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  • 1
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    Variable Phase Approach to Potential Scattering (1967)
    New York ; London : Academic Press
    Details
    Keywords: DDC 519.6 ; LC QA402.5
    Pages: Online-Ressource (x, 244 pages)
    ISBN: 9780121555504
    URL: http://www.sciencedirect.com/science/bookseries/00765392/35
    Language: English
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    E-BOOK
  • 2
    Electronic Resource
    Electronic Resource
    Nonlinear Schrödinger-type equations from multiscale reduction of PDEs. I. Systematic derivation (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 6399-6443 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this article we begin a systematic investigation via multiscale expansions of nonlinear evolution PDEs (partial differential equations). In this first article we restrict consideration to a single, autonomous, but otherwise generic, PDE in 1+1 variables (space+time), of first order in time, whose linear part is dispersive, and to solutions dominated by a single plane wave satisfying the linear part of the PDE. The expansion parameter is an, assumedly small, coefficient multiplying this plane wave. The main (indeed, asymptotically exact) effect of the (weak) nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics, which is generally described, in (appropriately defined) coarse-grained time and space variables, by evolution equations of nonlinear Schrödinger type. A systematic analysis of such equations is presented, corresponding to various assumptions on the "resonances" occurring for the first few harmonics. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1287644
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  • 3
    Electronic Resource
    Electronic Resource
    Partially solvable quantum many-body problems in D-dimensional space (D=1,2,3,...) (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 4208-4226 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A simple technique employed almost three decades ago to manufacture partially solvable quantum many-body problems is revisited. [A quantum problem is "partially solvable" if (only) some of its eigenvalues and eigenfunctions can be exhibited]. The models thereby generated are characterized by Hamiltonians of normal form, i.e., standard kinetic plus momentum-independent potential energy; in most cases the latter features three-body, in addition to two-body and one-body, interactions. The setting refers to D-dimensional space; the examples focus on D=1, D=2, and D≥2, and include generalizations of, and additional results on, cases recently discussed in the literature, as well as new models. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532961
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  • 4
    Electronic Resource
    Electronic Resource
    Integrable and solvable many-body problems in the plane via complexification (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 5268-5291 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A simple prescription allows us to transform, by appropriate complexification, any one-dimensional many-body problem (describing "motions on a line") with Newtonian ("acceleration equals forces") equations of motion featuring forces which depend analytically on the positions, and possibly on the velocities, of the particles, and which are moreover scaling- or translation-invariant, into a two-dimensional many-body problem (describing "motions in the plane") with rotation-invariant equations of motion. If the original (one-dimensional) model is Hamiltonian, the two-dimensional model is also Hamiltonian (in fact, bi-Hamiltonian). In this manner, starting from known integrable or solvable one-dimensional many-body problems, integrable or solvable many-body problems in the plane, generally featuring much richer dynamics, are obtained. Several examples are exhibited. Finally another, more direct, prescription is outlined, to transform by complexification almost any one-dimensional many-body problem into a rotation-invariant many-body problem in the plane. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532570
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  • 5
    Electronic Resource
    Electronic Resource
    C-integrable nonlinear partial differentiation equations. I (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 875-887 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A technique to generate C-integrable nonlinear partial differentiation equations (PDEs) (i.e., nonlinear PDEs linearizable by an appropriate Change of variables) is reported, and several examples of such PDEs are exhibited.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529346
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  • 6
    Electronic Resource
    Electronic Resource
    Lax pairs galore (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 72-74 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A formula that yields an (apparently—but only apparently—nontrivial) Lax pair for any nonlinear evolution PDE in 1+1 dimensions possessing a local conservation law is presented. Several examples are exhibited.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529096
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  • 7
    Electronic Resource
    Electronic Resource
    Solutions of certain integrable nonlinear PDE's describing nonresonant N-wave interactions (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 639-654 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The evolution equations that describe, in appropriately "coarse-grained'' and "slow'' variables, the evolution of the envelopes of N nonresonant dispersive waves, can be reduced to the "universal'' form (∂/∂t+vn ∂/∂x) un(x,t) =un(x,t) ∑Nm=1βnmum(x,t). In this paper the special case with βnm=(vm−vn) βm, which is integrable by quadratures, is investigated. The subclass of localized solutions (i.e., vanishing as x→±∞) gives rise to a novel solitonic phenomenology. The class of solutions that are asymptotically finite contains a richer solitonic phenomenology, including those of novel type (which move with the speeds vn, and can have any shape) and more standard kinks (which can move with any speed, and have standard shapes). The class of rational solutions, and the integrable dynamical systems naturally associated with these solutions, are also investigated; these dynamical systems include, and extend, known integrable systems. In this paper the treatment is confined to 1+1 dimensions; at the end, together with some other generalizations, a partially solvable multidimensional extension is reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528432
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  • 8
    Electronic Resource
    Electronic Resource
    Erratum: Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials [J. Math. Phys. 12, 419–436 (1971)] (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 3646-3646 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531804
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  • 9
    Electronic Resource
    Electronic Resource
    Solvable (nonrelativistic, classical) n-body problems on the line. II (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 1253-1258 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A solvable n-body problem is exhibited, which features equations of motion of Newtonian type, mjx¨j=Fj, j=1,...,n, with "forces'' Fj that are linear and quadratic in the particle velocities, Fj=x(overdot)j{∑nk=1[f(1)jk(x) +x(overdot)kf(2)jk(x)]}, and depend highly nonlinearly on the positions xk≡xk(t), k=1,...,n, of the n "particles'' on the line. Explicit expressions of the functions f(i)jk(x), in terms of elliptic functions, are given; they contain n+4 arbitrary constants, in addition to the n "masses'' mk and to n arbitrary functions gk(xk). Special cases in which the elliptic functions reduce to trigonometric or rational functions are of course included. The technique whereby this model has been arrived at entails that its initial-value problem is solvable by quadratures [for any n and arbitrary initial data x(0) and x(overdot)(0)]. A discussion of the actual behavior of the solution, and of special cases, is postponed to future papers. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531460
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  • 10
    Electronic Resource
    Electronic Resource
    C-integrable nonlinear partial differential equations. III (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 5810-5831 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A technique based on a change of dependent variables, used in a previous paper to generate C-integrable nonlinear partial differential equations (PDEs) (i.e., nonlinear PDEs linearizable by an appropriate Change of variables) in 1+1 dimensions (one time and one space variables), is extended to the case of more than one space dimension. Several examples of evolution C-integrable PDEs in 1+2 dimensions (one time and two space variables) are exhibited.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530284
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