Symmetry Group Analysis and Surfaces in Lie Algebras for Nonlinear Phenomena in Physics

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The proposed program has as its objective the development of new tools for constructing and investigating exact and approximate analytic solutions of systems of nonlinear differential equations appearing in various branches of mathematical physics. These new approaches involve adaptations of the symmetry reduction method and geometric studies of surfaces immersed in homogenous spaces. They will be applied to the analysis of physical phenomena described by nonlinear systems of field theory and fluid dynamics. The program includes the following projects. 1. Construction of surfaces in homogenous spaces for nonlinear field theory equations The differential geometrical study of many models of field theory has proven very useful, especially when focused on the analysis of surfaces representing the sets of solutions. In this project, new techniques for constructing surfaces immersed in homogenous spaces are developed and their properties are analyzed in connection with the physical features of the model. This study focuses on complex Grassmannian models, the associated surfaces and higher-dimensional submanifolds and their link with coherent state theory. It can have applications to many physical systems describing phenomena in which surface dynamics is of interest, e.g. quantum field theory. 2. Soliton surfaces associated with generalized CP^(N-1) sigma models This project is devoted to the study of an invariant formulation of integrable CP^(N-1) sigma models in two dimensions. It involves a systematic description of higher-rank projectors and leads to the construction of the corresponding soliton surfaces. The proposed original procedure produces multileaf soliton surfaces resulting from ''stacking'' the surfaces corresponding to lower rank projectors. This new systematic approach to the derivation of soliton surfaces for sigma models can have numerous physical applications, from superstrings and branes to biological membranes. 3. Stability analysis of invariant solutions via the variational method This project concerns the stability behaviour of group invariant solutions of nonlinear differential systems. A new way of constructing approximate solutions derivable from an action integral through a variational method (by introducing a variational parameter to group invariant solutions) is proposed. This allows for a stability analysis of the obtained solutions using the perturbative computations and may provide approximate analytical results where only numerical ones were known.