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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759442
• 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，用于构建和调查数学物理各个分支的非线性微分方程系统的精确和近似分析解决方案。这些新方法涉及对称性还原方法的适应和浸入同质空间的表面的几何研究。它们将应用于通过现场理论和流体动力学的非线性系统描述的物理现象的分析。该程序包括以下项目。 1。非线性场理论方程中均匀空间中表面的构建。事实证明，许多田间理论模型的差异研究研究非常有用，尤其是当专注于代表解决方案集的表面分析时。在这个项目中，开发了用于构建融入同质空间的表面的新技术，并根据模型的物理特征分析了它们的性能。这项研究的重点是复杂的硕士模型，相关的表面和较高维的亚曼福尔德及其与连贯的状态理论的联系。它可以在许多物理系统中应用，这些物理系统描述了表面动态感兴趣的现象，例如量子场理论。 2。与广义CP^（n-1）Sigma模型相关的孤子表面该项目致力于研究在两个维度中的可整合CP^（n-1）Sigma模型的不变公式。它涉及对高级投影仪的系统描述，并导致构建相应的孤子表面。所提出的原始过程产生的多叶孤子表面是由“堆叠”的表面产生的，与较低的投影仪相对应。这种新的系统方法用于Sigma模型的孤子表面推导，可以具有许多物理应用，从超弦和麸皮到生物膜。 3。通过变异方法对不变解决方案的稳定性分析涉及非线性差异系统组不变解决方案的稳定性行为。提出了一种通过变异方法从动作积分衍生的近似解决方案的新方法（通过将变异参数引入组不变解决方案）。这允许使用扰动计算对所获得的溶液进行稳定分析，并且在仅知道数值的结果的情况下可能会提供近似的分析结果。