对称分类、整体群表示和不变参数化格式研究

项目名称： 对称分类、整体群表示和不变参数化格式研究

项目编号： No.11501204

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 黄定江

作者单位： 华东理工大学

项目金额： 18万元

中文摘要： 非线性偏微分方程的群理论分析是数学物理和可积系统领域的重要课题。本项目拟研究含有多任意元多自变量的非线性偏微分方程的李点对称群分类、局部对称的整体群表示以及对称保持的参数化格式。具体包括：①通过扩展连续性等价群理论，定义新的扩展等价群，研究其代数结构和计算方法；②基于上述等价群，发展直接积分法和李代数与子群分析法，研究方程李点对称分类的新相容性方法；③利用抽象李群表示理论，研究非线性偏微分方程局部李点对称的整体群表示；④研究基于直接群分类的对称保持的参数化格式的构造；⑤应用这些理论和方法于数学物理中典型的非线性偏微分方程的研究以及不变解构造。由于完全分类是群分析领域的难题，而李群表示论和直接群分类思想却很少用于微分方程李对称和参数化格式的研究，因此本项课题对揭示含有多任意元多自变量的非线性偏微分方程的局部对称结构与整体表示理论之间的关系，探索方程对称保持的离散数值解法具有一定的科学意义。

中文关键词： 对称分类；整体群表示；不变参数化格式；等价群；非线性偏微分方程

英文摘要： Group analysis of nonlinear partial differential equations is an important issue in the fields of mathematical physics and integrable system. In this project, we will explore Lie symmetry group classification, global group representation of local symmetry and symmetry preserving parameterization schemes for nonlinear partial differential equations which contain several independent variables and arbitrary elements. The main contents consist of: ① we will first present some new allowed, conditional and additional equivalence groups by extending the continuous equivalence group theory, then study their algebraic structures and algorithmic methods for computing such groups; ② secondly, with the aid of the new extended equivalence group, we will propose some new compatible method for Lie symmetry classification of differential equations by extending direct integration， Lie algebra and subgroup analysis methods; ③ thirdly, we will study the global group representation of local symmetry of nonlinear partial equations by introducing the abstract Lie group representation theory; ④ fourthly, symmetry preserving parameterization schemes is investigated based on techniques of direct group classification; ⑤finally, we will apply these new theories and methods to some typical nonlinear partial equations in mathematical physics and construct their invariant solutions. The complete group classification of differential equation is a difficult problem in the field of group analysis, and furthermore Lie group representation theory and techniques of direct group classification are rarely used to the investigation of Lie symmetry and parameterization schemes of differential equations respectively. Therefore，this project is significant in revealing the relation between local symmetries and global representation theory for nonlinear partial equation with several arbitrary elements and several independents and is also important for designing symmetry preserving discrete numerical algorithms.

英文关键词： symmetry classification;global group representation;invariant parameterization schemes;equivalence group;nonlinear partial differential equations