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，用于构造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。这些新方法包括对对称约简法的适应和对沉浸在同质空间中的曲面的几何研究。它们将被应用于分析由非线性系统的场论和流体动力学所描述的物理现象。该计划包括以下项目。1. 许多场理论模型的微分几何研究已被证明是非常有用的，特别是在集中分析表示解集的曲面时。在这个项目中，开发了在均匀空间中构建表面的新技术，并根据模型的物理特征分析了它们的特性。本文主要研究了复杂格拉斯曼模型、相关曲面和高维子流形及其与相干态理论的联系。它可以应用于描述表面动力学感兴趣的现象的许多物理系统，例如量子场论。2. 本项目致力于研究二维可积CP^(N-1) σ模型的不变公式。它涉及对高阶投影的系统描述，并导致相应孤子表面的构造。提出的原始程序通过“堆叠”低秩投影对应的表面产生多叶孤子表面。这种新的系统方法推导了sigma模型的孤子表面，可以有许多物理应用，从超弦和膜到生物膜。3. 本课题研究非线性微分系统群不变解的稳定性行为。提出了一种用变分方法构造可由作用积分导出的近似解的新方法（在群不变解中引入变分参数）。这允许使用微扰计算对得到的解进行稳定性分析，并且可以在只知道数值解的情况下提供近似的解析结果。

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.