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  • 1. Basarab-Horwath, P.
    et al.
    Euler, Norbert
    Technische Hochschule Darmstadt.
    Euler, Marianna
    Amplitude-phase representation for solutions of nonlinear d'Alembert equations1995In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 28, no 21, p. 6193-6201Article in journal (Refereed)
    Abstract [en]

    We consider the nonlinear complex d'Alembert equation Square Operator Psi =F( mod Psi mod ) Psi with Psi represented in terms of amplitude and phase, in (1+n)-dimensional Minkowski space. We exploit a compatible d'Alembert-Hamilton system to construct new types of exact solutions for some nonlinearities.

  • 2. Duarte, L.G.S.
    et al.
    Euler, Norbert
    Moreira, I.C.
    Steeb, Willi-Hans
    Invertible point transformations, Painlevé analysis and anharmonic oscillators1990In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 23, p. 1457-1463Article in journal (Refereed)
    Abstract [en]

    The techniques of an invertible point transformation and the Painleve analysis can be used to construct integrable ordinary differential equations. The authors compare both techniques for anharmonic oscillators.

  • 3. Euler, Marianna
    et al.
    Euler, Norbert
    Köhler, A.
    On the construction of approximate solutions for a multidimensional nonlinear heat equation1994In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 27, no 6, p. 2083-2092Article in journal (Refereed)
    Abstract [en]

    Summary: We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation $\partial u/\partial x_0+ \Delta u= au^n+ \varepsilon f(u)$, where $a$ and $n$ are arbitrary real constants, $f$ is a smooth function, and $0< \varepsilon\ll 1$.

  • 4. Euler, Norbert
    et al.
    Shul'ga, Marianna W.
    Steeb, Willi-Hans
    Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation1992In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 25, no 18, p. 1095-1103Article in journal (Refereed)
    Abstract [en]

    The authors give the approximate symmetries for the multidimensional Landau-Ginzburg equation delta 2u/ delta x2i+ delta u/ delta x4=a1+a2u+ in un where n in R and 0( in <

  • 5. Euler, Norbert
    et al.
    Shul'ga, Marianna W.
    Steeb, Willi-Hans
    Lie symmetries and Painlevé test for explicitly space- and time-dependent nonlinear wave equations1993In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 26, p. 307-313Article in journal (Refereed)
    Abstract [en]

    The authors investigate the Lie symmetry vector fields of the wave equation Square Operator u+f(x1, . . ., xn, u)=0 where f is some nonlinear smooth function and n>or=2. The Painleve test is considered for the construction of explicitly space- and time-dependent integrable one-space-dimensional nonlinear wave equations.

  • 6. Euler, Norbert
    et al.
    Steeb, Willi-Hans
    Cyrus, K.
    On exact solutions for damped anharmonic oscillators1989In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 22, p. 195-199Article in journal (Refereed)
    Abstract [en]

    For arbitrary functions f1, f2 and f3 the anharmonic oscillator x+f1(t)x+f2(t)x+f3(t)x3=0 cannot be solved in closed form (i.e. the general solution cannot be expressed as elliptic functions). The authors apply the Painleve test to obtain the constraint on the functions f1, f2 and f3 for which the equation passes the test. The constraint on f1, f2 and f3 (i.e the differential equation which f1, f2, and f3 obey) is discussed and solutions are given.

  • 7. Euler, Norbert
    et al.
    Steeb, Willi-Hans
    Mulser, P.
    Symmetries of a nonlinear equation in plasma physics1991In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 24, no 14, p. 785-787Article in journal (Refereed)
    Abstract [en]

    The Lie symmetry vector fields are derived for a nonlinear equation in plasma physics.

  • 8. Hereman, W.
    et al.
    Steeb, Willi-Hans
    Euler, Norbert
    Comment on 'Towards the conservation laws and Lie symmetries for the Khokhlov-Zabolotskaya equation in three dimensions'1992In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 25, no 8, p. 2417-2418Article in journal (Other academic)
    Abstract [en]

    The authors comment on the Lie point symmetries for the Khokhlov-Zabolotskaya equation as calculated by Roy Chowdhury and Nasker (1986), and demonstrate that their result for the coefficients of the vector field is correct but incomplete.

  • 9.
    Kurasov, Pavel
    Luleå tekniska universitet.
    On the Coulomb potential in one dimension1996In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 29, no 8, p. 1767-1771Article in journal (Refereed)
    Abstract [en]

    A mathematically rigorous definition of the one-dimensional Schrödinger operator -d2/dx2 - γ/x is given. It is proven that the domain of the operator is defined by the boundary conditions connecting the values of the function on the left and right half-axes. The investigated operator is compared with the Schrödinger operator containing the Coulomb potential -γ/|x

  • 10.
    Kurasov, Pavel
    Luleå tekniska universitet.
    Response to ``Comment on `On the Coulomb potential in one dimension' '' by Fischer, Leschke and Müller1997In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 30, no 15, p. 5583-5589Article in journal (Other academic)
    Abstract [en]

    The differential operator -(d2/dx2) - (γ/x), γ ∈ ℝ, in one dimension is studied using distribution theory. It is proven that there exists a unique self-adjoint operator corresponding to the differential expression understood in the principle-value sense. Point interactions determined by the singular operator -(d2/dx2) - (γ/x) + αδ(x) are studied.

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