Symmetry Reduction Method and Surfaces in Lie Algebras for Nonlinear Phenomena in Physics

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The proposed research aims at constructing exact and approximate analytic solutions of systems of nonlinear partial differential equations and investigating their properties important for the description of physical phenomena. The project relies on methods of symmetry reduction and geometric studies of surfaces immersed in homogeneous spaces. These methods are applied to nonlinear systems of field theory and fluid dynamics. New tools are proposed to investigate these complex systems. The program includes the following projects: 1. Surfaces in Homogeneous Spaces and Nonlinear Field Theory Many models of field theory (e.g. sigma models) have deep and previously not expected relations with the differential geometrical study of surfaces immersed in homogeneous spaces. In this project, some techniques for constructing such surfaces are proposed and their properties are analyzed in connection with physical features of the model. This study has many applications to physical systems describing phenomena in which surface dynamics is of interest (e.g. in quantum field theory, phase transition theory, fluid dynamics and the theory of biological membranes). 2. Quantum Hamiltonian Systems and Soliton Surfaces This project offers a systematic description of integrable quantum Hamiltonian systems and their soliton solutions. The proposed approach combines the analysis of generalized symmetries of such systems and associated surfaces immersed in Lie algebras. It will be employed, among other, for constructing integrable quantum systems with a definite gap in the bounded spectrum which have interesting physical applications. 3. Extension of Invariant Solutions via the Variational Method This project involves a new way of constructing approximate solutions of nonlinear systems, derivable from an action integral by using variational methods (introducing variational parameters to group-invariant solutions). This approach can be useful for applications to many physical situations, especially dynamical processes which do not have a high degree of symmetry and may provide approximate analytical solutions to many problems which so far were accessible by numerical description only.