Classification problems of group analysis

The beginnings of the theory of Lie groups and Lie algebras were inseparably linked with group analysis of differential equations and, in particular, with problems of group classification of differential equations. Inspired by the idea of creating a universal theory of integration of ordinary differential equations similar to the Galois theory of solving algebraic equations, S. Lie developed the theory of continuous transformation groups, classified such locally non-singular groups acting on the complex and real planes, described their differential invariants and then carried out group classification of second order ordinary differential equations. Solving the problems of group classification is interesting not only from a purely mathematical point of view, but is also important for applications. In quantum and other physical models there often exist a priori requirements on symmetry groups which follow from physical laws (in particular, from Galilei or relativistic theory). Moreover, the modelling differential equations may contain parameters or functions which have been found experimentally and so are not strictly fixed. (Such parameters and functions are called arbitrary elements.) At the same time mathematical models have to be simple enough to analyse and solve them. Group classification is one of the most important symmetry methods used to choose physically relevant models from parametric classes of systems of (partial or ordinary) differential equations. Investigations of group classification of differential equations can be extended to studying further concepts concerned with differential equations. Symmetry transformations of a single system of differential equations and equivalence transformations in a class of such systems generate equivalence relations on the corresponding sets of non-classical and potential symmetries, exact solutions, conservation laws etc. Therefore, it is natural to formulate the problems on finding the above and similar objects as problems of classification. In the proposed project we plan to investigate a set of important interrelated problems of modern group analysis of differential equations by means of using the concepts described above. Both the theoretical background of group analysis and the particular problems to be studied are important for applications in physics, biology and other fields. The theoretical part of the project is aimed at a modification of the whole framework of group classification of differential equations (from the statement of problems to the requirements of formulating results) on the basis of the notion of normalized classes of differential equations and powerful new methods of constructing realizations of Lie algebras. Problems of classification of different concepts (non-classical symmetries, exact solutions, conservation laws and potential symmetries) related to differential equations will be studied. Connections between non-classical symmetries and characteristics of differential equations will be investigated. The developed methods and techniques will be applied to known equations of mathematical physics, in particular, to nonlinear Schrödinger equations and diffusion-convection equations.