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

本研究旨在建立非线性偏微分方程系统的精确和近似解析解，并研究其对描述物理现象的重要性质。该项目依赖于对称还原的方法和沉浸在均匀空间中的表面的几何研究。这些方法适用于非线性系统的场论和流体力学。提出了新的工具来研究这些复杂的系统。该方案包括以下项目：**1；齐次空间中的曲面与非线性场论**许多场论模型（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，提出了一些构造这种表面的技术，并结合模型的物理特征分析了它们的性质。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）**2。量子哈密顿系统和孤子曲面**本项目提供了可积量子哈密顿系统及其孤子解的系统描述。所提出的方法结合了这类系统的广义对称性分析和相关面浸入李代数。除其他外，它将用于构造在有界光谱中具有确定间隙的可积量子系统，这些系统具有有趣的物理应用。用变分方法扩展不变解**本项目涉及到用变分方法构造非线性系统的近似解的一种新方法，该非线性系统可由作用积分导出（在群不变解中引入变分参数）。这种方法可以用于许多物理情况的应用，特别是不具有高度对称性的动态过程，并且可以为许多迄今为止只能通过数值描述获得的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.