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

物理学中非线性现象的李代数的对称群分析和曲面

• 批准号: RGPIN-2019-03984
• 负责人: Grundland, AlfredMichel
• 金额: $ 1.75万
• 依托单位: Université du Québec à Trois-Rivières
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737320
• 关键词: Symmetry; Group; Analysis; Surfaces; Lie

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.

该计划的目标是开发新工具，用于构建和研究数学物理各个分支中出现的非线性微分方程组的精确和近似解析解。这些新方法涉及对称性约简方法的调整以及浸入均匀空间中的表面的几何研究。它们将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. 非线性场论方程的齐次空间中的曲面构造 许多场论模型的微分几何研究已被证明非常有用，特别是当重点分析代表解集的曲面时。在该项目中，开发了构建沉浸在均质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特性。本研究重点关注复杂的格拉斯曼模型、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. 与广义 CP^(N-1) sigma 模型相关的孤子曲面 该项目致力于研究二维可积 CP^(N-1) sigma 模型的不变公式。它涉及对更高阶投影仪的系统描述，并导致相应的孤子表面的构造。所提出的原始程序产生多叶孤子表面，该表面是通过“堆叠”与较低等级投影仪相对应的表面而产生的。这种推导西格玛模型孤子表面的新系统方法可以有许多物理应用，从超弦和膜到生物膜。 3. 通过变分法对不变解进行稳定性分析 该项目涉及非线性微分系统群不变解的稳定性行为。提出了一种通过变分方法（通过向不变解组引入变分参数）构造可从作用积分导出的近似解的新方法。这允许使用微扰计算对所获得的解进行稳定性分析，并且可以提供仅已知数值结果的近似分析结果。