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

所提出的研究旨在构造非线性偏微分方程组的精确和近似解析解，并研究它们对于描述物理现象很重要的性质。该项目依赖于对称性简化方法和沉浸在均匀空间中的表面的几何研究。这些方法应用于场论和流体动力学的非线性系统。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. 均匀空间中的表面和非线性场论 许多场论模型（例如 sigma 模型）与浸入均匀空间中的表面的微分几何研究有着深刻的、以前未曾预料到的关系。在该项目中，提出了一些构建此类表面的技术，并结合模型的物理特征分析了它们的特性。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（例如在量子场论、相变理论、流体动力学和生物膜理论中）。 2. 量子哈密顿系统和孤子表面 该项目提供了可积量子哈密顿系统及其孤子解决方案的系统描述。所提出的方法结合了对此类系统的广义对称性和沉浸在李代数中的相关表面的分析。除其他外，它将用于构建在有界光谱中具有明确间隙的可积量子系统，这具有有趣的物理应用。 3. 通过变分法扩展不变解 该项目涉及一种构造非线性系统近似解的新方法，可通过使用变分方法（将变分参数引入群不变解）从作用积分导出。这种方法对于许多物理情况的应用非常有用，特别是不具有高度对称性的动态过程，并且可以为迄今为止只能通过数值描述来获得的许多问题提供近似解析解。

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 Theory Many 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 Surfaces This 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 Method This 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.