Symmetry Reduction Method and Surfaces in Lie Algebras for Nonlinear Phenomena in Physics

物理非线性现象李代数中的对称性约简方法和曲面

• 批准号: RGPIN-2014-06401
• 负责人: Grundland, Alfred
• 金额: $ 1.82万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=633058
• 关键词: Symmetry; Reduction; Method; Surfaces; Lie

The proposed research aims at constructing exact and approximate analytic solutions of systems of nonlinear partial differential equations and investigating their properties important for the description of physical phenomena. The project relies on methods of symmetry reduction and geometric studies of surfaces immersed in homogeneous spaces. These methods are applied to nonlinear systems of field theory and fluid dynamics. New tools are proposed to investigate these complex systems. The program includes the following projects:1. Surfaces in Homogeneous Spaces and Nonlinear Field TheoryMany models of field theory (e.g. sigma models) have deep and previously not expected relations with the differential geometrical study of surfaces immersed in homogeneous spaces. In this project, some techniques for constructing such surfaces are proposed and their properties are analyzed in connection with physical features of the model. This study has many applications to physical systems describing phenomena in which surface dynamics is of interest (e.g. in quantum field theory, phase transition theory, fluid dynamics and the theory of biological membranes).2. Quantum Hamiltonian Systems and Soliton SurfacesThis project offers a systematic description of integrable quantum Hamiltonian systems and their soliton solutions. The proposed approach combines the analysis of generalized symmetries of such systems and associated surfaces immersed in Lie algebras. It will be employed, among other, for constructing integrable quantum systems with a definite gap in the bounded spectrum which have interesting physical applications.3. Extension of Invariant Solutions via the Variational MethodThis project involves a new way of constructing approximate solutions of nonlinear systems, derivable from an action integral by using variational methods (introducing variational parameters to group-invariant solutions). This approach can be useful for applications to many physical situations, especially dynamical processes which do not have a high degree of symmetry and may provide approximate analytical solutions to many problems which so far were accessible by numerical description only.

拟议的研究旨在构建精确和近似的非线性偏微分方程系统的解析解，并调查其重要的物理现象的描述属性。该项目依赖于对称性减少和沉浸在齐次空间中的表面的几何研究的方法。这些方法适用于非线性系统的场论和流体动力学。提出了新的工具来研究这些复杂的系统。该计划包括以下项目：1.齐次空间中的曲面和非线性场论许多场论模型（例如sigma模型）与齐次空间中曲面的微分几何研究有着深刻的、以前没有预料到的关系。在这个项目中，提出了一些技术来构建这样的表面，并分析其属性与模型的物理特征。这项研究在描述表面动力学感兴趣的现象的物理系统中有许多应用（例如量子场论，相变理论，流体动力学和生物膜理论）。量子哈密顿系统和孤子表面本项目提供了一个系统的描述可积量子哈密顿系统和它们的孤子解。所提出的方法结合了广义对称性的分析，这样的系统和相关的表面沉浸在李代数。它将被用来构造在有界谱中具有一定能隙的可积量子系统，这些系统具有有趣的物理应用。通过变分方法扩展不变解这个项目涉及一种新的方法来构造非线性系统的近似解，通过使用变分方法（将变分参数引入组不变解）从作用积分导出。这种方法可以是有用的应用程序的许多物理情况下，特别是动力学过程，不具有高度的对称性，并可能提供近似的解析解到目前为止，只有通过数值描述的许多问题。