Classification problems of group analysis

Classification problems form the core of group analysis of differential equations. They are the problems that motivated Sophus Lie to create his theory of continuous transformation groups and Lie algebras. The study of group classification problems in classes of differential equations initiated the modern stage of development of group analysis. Different standard techniques were created for solving such problems; some of these techniques were realized as packages in symbolic calculation systems. Nevertheless, the set of classes of differential equations which can be classified by the conventional methods does not include a number of important classes arising in mathematical applications. To extend this set, new tools as well as new notions have to be introduced and new approaches are called for. It is the aim of the proposed project to develop such approaches. Our investigations will be based on the results obtained in the course of the precursor Lise Meitner project. Fundamental concepts and algorithms of group classification problems of differential equations remain in the centre of the program. They will be supplemented with the classification of various concepts (conditional and potential symmetries, reduction operators, exact solutions, conservation laws, etc.) associated with differential equations. Investigations in this direction will be based on different equivalence relations in classes of differential equations and the notion of normalized classes. Special attention will be given to applications of group analysis to mathematical models of meteorology. An important part of the program is formed by algebraic problems related to group analysis of differential equations. These problems concern realizations, contractions and invariants of Lie algebras. The algorithm of finding realizations of Lie algebras will be enhanced. In fact, the description of realizations of Lie algebras up to an equivalence relation gives a basis for solving group classification problems in normalized classes of differential equations. The original approach to the computation of invariants (generalized Casimir operators) of Lie algebras, involving Cartan`s method of moving frames, will be extended and applied to new classes of Lie algebras. New necessary criteria of contractions will be proposed. Special examples of contractions, which are important for theory of contractions, will be constructed. The algebraic problems which will be considered are also interesting in their own right. Thus, generalized Casimir operators have numerous applications in different fields of mathematics and physics in which Lie algebras arise (representation theory, integrability of Hamiltonian differential equations, quantum numbers, etc.). Proper contractions between Lie algebras underlying physical theories induce the possibility of singular limiting processes between such theories. The best known example concerning these processes is given by the connection between relativistic and classical mechanics with their underlying Poincaré and Galilean symmetry groups. If the velocity of light is assumed to go to infinity, relativistic mechanics `transforms` into classical mechanics. This also induces a singular transition from the Poincaré algebra to the Galilean algebra. Another well-known example is the limit process from quantum mechanics to classical mechanics under the Planck constant approaching zero, which corresponds to the contraction of the Heisenberg algebras to the Abelian algebras of the same dimensions. It is one of the aims of the project to enhance the mathematical understanding of such processes.

The main goal of the project was a thorough investigation and a substantial further development of fundamental concepts and algorithms of group classification of differential equations. These studies were based on the central notion of normalized classes as well as on suitable equivalence relations in classes of differential equations. Several new concepts (a weakly normalized class, the equivalence groupoid of a class, complete and partial preliminary group classifications, etc.) were introduced. The method of preliminary group classification was rigorously defined, enhanced and related to the algebraic method of group classification of differential equations. A new version of the algebraic method for finding the complete point symmetry groups of differential equations was suggested. As a demonstration of the effectiveness of the algebraic techniques developed, we exhaustively solve the 20-years-old problem on group classification of a class of nonlinear wave equations arising in the theory of elasticity. Lie symmetries of a number of other classes of differential equations important for applications were also classified. This main line of research was supplemented by the classification of a variety of further objects (conditional symmetries, reduction operators, exact solutions, conservation laws, potential symmetries, etc.) related to differential equations. Thus, conservation laws of (1+1)-dimensional second-order evolution equations were exhaustively described up to contact equivalence. We also proved that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables. Introducing the notions of regular and singular reduction modules allowed us to extend the no-go results on nonclassical symmetries of (1+1)-dimensional evolution equations to arbitrary partial differential equations. Reductions of differential equations to algebraic equations and first-order ordinary differential equations were considered in detail within the framework proposed, and related no-go theorems were proved. Special attention was paid to applications of group analysis to mathematical models of meteorology. Along with the study of specific meteorological equations, we established a general relation between problems of invariant parameterization and group classification, introduced the framework of conservative parameterization and combined it with that of invariant parameterization. Using an original mimetic method of discretization, we constructed invariant conservative numerical schemes for the shallow water equations, which also gives the first example of invariant numerical schemes for a multidimensional system of differential equations. An important part of the project was formed by algebraic problems related to group analysis of differential equations. These problems concern, in particular, contractions of Lie algebras. Roughly speaking, limit processes between physical theories are reflections of limit processes (called contractions) between underlying symmetry groups or the corresponding Lie algebras, and a way to realize a contraction is essential as well. The simplest way to contract an algebra (so-called generalized Inönü-Wigner contractions) is to choose a basis, scale its elements and then take the limit of the transformed algebra multiplications, but not all contractions can be realized in this way. We found all contractions between four-dimensional Lie algebras that are not realized as generalized Inönü-Wigner contractions, and dimension four is the lowest where such contractions exist. The description of generalized Inönü-Wigner contractions of four-dimensional Lie algebras was completed by finding minimal integer exponents for such contractions.