Separation of variables in Kolmogorov's equation. I

Calculations are made of the coefficients of the nonstationary Kolmogorov equation for which a solution can be found by separation of variables using a complete set of differential symmetry operators of first order; the variables are separated in accordance with this procedure. A solution is found to the equation describing the motion of a dynamical system containing a random parameter.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 16–20, August, 1977.     

Separation of variables in the diffusion equation. III

The Separation of variables of type N = 2 is completed, and all complete sets of type N = 1 are constructed for the diffusion equation with a scalar matrix for the leading derivatives.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48–50, July, 1979.In conclusion, we take this opportunity to thank V. N. Shapovalov for continued interest.     

Separation of variables in kolmogorov's equation. II

This communication continues the investigation begun in [10] of the separation of variables (SV) in Kolmogorov's equation (KE). SV is discussed below with the help of a set of symmetry operators containing a second-order differential operator; a group of KE transformations is constructed with respect to which it is covariant.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 27–32, March, 1979.     

Separation of variables in the diffusion equation. II

The covariant and symmetry properties of the linear diffusion equation having a scalar matrix of variable diffusion coefficients are studied. By means of differential symmetry operators of order no higher than two, a complete separation of variables is effected for the stationary and nonstationary cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 40–45, December, 1976.     

Separation of variables in the diffusion equation. I

For both stationary and nonstationary cases, all the coefficients of the linear diffusion equation with a constant diffusion tensor are enumerated for which total separation of the variables is possible by means of differential symmetry operators of a definite structure.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 46–53, November, 1976.     

Separation of variables in the Klein-gordon equations. III

This paper supplements [1] and [2]. All kinds of external electromagnetic fields are found which contain arbitrary functions admitting complete separation of variables in the Klein-Gordon equations by using one first and two second order differential symmetry operators. The curvilinear coordinates in which the variables separate are presented, and equations in the separated variables are written down.     

Separation of variables in the Dirac equation

In this paper we investigate the possibility of separating variables in the Dirac equation by using complete sets, containing second-order differential operators and integral operators having a definite structure.Translated from Izvestiya VUZ. Fizika, No. 6, pp. 94–100, June, 1973.     

The axisymmetric equivalent of Kolmogorov's equation

A type of turbulence which is next to local isotropy in order of simplicity, but which corresponds more closely to turbulent flows encountered in practice, is locally axisymmetric turbulence. A representation of the second and third order structure function tensors of homogeneous axisymmetric turbulence is given. The dynamic equation relating the second and third order scalar structure functions is derived. When axisymmetry turns into isotropy, this equation is reduced to the well-known isotropic result: Kolmogorov's equation. The corresponding limiting form is also reduced to the well-known isotropic limiting form of Kolmogorov's equation. The new axisymmetric and theoretical results may have important consequences on several current ideas on the fine structure of turbulence, such as ideas developed by analysis based on the isotropic dissipation rate or such as extended self similarity (ESS) and the scaling laws for the n-order structure functions. Received 20 October 2000 and Received in final form 25 May 2001     

Separation of variables in the Klein — Gordon equation. I

All types of external electromagnetic fields containing arbitrary functions which admit of separation of variables in the Klein-Gordon equation by using three first-order differential symmetry operators, and stationary fields admitting separation of variables by using two first- and one second-order differential operators, are found. The curvilinear coordinates in which the variables are divided are presented and the equations are written down in the separated variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 66–72, November, 1973.     

Separation of variables in the stationary Schrödinger equation

The problem of separation of variables in the stationary Schrödinger equation is considered for a charge moving in an external electromagnetic field. On the basis of the definition formulated, necessary and sufficient conditions are found for separation of variables in equations of elliptic type to which the stationary Schrödinger equation belongs. Application of general theorems made it possible to enumerate all types of electromagnetic fields and systems of coordinates in which separation of variables in the stationary Schrödinger equation is possible. Systems of ordinary differential equations which the wave function in the separated variables satisfies are written down to explicit form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 45–50, August, 1972.     

Separation of variables in the nonstationary Schrödinger equation

Separation of variables in the Schrödinger equation is performed by using complete sets of differential operators of symmetry with operators not higher than second order, and all types of electromagnetic field potentials for which separation of variables is possible are listed.     

Separation of variables in the Klein-Gordon equations. II

Two kinds of external nonstationary electromagnetic fields are found containing arbitrary functions which admit of total separation of variables in the Klein-Gordon equations by using two differential symmetry operators and one second order operator. Curvilinear coordinates are presented in which the variables are divided, and equations are written down in the separated variables.Translated from Izvestiya VUZ, Fizika, No. 12, pp. 45–52, December, 1973.     

Separation of variables and symmetry operators for the conformally invariant Klein-Gordon equation on curved spacetime

The separability of the conformally invariant Klein-Gordon equation and the Laplace-Beltrami equation are contrasted on two classes of Petrov type D curved spacetimes, showing that neither implies the other. The second-order symmetry operators corresponding to the separation of variables of the conformally invariant Klein-Gordon equation are constructed in both classes and the most general second-order symmetry operator for the conformally invariant Klein-Gordon operator on a general curved background is characterized tensorially in terms of a valence two-symmetric tensor satisfying the conformal Killing tensor equation and further constraints.     

Separation of variables in Einstein spaces. II. No ignorable coordinates

We prove that the only Einstein spaces which admit a coordinate system with no ignorable coordinates which separates the Hamilton-Jacobi equation are certain symmetric spaces of Petrov typeD due to Kasner and the constant-curvature de Sitter spaces. We also show that a space admitting a coordinate system with no ignorable coordinates which separates the Helmholtz (Schrödinger) equation must be of Petrov type     

Separation of variables in the wave equation. Sets of the type (1.1) and the algebra SU(1.2)

It is shown that for the wave equation in Minkowski space all complete sets of symmetry operators that contain one istropic operator reduce to sets in which the isotropic operator has the form /x⁰+/x³.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 102–105, February, 1991.     

Two-dimensional imaginary lobachevsky space. Separation of variables and contractions

The Inönü-Wigner contraction from the SO(2, 1) group to the E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the corresponding two-dimensional homogeneous spaces: two-dimensional one sheeted hyperboloid and two-dimensional pseudo-Euclidean space. Here we consider the contraction limits of some basis functions for the subgroup coordinates only.     

Separation of variables in the two-point BBGKY equations

Under two particular closure conditions, the two-point BBGKY equation is shown to be separable into equations for one- point turbulent fluctuations, yielding, respectively, a linear equation anda nonlinear integro-differential equation of convolution type. Analogy with Schrödinger's equation is discussed.