扩展离散可积系统的构造、求解及应用

项目名称： 扩展离散可积系统的构造、求解及应用

项目编号： No.11471182

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 林润亮

作者单位： 清华大学

项目金额： 60万元

中文摘要： 可积系统理论是数学和物理中的重要研究领域。近年来，离散可积系统的研究得到越来越多的重视。扩展离散可积系统的构造及求解在可积系统理论及实际应用中都有十分重要的意义。本项目拟在扩展离散可积系统的构造、求解及应用方面进行研究，主要研究以下几个方面：1）拟利用平方特征函数对称与双Darboux变换的关系，提出构造扩展离散可积方程的一个系统方法；2）构造一些重要离散可积方程的扩展系统，并研究扩展系统的性质，如：离散Darboux方程，离散BKP方程，离散CKP方程；3）研究重要离散方程的约束系统的带源扩展问题，如：离散(modified) KdV方程，离散Gel'fand-Dikii型方程；4）研究离散扩展可积系统与连续系统之间的连续极限关系；5）利用代数几何或分析的方法得到扩展系统更多形式的解；6）讨论扩展可积系统在物理、几何等方面的应用。

中文关键词： 离散可积系统；达布变换；对称约化；孤立子；连续极限

英文摘要： The theory of integrable systems is one of the important subjects in the research of mathematics and physics. In recent years, the discrete integrable systems attracted more and more interests, the construction of an extended discrete integrable system (ext-DIS) and their solutions have many importance in the theory of integrable systems and also in the applications. In this project, we plan to investigate the construction of the ext-DIS, their solutions and applications, mainly on the following aspects: 1) to propose a systematic method to construct the ext-DIS on base of the squared eigenfunctions symmetry and binary Darboux transformation; 2) to construct the exntended system of some important discrete integrable systems, and to study their properties, e.g., the discrete Darboux equation, the discrete BKP equation, the discrete CKP equation; 3) to construct and to study the extended system for the constrained flows of some DIS, e.g., the discrete (modified) KdV equation, the discrete Gel'fand- - Dikii type equations; 4) to study the continuous limit relation between the ext-DIS and its continuous part; 5) to find more explicit solutions of the ext-DIS by the algebro-geometric or analytic techniques; 6) to study the physical and geometric applications of the ext-DIS.

英文关键词： discrete integrable systems;Darboux transformation;symmetry constraint;soliton;continuous limit