标题： 对称性、可积性与非线性系统的精确解 显示英文标题

标题： Symmetries, Integrability and Exact Solutions for Nonlinear Systems

摘要： 本文旨在对非线性动力系统中可积性的概念进行一般性概述，并说明对称方法如何应用于解决这一问题。 在第一部分中，处理了直接和间接对称方法或解的最佳系统等关键问题，讲座的第二部分对两个具体的非线性动力系统模型进行了有效研究，以说明该过程是如何进行的。 这两个模型分别是来自广义相对论的二维瑞奇流模型和二维对流-扩散方程。 部分结果，特别是关于解的最佳系统的结果，是新的。 关键词：李对称性，不变量，相似性约化。 显示英文摘要

摘要： The paper intends to offer a general overview on what the concept of integrability means for a nonlinear dynamical system and how the symmetry method can be applied for approaching it. After a general part where key problems as direct and indirect symmetry method or optimal system of solutions are tackled out, in the second part of the lecture two concrete models of nonlinear dynamical systems are effectively studied in order to illustrate how the procedure is working out. The two models are the 2D Ricci flow model coming from the general relativity and the 2D convective-diffusion equation . Part of the results, especially concerning the optimal systems of solutions, are new ones. Keywords: Lie symmetries, invariants, similarity reduction.