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  • 1
    Electronic Resource
    Electronic Resource
    On the general solution of the d'Alembert equation with a nonlinear eikonal constraint and its applications (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 7109-7127 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We construct the general solutions of the system of nonlinear differential equations (D'Alembertian)nu=0, uμuμ=0 in the four- and five-dimensional complex pseudo-Euclidean spaces. The results obtained are used to reduce the multidimensional nonlinear d'Alembert equation (D'Alembertian)4u=F(u) to ordinary differential equations and to construct its new exact solutions. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531142
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  • 2
    Electronic Resource
    Electronic Resource
    Lie symmetry and integrability of ordinary differential equations (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 6745-6756 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Combining a Lie algebraic approach that is due to Wei and Norman [J. Math. Phys. 4, 475 (1963)] and the ideas suggested by Drach [Compt. Rend. 168, 337 (1919)], we have constructed several classes of systems of linear ordinary differential equations that are integrable by quadratures. Their integrability is ensured by integrability of the corresponding stationary cubic Schrödinger, KdV, and Harry–Dym equations. Next, we obtain a hierarchy of integrable reductions of the Dirac equation of an electron moving in the external field. Their integrability is shown to be in correspondence with integrability of the stationary mKdV hierarchy. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532654
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  • 3
    Electronic Resource
    Electronic Resource
    On the non-Lie reduction of the nonlinear Dirac equation (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 3488-3490 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The method of construction of exact solutions of nonlinear spinor equations based on their conditional (non-Lie) symmetry is suggested. With the help of this method new ansätze that reduce the nonlinear Poincaré-invariant Dirac equation to ordinary differential equations are constructed. The new family of exact solutions of the nonlinear Dirac equation with scalar self-interaction is found.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529463
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  • 4
    Electronic Resource
    Electronic Resource
    Conditional symmetry and spectrum of the one-dimensional Schrödinger equation (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 3198-3217 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We develop an algebraic approach to studying the spectral properties of the stationary Schrödinger equation in one dimension based on its high-order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schrödinger operator by n×n matrices for any n∈N and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n×n matrices. The connection to so-called quasiexactly solvable models is discussed. It is established, in particular, that the case, when conditional symmetries reduce to high-order Lie symmetries, corresponds to exactly solvable Schrödinger equations. A symmetry classification of Schrödinger equation admitting nontrivial high-order Lie symmetries is carried out, which yields a hierarchy of exactly solvable Schrödinger equations. Exact solutions of these are constructed in explicit form. Possible applications of the technique developed to multidimensional linear and one-dimensional nonlinear Schrödinger equations are briefly discussed. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531588
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  • 5
    Electronic Resource
    Electronic Resource
    On the new approach to variable separation in the time-dependent Schrödinger equation with two space dimensions (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 5506-5521 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We suggest an effective approach to separation of variables in the Schrödinger equation with two space variables. Using it we classify inequivalent potentials V(x1,x2) such that the corresponding Schrödinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrödinger equation with the anisotropic harmonic oscillator potential V=k1x21+k2x22 and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schrödinger equation (V=0). © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531274
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  • 6
    Electronic Resource
    Electronic Resource
    Separation of variables in (1+2)-dimensional Schrödinger equations (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 1197-1217 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Using our classification of separable Schrödinger equations with two space dimensions published in J. Math. Phys. 36, 5506 (1995) we give an exhaustive description of the coordinate systems providing their separability. Furthermore, we apply these results to separate variables in the heat, Hamilton–Jacobi, and Fokker–Planck equations. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531874
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  • 7
    Electronic Resource
    Electronic Resource
    On Covariant Realizations of the Euclid Group (2000)
    Springer
    Communications in mathematical physics 212 (2000), S. 535-556 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We classify realizations of the Lie algebras of the rotation O(3) and Euclid E(3) groups within the class of first-order differential operators in arbitrary finite dimensions. It is established that there are only two distinct realizations of the Lie algebra of the group O(3) which are inequivalent within the action of a diffeomorphism group. Using this result we describe a special subclass of realizations of the Euclid algebra which are called covariant ones by analogy to similar objects considered in classical representation theory. Furthermore, we give an exhaustive description of realizations of the Lie algebra of the group O(4) and construct covariant realizations of the Lie algebra of the generalized Euclid group E(4).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200000222
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  • 8
    Electronic Resource
    Electronic Resource
    Non-lie symmetry of Galileo-invariant equation for a particle with spin s=1/2 (1991)
    Springer
    Theoretical and mathematical physics 89 (1991), S. 1305-1309 
    Details
    ISSN: 1573-9333
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01017826
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  • 9
    Electronic Resource
    Electronic Resource
    Reduction of the self-dual Yang-Mills equations I. Poincaré group (1995)
    Springer
    Ukrainian mathematical journal 47 (1995), S. 528-536 
    Details
    ISSN: 1573-9376
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For the vector potential of the Yang-Mills field, we give a complete description of ansatzes invariant under three-parameterP (1, 3) -inequivalent subgroups of the Poincaré group. By using these ansatzes, we reduce the self-dual Yang-Mills equations to a system of ordinary differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01056038
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  • 10
    Electronic Resource
    Electronic Resource
    Conditional symmetry and reduction of partial differential equations (1992)
    Springer
    Ukrainian mathematical journal 44 (1992), S. 875-886 
    Details
    ISSN: 1573-9376
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Sufficient reduction conditions for partial differential equations possessing nontrivial conditional symmetry are established. The results obtained generalize the classical reduction conditions of differential equations by means of group-invariant solutions. A number of examples illustrating the reduction in the number of independent and dependent variables of systems of partial differential equations are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01056141
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