Algebraic properties of the Dirac equation

The first-order symmetry operators of the Dirac equation are classified according to their tensor properties under transformations of the homogeneous Lorentz group; a minimal system of generators for the ring of symmetry operators of the free Dirac equation is obtained, and the physical meaning of the spin operators is considered; fields are found which admit symmetry operators of first order.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 84–89, February, 1972.The author is grateful to V. N. Shapovalov for discussions and valuable suggestions.     

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.     

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.     

Symmetry and Conservation Laws for a Dirac Equation in a Riemannian Space

The linkage of the linear symmetry operators of the Dirac–Fock equation for electrons and massless particles with the differential conservation laws and the symmetry of equations in conformal spaces are studied.     

Noncommutative solutions of the d'Alembert equation

All the subalgebras of first-order symmetry operators for the d'Alembert equation, generating the bases of solutions in the method of noncommutative integration of linear differential equations, which cannot be constructed in the method of separation of variables, are found. These bases themselves are then given in explicit form. The complete systems of solutions of the d'Alembert equation, determined by noncommutative sets of first-order symmetry operators, are thereby classified. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 25–30, June, 1998.     

On askey-wilson algebra

The ladder representations of Askey-Wilson algebra (AWA) is investigated and the over-lap problem is discussed. The full class of exactly solvable potentials for one-dimensional ordinary Schrödinger equation with the AWA as an algebra of dynamical symmetry is found for creation-annihilation operators of third order. The generating spectrum algebra for latticeSchrödinger equation as AWA is examined. All exactly solvable potentials with this dynamical symmetry are found. Some generalizations of obtained results are discussed.     

Complete sets of symmetry operators containing a second-order operator and the problem of separation of variables in the wave equation

For the wave equation in Minkowski space, a space is defined of nontrivial local second-order differential symmetry operators. The algebraic conditions, which, in accordance with the general theorems on the separation of variables, must be satisfied by the commutative subalgebras, including two first-order operators and second-order operator, are formulated in coordinate-free form. On the basis of these subalgebras, there are obtained all the complete sets of symmetry operators of types (2.0), (2.1). Sets are presented which do not have analogues in papers of other authors.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 115–119, April, 1991.     

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.     

Dark Sharma-Tasso-Olver Equations and Their Recursion Operators

A complete scalar classification for dark Sharma-Tasso-Olver's(STO's) equations is derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark STO systems, thus some special equations including symmetry equation and dual symmetry equation are obtained by selecting a free parameter. Furthermore, the recursion operators of STO equation and dark STO systems are constructed by a direct assumption method.     

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.     

Some Problems of the Existence of Symmetry Spinor Operators for the Dirac Equation

Conditions necessary for the existence of a class of fields that can be used to construct the spinor symmetry operators for the Dirac equation in Riemannian space are specified in the present paper. The metrics of spaces with four-dimensional groups of motions in which these fields exist are indicated. A class of spaces is identified in which the Dirac equation admits no separation of variables within the framework of the definition adopted, but the algebra of symmetry of the Dirac equation satisfies the conditions of theorems of the noncommutative intergrability.     

Methods of generating integrable potentials for the Sochrödinger equation and nonlocal symmetries

Methods of generating exactly integrable potentials for the Schrödinger equation are consolidated within the framework of a simple construction. The Abraham-Moses method is generalized to the case of the nonstationary Schrödinger equation. An algorithm is proposed for solving the Schrödinger equation based on nonlocal symmetry operators.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 19–25, September, 1991.     

Conditional Symmetry of Equations of Nonstationary Filtration and of the Nonlinear Heat Equation

Abstract

Conditional symmetry of the nonlinear gas filtration equation is studied. The operators obtained enabled to constract ansatzes reducing this equation to ordinary differential equations and to obtain its exact solutions.     

Classification of the Dirac equation with an external SU(3) gauge field admitting a first-order symmetry operator of special type

A classification is performed of massless gauge fields admitting one first-order symmetry operator of special type for the Dirac equation in Minkowski space. The gauge group is chosen to be SU(3). The factors multiplying the derivatives of the symmetry operator do not contain generators of the gauge group, which allows us to classify the fields according to symmetry operators of the free Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 22–21, June, 1989.     

On Symmetry of the Generalized Breit Equation

Abstract

In this paper we find the complete set of symmetry operators for the two-particle Breit equation in the class of first-order differential operators with matrix coefficients. A new integral of motion is obtained.     

Symmetry Reduction of Two-Dimensional Damped Kuramoto-Sivashinsky Equation

In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system: the two-dimensional damped Kuramoto-Sivashinsky ((2D) DKS ) equation is studied. By applying the basic Lie symmetry method for the (2D) DKS equation, the classical Lie point symmetry operators are obtained. Also, the optimal system of one-dimensional subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The nonclassical symmetries of the (2D) DKS equation are also investigated.     

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 83–87, May, 1995.     

The Symmetry Structure of the Heavenly Equation

We show that excitations of physical interest for the heavenly equation are generated by symmetry operators which yield two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi–Hanson and Gibbons–Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the so-called Schröder equation, which plays a crucial role in the bootstrap model and in renormalization theory.     

Symmetries of theq-difference heat equation

The symmetry operators of aq-difference analog of the heat equation in one space dimension are determined. They are seen to generate aq-deformation of the semidirect product of sl(2, ) with the three-dimensional Weyl algebra. It is shown that this algebraic structure is preserved if differentq-analogs of the heat equation are considered. The separation of variables associated to the dilatation symmetry is performed and solutions involving discreteq-Hermite polynomials are obtained.     

Generalized renormalization group equation for systems of arbitrary symmetry

An exact renormalization group equation for the Ginzburg-Landau-Wilson functional of an arbitrary symmetry is obtained. The equation derived does not contain redundant operators which must be transformed away.