Search Results - (Author, Cooperation:Winternitz)

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  1. 1
    Staff View
    ISSN:
    1432-069X
    Source:
    Springer Online Journal Archives 1860-2000
    Topics:
    Medicine
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1007/BF01824398

    Articles: DFG German National Licenses
  2. 2
    Staff View
    ISSN:
    1432-069X
    Source:
    Springer Online Journal Archives 1860-2000
    Topics:
    Medicine
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1007/BF02111306

    Articles: DFG German National Licenses
  3. 3
    Boyer, C. P. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1989
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    This is the first of two papers in which the authors give a complete classification of symmetry reduced solutions of Plebanski's potential equation for self-dual Einstein spaces. In this first part the infinite pseudogroup of symmetries of Plebanski's equation is described, and the conjugacy classes of all local subgroups of dimensions one, two, and three over both the real and complex numbers are classified. Then in the second paper, this classification is used to obtain all symmetry-reduced solutions.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.528379

    Articles: DFG German National Licenses
  4. 4
    Rideau, G. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1990
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    All realizations of the Lie algebras p(1,1), sim(1,1), and conf(1,1) are classified under the action of the group of local diffeomorphisms of R3. The result is used to obtain all second-order scalar differential equations, invariant under the corresponding Poincaré, similitude, and conformal groups. The invariant equations are, in general, nonlinear, and the requirement of linearity turns out to be very restrictive. Group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.528789

    Articles: DFG German National Licenses
  5. 5
    Beckers, J. ; Gagnon, L. ; Hussin, V. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1990
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    Nonlinear superequations, for which the general solution can be expressed algebraically in terms of a finite number of particular solutions, are obtained. They are based on the orthosymplectic supergroup OSP(m,2n) and its action on a homogeneous superspace. Superposition formulas are discussed for the cases m=1, n arbitrary, and m=2, n=1. For OSP(2,2) the number of particular solutions needed to reconstruct the general solution depends on the dimension of the underlying Grassmann algebra, whereas for OSP(1,2n) it does not.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.528997

    Articles: DFG German National Licenses
  6. 6
    Gagnon, L. ; Hussin, V. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1988
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    Superposition formulas are derived expressing the general solution of several different systems of nonlinear ordinary differential equations in terms of a fundamental set of particular solutions. The equations, as well as the superposition formulas, are induced by the action of the exceptional Lie group G2 (complex or real) on a homogeneous space G2/G, where G⊆G2 is a maximal subgroup of G2. When G is either parabolic, or simple, three particular solutions are needed. When G is SL(2,C)×SL(2,C) (or one of its real forms), then two particular solutions suffice.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.528141

    Articles: DFG German National Licenses
  7. 7
    Champagne, B. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1988
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE's) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)-dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE's to various equations involving only two independent variables.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.528173

    Articles: DFG German National Licenses
  8. 8
    Beckers, J. ; Hussin, V. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1987
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    Nonlinear equations with superposition formulas are obtained, corresponding to the action of the complex and real forms of the exceptional Lie group G2 on the homogeneous spaces G2/H. The isotropy group of the origin H is taken as one of the maximal parabolic subgroups of G2, or as one of the maximal reductive subgroups, leaving some vector space V∈C7 (or V∈R7) invariant. The parabolic subgroups, as well as the simple subgroups SL(3,C), SU(3), SL(3,R) or SU(2,1) lead to equations with quadratic or quartic nonlinearities. The semisimple subgroups SL(2,C)⊗SL(2,C), SU(2)⊗SU(2), and SU(1,1)⊗SU(1,1) lead to equations with quadratic nonlinearities and additional nonlinear constraints on the independent variables.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.527636

    Articles: DFG German National Licenses
  9. 9
    del Olmo, M. A. ; Rodríguez, M. A. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1987
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    A system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group–subgroup pair G&supuline;G0. We consider the case when G=SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k-dimensional vector space invariant (1≤k≤n). The nonlinear ordinary differential equations (ODE's) in this case are rectangular matrix Riccati equations for a matrix W(t)∈Cn×k. The special case n=rk (n,r,k∈N) is considered and a superposition formula is obtained, expressing the general solution in terms of r+3 particular solutions for r≥2, k≥2. For r=1 (square matrix Riccati equations) five solutions are needed, for r=n (projective Riccati equations) the required number is n+2.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.527637

    Articles: DFG German National Licenses
  10. 10
    Couture, M. ; Patera, J. ; Sharp, R. T. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1991
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The Wigner–Inönü contraction is generalized to include gradings other than Z2 and the formalism is applied to obtain all toroidal contractions of sl(3,C). Examples of contractions based on nontoroidal gradings are also given.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.529154

    Articles: DFG German National Licenses
  11. 11
    Melkonian, S. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1991
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The Lie point symmetries are obtained for each one of two, nonlinear, dispersive thin-film equations in 2+1 variables. By means of the invariants of the one- and two-dimensional subgroups, all reductions to ordinary differential equations and to partial differential equations in two variables are obtained. Solutions are provided for a number of the reduced equations. It is also ascertained that neither one of the two considered thin-film equations is integrable. In some cases, solitary- and periodic-wave solutions having curved wavecrests are obtained.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.529481

    Articles: DFG German National Licenses
  12. 12
    Rubin, J. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1990
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The Lie point symmetries of the first two equations in the Kadomtsev–Petviashvili (KP) hierarchy, introduced by Jimbo and Miwa, are investigated. The first is the potential KP equation, the second involves four independent variables and is called the Jimbo–Miwa (JM) equation. The joint symmetry algebra for the two equations is shown to have a Kac–Moody–Virasoro structure, whereas the symmetry algebra of the JM equation alone does not. Subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.528660

    Articles: DFG German National Licenses
  13. 13
    Levi, D. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1996
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    Differential–difference equations of the form ün=Fn(t,un−1,un,un+1) are classified according to their continuous Lie point symmetry groups. It is shown that for nonlinear equations, the symmetry group can be at most seven-dimensional. The integrable Toda lattice is a member of this class and has a four-dimensional symmetry group. © 1996 American Institute of Physics.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.531722

    Articles: DFG German National Licenses
  14. 14
    Tenenblat, K. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1993
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The intrinsic generalized sine–Gordon and wave equations are integrable n-dimensional generalizations of the sine–Gordon and wave equations, respectively. It is shown that their Lie-point symmetry groups are finite-dimensional. They consist only of translations in the first case and translations and dilations in the second. The symmetries are used to obtain classes of exact invariant solutions.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.530042

    Articles: DFG German National Licenses
  15. 15
    Rideau, G. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1993
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    Representations of the quantum algebra suq(2) are constructed in terms of q-special functions, defined on real O(3) spheres and real planes. They include q-Vilenkin functions, related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.530248

    Articles: DFG German National Licenses
  16. 16
    Martina, L. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1992
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations. Using a general and systematic approach based on subgroup classification methods, nontrivial partially invariant solutions for specific equations are explicitly obtained, belonging to the NLKGE and NLLE classes in two dimensions. It is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing PDEs to ODEs.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.529592

    Articles: DFG German National Licenses
  17. 17
    Lafortune, Stéphane ; Winternitz, Pavel

    College Park, Md. : American Institute of Physics (AIP)
    Published 1996
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The purpose of this article is to derive a superposition formula for the pseudounitary matrix Riccati equation of dimension N≥2. The superposition formula will be written in closed form in terms of five particular solutions satisfying certain well-specified conditions defining a fundamental set. Examples will be studied in order to show how the superposition formula works and how it can be used in numerical calculations. © 1996 American Institute of Physics.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.531448

    Articles: DFG German National Licenses
  18. 18
    Ayari, M. A. ; Hussin, V. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1999
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The method of symmetry reduction is used to solve Grassmann-valued differential equations. The (N=2) supersymmetric Korteweg–de Vries equation is considered. It admits a Lie superalgebra of symmetries of dimension 5. A two-dimensional subsuperalgebra is chosen to reduce the number of independent variables in this equation. We are then able to give different types of exact solutions, in particular soliton solutions. © 1999 American Institute of Physics.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.532842

    Articles: DFG German National Licenses
  19. 19
    Grundland, A. M. ; Winternitz, P. ; Zakrzewski, W. J.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1996
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    We use the methods of group theory to reduce the equations of motion of the CP1 model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple and solve many of these equations in terms of elementary functions, elliptic functions, and Painlevé transcendents. Some of the reduced equations do not have the Painlevé property. The existence of a Lax pair, making the model integrable, is hence very unlikely, even though it possesses many properties of integrable systems (such as stable "numerical solitons''). © 1996 American Institute of Physics.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.531446

    Articles: DFG German National Licenses
  20. 20
    Rideau, G. ; Winternitz, P.

    College Park, Md. : American Institute of Physics (AIP)
    Published 1993
    Staff View
    ISSN:
    1089-7658
    Source:
    AIP Digital Archive
    Topics:
    Mathematics
    Physics
    Notes:
    The most general second order evolution equation ψt+F(x,t,ψψ*,ψx,ψx*,ψxx, ψxx*)=0, invariant under the Galilei, Galilei-similitude, and Schrödinger groups in two dimensions, is constructed. A preliminary step is a classification of all possible realizations of the corresponding Lie algebras of vector fields in R2×C parametrized by x, t, ψ, and ψ*. Applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories. Among the Schrödinger invariant equations, in particular, are found integrable equations, linearizable by contact transformations.
    Type of Medium:
    Electronic Resource
    URL:
    http://dx.doi.org/10.1063/1.530261

    Articles: DFG German National Licenses