Applications of Symmetry Methods in Continuum Mechanics

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Many physical phenomena are modeled by differential equations, which relate the physical parameters of interest and their rates of change. The majority of contemporary models are nonlinear, that is, a sum of two solutions to the governing equations is not a solution. A simple example of a nonlinear phenomenon is provided by interacting water waves, where the height of the resulting wave is not equal to the sum of elevations of each of the waves. Nonlinear effects make the exact general solution of nonlinear differential equations virtually impossible. Specific solution techniques have been developed for some classes of equations, but such techniques lack generality; numerical solutions lack precision and flexibility, especially in three-dimensions, where extra large data structures have to be used to achieve adequate precision.The theory of symmetries of differential equations considered in the current proposal is applicable to wide classes of nonlinear models; it is known to be able to yield useful exact solutions and analytical information about the underlying phenomenon. A symmetry is any transformation that transforms variables, but preserves the equations describing the model. Under the action of a symmetry transformation, the equations are simply mapped into themselves. Groups of local symmetries of differential equations can be systematically computed. When known, they are used for the reduction of order and complete solution for some classes of models (given by ordinary differential equations), and for obtaining self-similar and other types of exact solutions of other models (given by partial differential equations).The current proposal consists of theoretical and applied parts.The first theoretical part is devoted to the derivation and study of symmetries of differential equations which do not have, or have too few, local symmetries. The approach taken here is to seek extra symmetries of augmented (potential) systems of equations, which contain additional nonlocal (potential) variables. This approach has recently yielded new results for some applications of interest; it is intended to further generalize and extend the nonlocal symmetry framework. The second theoretical part is devoted to the approximate symmetries, that is, symmetries that hold approximately, for equations that involve small parameters and/or different time/space scales. To date, there is no unified theory of approximate symmetries. Within the proposed research project, it is planned to work on a consistent extension of the symmetry theory to include approximate symmetries, and on systematic methods of construction of approximate solutions for nonlinear models. The applied part of the proposed research consists in seeking symmetries, exact solutions, and quantities conserved by the dynamics of nonlinear problems in continuum mechanics, including dynamics of fluids, gases, plasmas, and nonlinear elastic media. In particular, with two groups of collaborators, we plan to study models of wave propagation in elastic solids, and equations pertaining to turbulence modelling in incompressible fluids. It is also planned to work on the extension and improvement of the symbolic symmetry computation software package GeM for Maple, written by the applicant. The package is currently used by a significant number of researchers around the globe.