V. Lahno

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.

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.

Group classification of the general second-order evolution equation: semi-simple invariance groups

In this paper, we consider the problem of group classification of the generic second-order evolution equation in one spatial variable. We construct all inequivalent evolution equations whose invariance groups are either semi-simple or semi-direct products of semi-simple and solvable Lie groups. The obtained lists of invariant equations contain both already known equations and the broad classes of new evolution equations possessing nontrivial Lie symmetry.

Group classification of nonlinear wave equations

We perform complete group classification of the general class of quasilinear wave equations in two variables. This class may be seen as a generalization of the nonlinear d’Alembert, Liouville, sin∕sinh-Gordon and Tzitzeica equations. We derive a number of new genuinely nonlinear invariant models with high symmetry properties. In particular, we obtain four classes of nonlinear wave equations that admit five-dimensional invariance groups.

Group Classification of the General Evolution Equation: Local and Quasilocal Symmetries

We give a review of our recent results on group classification of the most general nonlinear evolution equation in one spatial variable. The method applied relies heavily on the results of our paper Acta Appl. Math., 69, 2001, in which we obtain the complete solution of group classification problem for general quasilinear evolution equation.

Preliminary group classification of a class of fourth-order evolution equations

We perform preliminary group classification of a class of fourth-order evolution equations in one spatial variable. Following the approach developed by Basarab-Horwath et al. [Acta Appl. Math. 69, 43 (2001)], we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple Lie groups. In addition, we describe all fourth-order evolution equations from the class under consideration which are invariant under solvable Lie groups of dimension n<=4. We have constructed all Galilei-invariant equations belonging to the class of evolution differential equations under study. The list of so obtained invariant equations contains both the well-known fourth-order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics, chemistry, and biology.

Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras

We give a complete point-symmetry classification of all third-order evolution equations of the form ut=F(t,x,u,ux,uxx)uxxx+G(t,x,u,ux,uxx) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.

Classifying evolution equations

A Lie point symmetry classification of evolution equations in 1+1 time-space dimensions was presented. A combination of the standard Lie algorithm for point symmetry and the equivalence group of th ...