dorsal/arxiv
View SchemaGeometric B\"acklund--Darboux transformations for the KP hierarchy
| Authors | G. F. Helminck, J. W. van de Leur |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9806009 |
| URL | https://arxiv.org/abs/solv-int/9806009 |
Abstract
We shown that, if you have two planes in the Segal-Wilson Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from elementary BDT's and is given here in a closed form. The geometric description of elementary DBT's requires that one has a geometric interpretation of the dual wavefunctions involved. This is done here with the help of a suitable algebraic characterization of the wavefunction. The BDT's also induce transformations of the tau-function associated to a plane in the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric characterization of the BDT'ss that preserves these subsystems of the KP hierarchy. This generalizes the classical Darboux-transformations. we also determine an explicit expression for the squared eigenfunction potentials. Next a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy. It is shown that infinite flags in the Grassmannian yield solutions of the latter hierarchy. these flags can be constructed by means of BDT's, starting from some plane. Other applications of these BDT's are a geometric way to characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy and the construction of a vast collection of orthogonal polynomials, playing a role in matrix models.
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"abstract": "We shown that, if you have two planes in the Segal-Wilson Grassmannian that\nhave an intersection of finite codimension, then the corresponding solutions of\nthe KP hierarchy are linked by B\\\"acklund-Darboux transformations (BDT). The\npseudodifferential operator that performs this transformation is shown to be\nbuilt up in a geometric way from elementary BDT\u0027s and is given here in a closed\nform. The geometric description of elementary DBT\u0027s requires that one has a\ngeometric interpretation of the dual wavefunctions involved. This is done here\nwith the help of a suitable algebraic characterization of the wavefunction. The\nBDT\u0027s also induce transformations of the tau-function associated to a plane in\nthe Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric\ncharacterization of the BDT\u0027ss that preserves these subsystems of the KP\nhierarchy. This generalizes the classical Darboux-transformations. we also\ndetermine an explicit expression for the squared eigenfunction potentials. Next\na connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy.\nIt is shown that infinite flags in the Grassmannian yield solutions of the\nlatter hierarchy. these flags can be constructed by means of BDT\u0027s, starting\nfrom some plane. Other applications of these BDT\u0027s are a geometric way to\ncharacterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy\nand the construction of a vast collection of orthogonal polynomials, playing a\nrole in matrix models.",
"arxiv_id": "solv-int/9806009",
"authors": [
"G. F. Helminck",
"J. W. van de Leur"
],
"categories": [
"solv-int",
"hep-th",
"math.QA",
"nlin.SI"
],
"title": "Geometric B\\\"acklund--Darboux transformations for the KP hierarchy",
"url": "https://arxiv.org/abs/solv-int/9806009"
},
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