Renat Zhdanov

The structure of lie algebras and the classification problem for partial differential equations

The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form ut=F(t,x,u,ux)uxx+G(t,x,u,ux). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.

Group classification of heat conductivity equations with a nonlinear source

We suggest a systematic procedure for classifying partial differential equations (PDEs) invariant with respect to low-dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie method, the technique of equivalence transformations and the theory of classification of abstract low-dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are 3, 7, 28 and 12 inequivalent classes of PDEs of the type considered that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any PDE belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation. This classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modelling of, say, nonlinear heat transfer processes.

A precise definition of reduction of partial differential equations

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 + 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.

Symmetry and exact solutions of nonlinear spinor equations

Abstract This review is devoted to the application of algebraic-theoretical methods to the problem of constructing exact solutions of the many-dimensional nonlinear systems of partial differential equations for spinor, vector and scalar fields widely used in quantum field theory. Large classes of nonlinear spinor equations invariant under the Poincare group P(1, 3), Weyl group (i.e. Poincare group supplemented by a group of scale transformations), and the vonformal group C(1, 3) are described. Ansatze invariant under the Poincare and the Weyl groups are constructed. Using these we reduce the Poincare-invariant nonlinear Dirac equations to systems of ordinary differential equations and construct large families of exact solutions of the nonlinear Dirac-Heisenberg equation depending on arbitrary parameters and functions. In a similar way we have obtained new families of exact solutions of the nonlinear Maxwell-Dirac and Klein-Gordon-Dirac equations. The obtained solutions can be used for quantization of nonlinear equations.

Conditional symmetry of a porous medium equation

Abstract We obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension. It is proved that solving the determining equations for the conditional symmetries either yields classical (Lie symmetries) or is equivalent to solving the porous medium equation itself. The second possibility is shown to be typical for an arbitrary evolution-type partial differential equation in one spatial variable. We show that this very property is responsible for the well-known phenomenon when using non-classical symmetries yield exact solutions which prove to be invariant ones.

Symmetry Classification of KdV-Type Nonlinear Evolution Equations

Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation.

Separation of variables in the nonlinear wave equation

We develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations. Using it we obtain a number of new two-dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions.

Conditional symmetry and reduction of partial differential equations

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.

AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

Abstract: A new family of AN-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians.

Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries

We suggest a new approach to the problem of dimensional reduction of initial/ boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (general ...