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.

No additional funding sources recorded.