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  • 1
    Digitale Medien
    Digitale Medien
    On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy (1997)
    Springer
    Communications in mathematical physics 188 (1997), S. 305-325 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract: We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002200050166
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  • 2
    Digitale Medien
    Digitale Medien
    Darboux Coverings and Rational Reductions of the KP Hierarchy (1997)
    Springer
    Letters in mathematical physics 41 (1997), S. 291-305 
    Details
    ISSN: 1573-0530
    Schlagwort(e): KP hierarchy ; Darboux transformation ; rational hierarchies.
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We use the method of Darboux coverings to discuss the invariant submanifolds of the KP equations presented as conservation laws in the space of monic Laurent series in the spectral parameter (the space of the Hamiltonian densities). We identify a special class of these submanifolds with the rational invariant submanifolds entering matrix models of two-dimensional gravity recently characterized by Dickey and Krichever. Four examples of the general procedure are provided.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1007315810243
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  • 3
    Digitale Medien
    Digitale Medien
    Bihamiltonian Geometry, Darboux Coverings,¶and Linearization of the KP Hierarchy (1998)
    Springer
    Communications in mathematical physics 197 (1998), S. 303-324 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract: We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, to study the KP equations. In this approach they have the form of local conservation laws, and can be traded for a system of ordinary differential equations of Riccati type, which we call the Central System. We show that the latter can be linearized by means of a Darboux covering, and we use this procedure as an alternative technique to construct rational solutions of the KP equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002200050452
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  • 4
    Digitale Medien
    Digitale Medien
    Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV (1991)
    Springer
    Communications in mathematical physics 141 (1991), S. 329-351 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We study a certain family of Schrödinger operators whose eigenfunctions ϕ(χ, λ) satisfy a differential equation in the spectral parameter λ of the formB(λ,∂ λ)ϕ=Θ(x)ϕ. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02101509
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