### 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 |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 22 |

Publication status | Published - 2002 |

### Publication series

Name | Memorandum Faculty Mathematical Sciences |
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Publisher | University of Twente, Department of Applied Mathematics |

No. | 1645 |

ISSN (Print) | 0169-2690 |

### Keywords

- 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.