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

该计划的目标是开发新的工具，用于构建和研究出现在数学物理各个分支中的非线性微分方程系统的精确和近似解析解。这些新的方法涉及适应的对称性降低方法和几何研究的表面沉浸在均匀空间。它们将被应用于分析由场论和流体动力学的非线性系统描述的物理现象。该计划包括以下项目。 1.齐次空间中非线性场论方程曲面的构造 对场论的许多模型的微分几何研究已被证明是非常有用的，特别是当集中于分析代表解集的曲面时。在该项目中，开发了用于构建浸入均匀空间中的表面的新技术，并结合模型的物理特征分析了它们的性质。本研究的重点是复杂的格拉斯曼模型，相关的表面和高维子流形和他们的联系与相干态理论。它可以应用于许多描述表面动力学感兴趣的现象的物理系统，例如量子场论。 2.广义CP^（N-1）sigma模型中的孤子面 本计画致力于研究二维可积CP^（N-1）sigma模型的不变式。它涉及到一个系统的描述高阶投影仪，并导致相应的孤子表面的建设。所提出的原始程序产生多叶孤子表面导致的“堆叠”的表面对应于较低的排名投影仪。这种新的系统方法推导孤子表面的西格玛模型可以有许多物理应用，从超弦和膜生物膜。 3.用变分法分析不变解的稳定性 本计画研究非线性微分系统群不变解的稳定性。提出了一种通过变分方法（在群不变解中引入变分参数）构造近似解的新方法。这允许使用微扰计算获得的解决方案的稳定性分析，并可能提供近似的分析结果，只有数值的是已知的。

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.