Přehled o publikaci

The beginnings of the theory of Lie groups and Lie algebras were inseparably linked to the group analysis of di erential equations and, in particular, to problems of group classi cation of di erential equations. Inspired by the idea of creating a universal theory of integration of ordinary di erential equations similar to the Galois theory of solving algebraic equations, S. Lie developed the theory of continuous transformation groups, classi ed locally non-singular transformation groups acting on the complex and real plane, described their di erential invariants and then performed the group classi cation of second-order ordinary di erential equations. S. Lie also solved the problems of group classi cation for two-dimensional linear partial di erential equations and for nonlinear Klein{Gordon equations. Therefore, the main objects of study within the framework of group analysis of di erential equations in Lie's time were continuous (both point and contact) symmetry or equivalence transformations of di erential equations as well as algebraic and geometric structures related to such transformations that are called, in the modern terminology, local transformation groups and Lie algebras of (local) vector  elds. In the seminal paper [41], Noether was the  rst to consider generalized symmetries of di erential equations and to relate variational symmetries of a Lagrangian to local conservation laws of the associated system of Euler{Lagrange equations, thus extending the scope of group analysis of di erential equations to these kinds of mathematical objects. Later, other kinds of symmetries of di erential equations arose in the literature, including approximate, conditional, nonclassical and nonlocal symmetries, some of which are at most indirectly related to Lie groups and Lie algebras. For this reason this branch of mathematics is now often called symmetry analysis instead of group analysis. In addition to symmetries, many other objects encoding geometric properties of di erential equations, like local and potential conservation laws, coverings, recursion operators, Lagrangian and Hamiltonian structures, are studied within the framework of symmetry analysis of differential equations. The subject of this habilitation thesis  ts into the scope of symmetry analysis of di erential equations. The main part of the thesis consists of the following  ve papers: [T1] Kunzinger M. and Popovych R.O., Singular reduction operators in two dimensions, J. Phys. A: Math. Theor. 41 (2008), 505201, 24 pp., arXiv:0808.3577. [T2] Popovych R.O., Reduction operators of linear second-order parabolic equations, J. Phys. A: Math. Theor. 41 (2008), 185202, 31 pp., arXiv:0712.2764. [T3] Kunzinger M. and Popovych R.O., Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl. 379 (2011), 444{460, arXiv:1011.0277. [T4] Kunzinger M. and Popovych R.O., Potential conservation laws, J. Math. Phys. 49 (2008), 103506, 34 pp., arXiv:0803.1156. [T5] Popovych R.O. and Bihlo A., Symmetry preserving parameterization schemes, J. Math. Phys. 53 (2010), 073102, 36 pp., arXiv:1010.3010. As one could infer from the titles of these papers, they are devoted to   the study of nonclassical (or conditional) symmetries (including generalized ones) and nonclassical reductions of di erential equations, especially, the analysis of no-go cases in  nding such symmetries and reductions [T1, T2, T3];   the development the general theory of potential conservation laws and  nding criteria for determining whether a potential conservation law is nontrivial [T4];   the application of methods of group classi cation of di erential equation to construction of invariant parameterization schemes [T5].