Lokaziuk O. Realizations of Lie algebras on line and group classification of differential equations

The foundation of the dissertation research is the following two problems: group classification (1+1)-dimensional generalized nonlinear Klein–Gordon equations and group classification of normal linear systems of ordinary second-order differential equations. Despite the fact that these problems have a very long history of research, new results have been obtained in the framework of the algebraic approach after a detailed study of the transformational properties of the considered classes and their subclasses. The unifying tool in both of these problems was the efficient use of the classical Lie theorem on the realization of finite-dimensional Lie algebras by vector fields on the line. The methods and approaches proposed in the dissertation allowed us not only to essentially improve the previous results on Lie symmetries of equations from these classes, but also to significantly simplify the proof of classification results and verification their correctness. Key words: group classification of differential equations, nonlinear Klein– Gordon equations, linear systems of second-order ordinary differential equations, Lie symmetries, equivalence group, equivalence groupoid, equivalence algebra, algebraic method of group classification, realizations of Lie algebras by vector fields, maximal Lie invariance algebra.