# Spaces of boundary values related to a multipoint version of the KP-hierarchy

G.F. Helminck

Research output: Book/ReportReportProfessional

### Abstract

In this paper one considers a finite number of points in the complex plane and various spaces of boundary values on circles surrounding these points. To this geometric configuration one associates a Grassmann manifolds that is shown to yield solutions of a multipoint version of the linearization of the $KP$-hierarchy. These Grassmann manifolds are built in such a way that the determinant line bundle and its dual over them still make sense. The same holds for the so-called $\tau$-functions, determinants of certain Fredholm operators that measure the failure of equivariance at lifting the commuting flows of ths hierarchy to these bundles. Solutions of the linearization are described by wave functions of a certain type. They are perturbations of the trivial solution with the leading term of the perturbation determining the type. One concludes with showing that, if a plane in the Grassmann manifold yields wave functions of different types, they are connected by a differential operator in the coordinates of the flows.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 16 Published - 2004

### Publication series

Name Memorandum University of Twente, Department of Applied Mathematics 1744 0169-2690

### Keywords

• MSC-22E70
• MSC-58B25
• IR-65928
• MSC-35Q58
• MSC-22E65
• METIS-227110
• EWI-3564

### Cite this

Helminck, G. F. (2004). Spaces of boundary values related to a multipoint version of the KP-hierarchy. (Memorandum; No. 1744). Enschede: University of Twente, Department of Applied Mathematics.
Helminck, G.F. / Spaces of boundary values related to a multipoint version of the KP-hierarchy. Enschede : University of Twente, Department of Applied Mathematics, 2004. 16 p. (Memorandum; 1744).
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abstract = "In this paper one considers a finite number of points in the complex plane and various spaces of boundary values on circles surrounding these points. To this geometric configuration one associates a Grassmann manifolds that is shown to yield solutions of a multipoint version of the linearization of the $KP$-hierarchy. These Grassmann manifolds are built in such a way that the determinant line bundle and its dual over them still make sense. The same holds for the so-called $\tau$-functions, determinants of certain Fredholm operators that measure the failure of equivariance at lifting the commuting flows of ths hierarchy to these bundles. Solutions of the linearization are described by wave functions of a certain type. They are perturbations of the trivial solution with the leading term of the perturbation determining the type. One concludes with showing that, if a plane in the Grassmann manifold yields wave functions of different types, they are connected by a differential operator in the coordinates of the flows.",
keywords = "MSC-22E70, MSC-58B25, IR-65928, MSC-35Q58, MSC-22E65, METIS-227110, EWI-3564",
author = "G.F. Helminck",
note = "Imported from MEMORANDA",
year = "2004",
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series = "Memorandum",
publisher = "University of Twente, Department of Applied Mathematics",
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Helminck, GF 2004, Spaces of boundary values related to a multipoint version of the KP-hierarchy. Memorandum, no. 1744, University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2004. 16 p. (Memorandum; No. 1744).

Research output: Book/ReportReportProfessional

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AB - In this paper one considers a finite number of points in the complex plane and various spaces of boundary values on circles surrounding these points. To this geometric configuration one associates a Grassmann manifolds that is shown to yield solutions of a multipoint version of the linearization of the $KP$-hierarchy. These Grassmann manifolds are built in such a way that the determinant line bundle and its dual over them still make sense. The same holds for the so-called $\tau$-functions, determinants of certain Fredholm operators that measure the failure of equivariance at lifting the commuting flows of ths hierarchy to these bundles. Solutions of the linearization are described by wave functions of a certain type. They are perturbations of the trivial solution with the leading term of the perturbation determining the type. One concludes with showing that, if a plane in the Grassmann manifold yields wave functions of different types, they are connected by a differential operator in the coordinates of the flows.

KW - MSC-22E70

KW - MSC-58B25

KW - IR-65928

KW - MSC-35Q58

KW - MSC-22E65

KW - METIS-227110

KW - EWI-3564

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Helminck GF. Spaces of boundary values related to a multipoint version of the KP-hierarchy. Enschede: University of Twente, Department of Applied Mathematics, 2004. 16 p. (Memorandum; 1744).