Extended group analysis of differential equations

The value of symmetries in science can hardly be overestimated. Today, symmetries form a cornerstone of various physical disciplines, including classical and quantum mechanics, relativity and particle physics. More specifically, symmetries of differential equations allow the computation of exact solutions and conservation laws and thus can provide important information whether or not it might be possible to integrate the given equations. Although these computations are classical applications of symmetry methods, there exist more recent methods that owing to their potential practical relevance deserve a more exhaustive study than presently available. In particular, symmetries can be used for the design of invariant parameterization and discretization schemes. The aim of this project is a substantial advancement of group analysis of differential equations in the interplay of mathematics and applications, with a special focus on the atmospheric sciences. The research program is essentially built on the results of the antecedent FWF project Classification problems of group analysis. We intend to continue an extension and application of the modern perception of group analysis in the algebraic language. The algebraic formalization of various existing techniques of group analysis includes a reformulation of the framework of admissible transformations in classes of differential equations in terms of groupoids. We will rigorously define the notion of equivalence algebroids and infinitesimal normalization and will develop a universally applicable toolbox for the algebraic approach to the construction of point symmetry groups of single differential equations, equivalence groups and equivalence groupoids of classes of differential equations. The definition of reduction modules for systems of differential equations will be formalized using tools of formal compatibility theory. The framework of singular reduction modules will be developed and extended to systems of differential equations. As a practical demonstration of the utility of the theoretical concepts to be established we plan to systematically investigate wide classes of differential equations which naturally arise in the course of constructing physical parameterization schemes of unresolved processes in numerical models of geophysical fluid dynamics that admit prescribed symmetries and/or conservation laws. Specific core deliverables of the present project will be the description of the universal Abelian covering of second-order evolution equations up to contact equivalence, the study of low-order conservation laws of (1+1)- dimensional evolution equations, the construction of the linear potential frame for (1+1)-dimensional linear evolution equations of arbitrary order, potential symmetries and potential conservation laws associated with this potential frame, no-go statements on nonclassical symmetries of second-order linear partial differential equations with two independent variables and the classification of nonlinear reduction operators of (1+1)-dimensional nonlinear heat equations with sources depending on all variables.

In the course of the implementation of the project, we further developed the theoretical background of symmetry analysis of differential equations in its extended interpretation. The application to well-known models of fluid dynamics and meteorology, including their invariant and conservative parameterizations, was parallelly continued for demonstrating the ability of modern group analysis to contribute to the understanding of geophysical fluid dynamics. The above theoretical studies were supplemented by the classification of a variety of objects (Lie and discrete symmetries, admissible and equivalence transformations, conditional symmetries, reduction operators, exact solutions, conservation laws, potential symmetries, etc.) for particular systems of differential equations or classes of such systems that are important for real-world applications and for refining the theory of extended group analysis of differential equations. A number of original and modified techniques were created and used in the course of these computations. Some project findings were really surprising. In particular, continuing the study of equivalence groupoids of classes of differential equations, we introduced the notion of uniformly semi-normalized class of differential equations. The theorem on splitting symmetry groups in uniformly semi-normalized classes allowed us to ex- tend the range of applicability of the algebraic method of group classification to classes that are not normalized. We happened to construct for the first time several examples of nontrivial generalized equivalence groups such that equivalence- transformation components associated with equation variables locally depend on nonconstant arbitrary elements of the corresponding classes. Note that for more than 20 years after the first discussion of the notion of generalized equivalence groups, no examples of nontrivial generalized equivalence groups had been known in the literature, except classes for which some of arbitrary elements are constants. This is why certain doubts had started to circulate in the symmetry community whether this notion is valuable at all. We explicitly constructed, in a rigorous way, extended generalized equivalence groups of several classes of differential equations (both ordinary and partial ones). These are also the first examples of such a construction in the literature. We discovered classes of differential equations with nontrivial generalized equivalence groups containing proper subgroups that generate the same subgroupoids of the corresponding equivalence groupoids as the entire groups do. Minimal among such subgroups were called effective generalized equivalence groups of the associated classes of differential equations, and they have quite unusual properties. There exist classes of differential equations whose effective generalized equivalence groups are not normal subgroups of the corresponding generalized equivalence groups and are hence not unique. We also found a class of differential equations each of whose effective generalized equivalence groups does not contain its usual equivalence group. Note that the generalized equivalence group of a class necessarily contains its usual equivalence group. Noethers second theorem was enhanced and generalized to systems of differential equations that are not necessarily EulerLagrange equations. The exhaustive solution of the general in- verse problem on conservation laws for (1+1)-dimensional evolution equations allowed us to describe local conservation laws of even-order (1+1)-dimensional evolution equations. Refining the definition of nonclassical symmetries for single differential equations and introducing the notion of singular reduction modules led to revisiting and enhancing the entire theory of nonclassical symmetries of differential equations.