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  • 1
    Electronic Resource
    Electronic Resource
    Exact solutions for the nonlinear Klein–Gordon and Liouville equations in four-dimensional Euclidean space (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 2317-2322 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A systematic method for constructing particular solutions of the nonlinear Klein–Gordon and Liouville equations in four-spatial dimensions is developed. The method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations (ODE's) by introducing symmetry variables and then seeking exact solutions for more tractable ODE's. Various exact solutions are presented, in which new solutions with nonspherical symmetries are included. Furthermore, the exact method is applied to the above equations in general n-spatial dimensions. Among them, a conformally invariant nonlinear Klein–Gordon equation is particularly interesting from the viewpoint of field theories. The exact solutions for these equations are generalizations of those for the corresponding equations in four-spatial dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527764
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  • 2
    Electronic Resource
    Electronic Resource
    Linearization of novel nonlinear diffusion equations with the Hilbert kernel and their exact solutions (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 120-126 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Two novel nonlinear diffusion equations with the Hilbert kernel are proposed. The equations can be linearized by introducing appropriate dependent variable transformations. The initial value problems for the proposed equations are then solved exactly through the linearization and explicit nonperiodic and periodic solutions are constructed. The properties of solutions are investigated in detail. It is found that the blow up of solutions occurs at a finite time for both the nonperiodic and periodic cases due to the breakdown of certain analytic conditions imposed on the dependent variables.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529134
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  • 3
    Electronic Resource
    Electronic Resource
    Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2904-2916 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Bäcklund transformation (BT), an infinite number of conservation laws, and the inverse scattering transform (IST) of a model integrodifferential equation for water waves in fluids of finite depth [Y. Matsuno, J. Math. Phys. 29, 49(1989)] are constructed by employing the bilinear transformation method. The model equation is also shown to pass the Painlevé test. These facts prove the complete integrability of the equation. Both the deep- and shallow-water limits of various results thus obtained are then investigated in detail. In addition, a new method to evaluate conserved quantities for pure N-soliton is developed by utilizing actively the time part of the BT. It is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the Benjamin–Ono and the Korteweg–de Vries equations. The most important problem left open in this paper is to solve various IST equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528943
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  • 4
    Electronic Resource
    Electronic Resource
    A class of exact solutions of Yang's K gauge equation for SU(2) gauge fields (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 936-938 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Using a simple ansatz, Yang's K gauge equation for SU(2) gauge fields is reduced to a system of nonlinear ordinary differential equations. Exact solutions for the equations are obtained together with corresponding gauge potentials.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529027
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  • 5
    Electronic Resource
    Electronic Resource
    Erratum: New integrable nonlinear integrodifferential equations and related solvable finite-dimensional dynamical systems [J. Math. Phys. 29, 49 (1988)] (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 241-241 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528575
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    New integrable nonlinear integrodifferential equations and related solvable finite-dimensional dynamical systems (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 49-56 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A new integrable nonlinear integrodifferential equation (NIDE) is proposed. This equation may be interpreted as a model equation for deep-water waves. The N-periodic and N-soliton solutions for the equation are constructed by means of the bilinear transformation method. These solutions have the same structure as that for the Benjamin–Ono equation which describes internal waves in stratified fluids of great depth. Furthermore, it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite-dimensional dynamical systems described by first-order nonlinear ordinary differential equations. The discussion is also made on a more general NIDE that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528134
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Two-dimensional dynamical system associated with Abel's nonlinear differential equation (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 412-421 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A two-dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations(ODEs) with a complex parameter. It reduces to Abel's nonlinear ODE of the first kind by an appropriate transformation. Using this fact the properties of solutions are investigated in detail with the aid of numerical computations. It is found that various types of bifurcation phenomena occur depending on the values of the parameter. In particular, the solution is shown to blow up in finite time under certain conditions. In order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane. Finally, a discussion is made on some generalizations of the proposed system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529923
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  • 8
    Electronic Resource
    Electronic Resource
    Pulse formation in a dissipative nonlinear system (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3039-3045 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A nonlocal nonlinear evolution equation is proposed that describes pulse formation in a dissipative system. A novel feature of the equation is that it can be solved exactly through a linearization procedure. The solutions are constructed under appropriate initial and boundary conditions and their properties are investigated in detail. Of particular interest is pulse formation, which is caused by a balance between nonlinearity and dissipation. The asymptotic behavior of the solution for large time is then represented by a train of moving pulses with equal amplitudes. The corresponding position of each pulse is shown to be characterized by the zero of the Hermite polynomial, irrespective of initial conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529523
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  • 9
    Electronic Resource
    Electronic Resource
    Dynamics of solitons in a damped sine-Hilbert equation (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 2754-2764 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A damped sine-Hilbert (sH) equation is proposed. It can be linearized by a dependent variable transformation which enables one to solve an initial value problem of the equation. The N-soliton solution is obtained explicitly and its properties are investigated in comparison with those of the N-soliton solution of the sH equation. In particular the interaction of the two solitons is explored in detail with the aid of the pole representation. It is found that the interaction process is classified into the two types according to the initial amplitudes and positions of both solitons. In the general N-soliton case the long-time behavior of the solution is shown to be characterized by the positive N zeros of the Hermite polynomial of degree 2N. Finally, a linearized version of the damped sH equation is briefly discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529544
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  • 10
    Electronic Resource
    Electronic Resource
    Initial value problem of the linearized Benjamin–Ono equation and its applications (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 5198-5224 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We consider the initial value problem of the Benjamin–Ono (BO) equation linearized about the N-soliton solution. By establishing the completeness relation for the eigenfunctions of the linearized BO equation, we construct the explicit solution to this problem. As an application of the above result, we investigate the linear stability of the N-soliton solution. We show that the wave under consideration is stable against infinitesimal perturbations. Thus we have a direct multisoliton perturbation theory for the BO equation without recourse to the inverse scattering transform. In particular, we can handle the first-order solution beyond the adiabatic approximation. The completeness relation established here enables us to give a general scheme for evaluating the first-order correction to the leading-order N-soliton solution. We also demonstrate that the first-order solution satisfies an infinite set of conservation laws modified by the perturbations. Finally, in the one-soliton case, we perform explicit calculations of the first-order corrections for two different dissipative perturbations that arise in real physical systems and analyze their large time asymptotics. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531953
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