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  • 1
    Digitale Medien
    Digitale Medien
    On a zero curvature problem related to the ZS–AKNS operator (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3666-3675 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: All the ZS–AKNS operators, L = i[∂x−g(x)r(x)−∂x], which admit two linearly independent families of eigenfunctions, satisfying a differential equation in the spectral parameter λ of the form a(λ)∂λV+B(λ)V=aitch-theta(x)V, are characterized. Here, a(λ)∈C, B, and aitch-theta take values in the space of 2×2 complex matrices.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.529861
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  • 2
    Digitale Medien
    Digitale Medien
    Rational Solutions of the Master Symmetries¶of the KdV Equation (2000)
    Springer
    Communications in mathematical physics 211 (2000), S. 85-109 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract: In this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for KdV. The result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master-symmetry KdV hierarchy are bispectral potentials for the Schrödinger operator. By bispectral potentials we mean that the corresponding Schrödinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable. This complements certain results of Airault–McKean–Moser [4], Duistermaat–Grünbaum [10], and Magri–Zubelli [40]. As a consequence of bispectrality, the rational solutions of the master symmetries turn out to be solutions of a (generalized) string equation.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002200050803
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  • 3
    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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  • 4
    Digitale Medien
    Digitale Medien
    On the polynomial τ-functions for the KP hierarchy and the bispectral property (1992)
    Springer
    Letters in mathematical physics 24 (1992), S. 41-48 
    Details
    ISSN: 1573-0530
    Schlagwort(e): 34L25 ; 35Q51 ; 35Q52
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We show that the τ-functions obtained from Schur polynomials lead to wave functions w(x 1, x 2, ... ; k) that possess the following bispectral property: There exists a differential operator B{k,∂k}, independent of x 1 , such that B{k,∂k}w = Θ {x 1}w, where {x 1} is independent of k. This extends for the KP hierarchy some earlier results of J. J. Duistermaat and F. A. Grünbaum for the rational solutions of KdV and of P. Wright for certain rational solutions of the generalized KdV equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00430001
    Bibliothek Standort Signatur Band/Heft/Jahr Verfügbarkeit
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  • 5
    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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