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  • 1
    Electronic Resource
    Electronic Resource
    Schlesinger transformations of Painlevé II-V (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 2031-2045 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The explicit form of the Schlesinger transformations for the second, third, fourth, and fifth Painlevé equations is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529626
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  • 2
    Electronic Resource
    Electronic Resource
    The bi-Hamiltonian formulation of the Landau–Lifshiftz equation (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 2628-2633 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Landau–Lifshitz (LL) equation is a universal model for integrable magnetic systems. It contains the sine–Gordon (SG), nonlinear Schrödinger (NLS), and the Heisenberg model (HM) equations as particular or limiting cases. It is well known that the NLS, SG, and HM equations possess recursion operators. A recursion operator of an equation in Hamiltonian form generates (a) a hierarchy of integrable equations, and (b) a second Hamiltonian operator and more generally a hierarchy of Poisson structures. Here the recursion operator of the LL equation is obtained algorithmically, and hence its bi-Hamiltonian formulation is established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528053
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  • 3
    Electronic Resource
    Electronic Resource
    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
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527999
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  • 4
    Electronic Resource
    Electronic Resource
    Exact interaction of solitary waves for certain nonintegrable equations (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 324-345 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Physically interesting exact solutions are constructed for a large class of nonlinear nonintegrable evolution equations. These solutions describe the interaction of traveling waves. They exhibit rich phenomenology including breaking of solitary waves. Generalized conditional symmetries and bilinearity are used to derive these exact results. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531393
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  • 5
    Electronic Resource
    Electronic Resource
    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
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527763
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  • 6
    Electronic Resource
    Electronic Resource
    An inverse problem for multidimensional first-order systems (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1737-1746 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An inverse problem associated with N first-order equations in n+1 dimensions, n〉1, is considered: Given appropriate inverse data T reconstruct the potential q(x0,x), where q is an N×N off-diagonal matrix. Although q depends on n+1 variables, it turns out that T depends on 3n−1 variables. This necessitates imposing certain constraints on T, i.e., T must be suitably characterized. The characterization problem for T is solved explicitly. Furthermore, the problem of reconstructing q is reduced to one for reconstructing a 2×2 matrix potential in two dimensions. The inverse data needed for the reduced problem are obtained in closed form from T. A method for solving two-dimensional inverse problems has recently appeared in the literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527089
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  • 7
    Electronic Resource
    Electronic Resource
    On the integrability of linear and nonlinear partial differential equations (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 4188-4237 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE's) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE's and several of its variations such as the Wiener–Hopf technique. (b) The integral representation of the solution of linear PDE's in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrödinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533339
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  • 8
    Electronic Resource
    Electronic Resource
    Hamiltonian theory over noncommutative rings and integrability in multidimensions (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 2504-2514 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A generalization of the Adler–Gel'fand–Dikii scheme is used to generate bi-Hamiltonian structures in two spatial dimensions. In order to implement this scheme, a Hamiltonian theory is built over a noncommutative ring, namely the ring of formal pseudodifferential operators. Bi-Hamiltonian structures generated in this way can be used for the Kadomtsev–Petviashvili equation as well as other integrable equations in 2+1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529621
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  • 9
    Electronic Resource
    Electronic Resource
    On the asymptotic integrability of a higher-order evolution equation describing internal waves in a deep fluid (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 3415-3421 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A higher-order nonlocal evolution equation describing internal waves in a deep fluid is shown to be asymptotically integrable only if the coefficients of the higher-order terms satisfy certain constraints. In this case, the nonlocal equation can be transformed to the integrable Benjamin–Ono equation. The asymptotic integrability of the reductions of the higher-order evolution equation to a complex Burgers equation, to an envelope-wave equation, and to a finite-dimensional dynamical system is also considered. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531572
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  • 10
    Electronic Resource
    Electronic Resource
    Quadratic Poisson algebras and their infinite-dimensional extensions (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 3117-3131 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A certain generalization of the classical modified Yang–Baxter equation is presented. Also certain Poisson algebras arising in the calculus of variations are investigated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530457
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