF, van de Leur, JW & van de Leur, JW 2002, Elementary Darboux transformations for the $n$-component $KP$-hierarchy. Memorandum Faculty Mathematical Sciences, no. 1645, University of Twente, Department of Applied Mathematics, Enschede.

Elementary Darboux transformations for the $n$-component $KP$-hierarchy. / Helminck, G.F.; van de Leur, J.W.; van de Leur, J.W.

Enschede : University of Twente, Department of Applied Mathematics, 2002. 22 p. (Memorandum Faculty Mathematical Sciences; No. 1645).

Research output: Book/ReportReportProfessional

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AU - van de Leur, J.W.

AU - van de Leur, J.W.

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N2 - In this paper a purely algebraic setting is described in which a characterization of the dual wavefunctions of the multicomponent $KP$-hierarchy and an interpretation of the bilinear form of this system of nonlinear equations can be given. The framework enables the construction of solutions starting from a matrix version of the Sato Grassmannian and the expression in formal power series determinants, the so-called $\tau$-functions. This leads to a geometric description of the elementary Darboux transformations for the $n$-component $KP$-hierarchy and one concludes with showing how to construct them, both at the differential operator level as at the $\tau$-function level.

AB - In this paper a purely algebraic setting is described in which a characterization of the dual wavefunctions of the multicomponent $KP$-hierarchy and an interpretation of the bilinear form of this system of nonlinear equations can be given. The framework enables the construction of solutions starting from a matrix version of the Sato Grassmannian and the expression in formal power series determinants, the so-called $\tau$-functions. This leads to a geometric description of the elementary Darboux transformations for the $n$-component $KP$-hierarchy and one concludes with showing how to construct them, both at the differential operator level as at the $\tau$-function level.

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KW - MSC-22E65

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KW - EWI-3465

KW - MSC-58B25

KW - IR-65831

KW - MSC-22E70

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Helminck GF, van de Leur JW, van de Leur JW. Elementary Darboux transformations for the $n$-component $KP$-hierarchy. Enschede: University of Twente, Department of Applied Mathematics, 2002. 22 p. (Memorandum Faculty Mathematical Sciences; 1645).