Symmetry methods and their applications to the analysis of modern mathematical models

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Mathematical models of physical phenomena are often stated in terms of partial differential equations (PDE), relating physical parameters to their rates of change in time and space. PDE models arise in a variety of contexts, including the description of physical continua (e.g., fluids, plasmas, mixtures, gases, and solid bodies). The majority of mathematical models of complex systems are nonlinear: a sum of two solutions of governing equations is not a solution. Examples of nonlinear phenomena include water waves, large deformations of elastic solids, and turbulent fluid flows. The proposed research is concerned with the development of mathematical methods that can be systematically applied to analyze, simplify, and solve nonlinear PDEs. For a PDE model, one ideally would like to have a solution in terms of explicit formulas. This would allow to make computations, plot graphs, and analyze solution behaviour in a direct way. For linear models, many classical solution methods exist, but for nonlinear PDEs, usual methods are not applicable; researchers often must come up with special approaches, or use numerical simulations, which can be time consuming, and may not provide a "big picture". The theoretical part of this proposal is devoted to the development and extension of the theory of symmetries of PDEs. Applicable to wide classes of linear and nonlinear models, symmetry methods are used to systematically extract important information about the problem, simplify it, construct exact solutions, and in some cases, completely solve a given nonlinear problem. For some PDE systems of interest, however, classical symmetry methods yield few useful results. In these cases, extensions of symmetry methods prove beneficial. The applicant will capitalize on expertise and research momentum in theory of local/nonlocal symmetries, exact solutions, and symbolic computations, as well as on Canadian and international collaborations and HQP, to develop further systematic symmetry-related methods, in particular, the theory of nonlocal and approximate symmetries, to analyze and solve previously intractable nonlinear problems. The second part of this proposal has to do with the problem of symmetry-related computations for complex PDE systems, where computer-aided calculations are required. The applicant is the author of a Maple-based symbolic package GeM, used by researchers in many countries. In the upcoming grant cycle, it is intended to make GeM object-oriented, allowing for higher efficiency and simpler syntax, and also to implement additional routines to compute symmetry-related mathematical objects, including linearlization operators, adjoint and approximate symmetries, and Lagrangians. The current proposal includes an applied part. With collaborators, the applicant will work on nonlinear models of wave propagation in anisotropic elastic solids, and models pertaining to turbulent flows, and study the corresponding PDEs using advanced symmetry methods.