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  • 1985-1989  (7)
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  • 1
    Digitale Medien
    Digitale Medien
    Bi-Hamiltonian formulation of the Kadomtsev–Petviashvili and Benjamin–Ono equations (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 604-617 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: It was shown recently that the Kadomtsev–Petviashvili (KP) equation (an integrable equation in 2+1, i.e., in two-spatial and one-temporal dimensions) admits a bi-Hamiltonian formulation. This was achieved by considering KP as a reduction of a (3+1)-dimensional system (in the variables x,y1, y2,t). It is shown here, using the KP as a concrete example, that equations in 2+1 possess two bi-Hamiltonian formulations and two recursion operators. Both Hamiltonian operators associated with the x direction are local; in contrast only one of the Hamiltonian operators associated with the y direction is local. Furthermore, using the Benjamin–Ono equation as a concrete example, it is shown that intergrodifferential equations in 1+1 admit an algebraic formulation analogous to that of equations in 2+1.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.527999
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  • 2
    Digitale Medien
    Digitale Medien
    Recursion operator and bi-Hamiltonian structure for integrable multidimensional lattices (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 1593-1603 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: The recursion operator and the bi-Hamiltonian structure for an integrable two-dimensional version of the Toda chain are algorithmically derived from the corresponding linear problem. The intimate relation between two-dimensional theories and non-Abelian one-dimensional theories is emphasized. The analogies and differences between discrete and continuous integrable two-dimensional systems are pointed out and discussed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.527907
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  • 3
    Digitale Medien
    Digitale Medien
    Nonlinear evolution equations associated with a Riemann–Hilbert scattering problem (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 2469-2472 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: In an earlier paper nonlinear evolution equations associated with a Riemann–Hilbert scattering problem, which reduces, in an appropriate limit, to the Zakharov–Shabat–AKNS scattering problem, were considered. Here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations. Moreover, the direct linearization of the nonlinear evolution equations and an algorithm to construct anN-soliton solution are given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.526760
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  • 4
    Digitale Medien
    Digitale Medien
    An example of ∂¯ problem arising in a finite difference context: Direct and inverse problem for the discrete analog of the equation ψxx+uψ=σψy (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 777-780 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: The direct and inverse spectral problem for the discrete analog of the equation ψxx+uψ=σψy is solved in the framework of "∂¯'' theory. The time evolution of the spectral data for the simplest nonlinear differential-difference equations associated to this linear problem is derived.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.527618
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  • 5
    Digitale Medien
    Digitale Medien
    On the initial value problem for a class of nonlinear integral evolution equations including the sine–Hilbert equation (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 2310-2316 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: A method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented. The underlying scattering problem is a matrix Riemann–Hilbert problem. Scattering analysis shows that the spectrum is purely discrete. An application is to the so-called sine–Hilbert equation Hθt =−c sin θ, where c is a constant and H denotes the Hilbert transform.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.527763
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  • 6
    Digitale Medien
    Digitale Medien
    Recursion operators and bi-Hamiltonian structures in multidimensions. I (1988)
    Springer
    Communications in mathematical physics 115 (1988), S. 375-419 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract The algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied. In particular, the factorization of certain operators, called recursion operators, establishes the bi-Hamiltonian nature of all these equations. Recently, we have presented the recursion operator and the bi-Hamiltonian formulation of the Kadomtsev-Petviashvili equation, a two spatial dimensional analogue of the Korteweg-deVries equation. Here we present the general theory associated with recursion operators for bi-Hamiltonian equations in two spatial and one temporal dimensions. As an application we show that general classes of equations, which include the Kadomtsev-Petviashvili and the Davey-Stewartson equations, possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct Poisson brackets. Furthermore, we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01218017
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  • 7
    Digitale Medien
    Digitale Medien
    Recursion operators and bi-Hamiltonian structures in multidimensions. II (1988)
    Springer
    Communications in mathematical physics 116 (1988), S. 449-474 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We analyze further the algebraic properties of bi-Hamiltonian systems in two spatial and one temporal dimensions. By utilizing the Lie algebra of certain basic (starting) symmetry operators we show that these equations possess infinitely many time dependent symmetries and constants of motion. The master symmetries τ for these equations are simply derived within our formalism. Furthermore, certain new functionsT 12 are introduced, which algorithmically imply recursion operators Φ12. Finally the theory presented here and in a previous paper is both motivated and verified by regarding multidimensional equations as certain singular limits of equations in one spatial dimension.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01229203
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