Applications of Lie Pseudogroups

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This project will develop new computational tools for studying infinite-dimensional symmetry groups of nonlinear partial differential equations. The approach is based on an adaptation of the finite-dimensional equivariant moving frame method previously developed by the principal investigator. Algorithms for directly determining the structure of such symmetry groups and their algebra of differential invariants, for finding explicit, non-invariant solutions, for testing for linearizability and integrability, and for determining conservation laws and integrability constraints for problems arising in the calculus of variations will be developed. Specific applications will include boundary layer and other fluid models, solitons in higher space dimensions, and quasi-geostrophic systems in meteorology. Advances in the geometric foundations of infinite-dimensional Lie pseudo-groups are also anticipated. The design and application of general-purpose computer algebra packages for handling the required computations will be an integral part of the project.Exploitation of symmetry and invariance properties remains one of the most powerful general-purpose mathematical tools for studying and solving the complex nonlinear systems that arise in all areas of applications. The focus of this project will be to develop and exploit new methods for handling infinite-dimensional symmetry groups, which arise in a broad range of applications, including fluid and plasma mechanics, magneto-hydrodynamics, meteorology and climate modeling, relativity and gauge theories from fundamental physics, elasticity, and integrable systems/soliton models, and so on. The results of this project are thus anticipated to have a significant impact on the solution methods and consequent understanding of nonlinear phenomena in physics, engineering, and elsewhere.